On the Cauchy Problem and Solitons for a Class of 1D Boussinesq Systems

Abstract

In this paper we show the local and global well-posedness for the Cauchy problem associated with a special class of 1D-Boussinesq systems that emerges in the study of the evolution of long water waves with small amplitude in the presence of surface tension. We also show the existence of solitons (finite energy travelling wave solutions) in the case of wave speed $0<|\omega |<\omega _0$, for some $\omega _0>0$.

Introduction

In this work we consider the study of the one-dimensional Boussinesq type system

$$\begin{aligned} \left\{ \begin{array}{l} (I-a\mu \partial _x^2) \eta _t +\partial _x^2\Phi -b\mu \partial _x^4\Phi +\epsilon \partial _x(\eta \left( \partial _x\Phi \right) ^p) =0, \\ (I- c\mu \partial ^2_x) \Phi _t + \eta -d\mu \partial _x^2\eta +\frac{\epsilon }{p+1}\left( \partial _x\Phi \right) ^{p+1}=0, \end{array}\right. \end{aligned}$$

(1)

where $\eta =\eta (x,t)$ and $\Phi =\Phi (x,t)$ are real-valued functions, $\mu$ and $\epsilon$ are small positive parameters, p is a rational number of the form $p=\frac{p_1}{p_2}$ with $(p_1, p_2)=1$ and $p_2$ an odd number, and the constants $a\ge 0$, $c\ge 0$, $b> 0$, and $d> 0$ are such that

$$\begin{aligned} a+c-(b+d)=\frac{1}{3}-\sigma , \end{aligned}$$

where $\sigma ^{-1}$ is known as the Bond number. Regarding these models, it can be established that the evolution of long water waves with small amplitude is reduced to studying the solution $(\eta , \Phi )$ of the system (1) in the case of $p=1$, where ${\epsilon }$ is the amplitude parameter (nonlinearity coefficient), $\mu$ is the long-wave parameter (dispersion coefficient) and $\sigma$ is the inverse of the Bond number (associated with the surface tension). The variable $\Phi$ represents the rescale nondimensional velocity potential on the bottom $z=0$, and the variable $\eta$ corresponds the rescaled free surface elevation. The model considered in the paper is the 1D version of some Boussinesq system obtained by Quintero and Montes [8] in the case $a=c=\frac{1}{2}, b=\frac{2}{3}, d=\sigma$ (see also Montes [6]) and by Quintero [9] in the case $a=c=0, b=\frac{1}{6}, d=\sigma -\frac{1}{2}$, which appear when looking at the evolution of long water waves with small amplitude in the presence of surface tension. Results for the two-dimensional version of the Boussinesq system (1), we want to mention [6–9]. For instance, in the cases $a=\frac{1}{2}=c, b=\frac{2}{3}, d=\sigma$ and $a=c=0, b=\frac{1}{6}, d=\sigma -\frac{1}{2}$, well-posedness for the Cauchy problem for $s\ge 2$ and $p\ge 1$ were obtained by Quintero and Montes in work in revision and by Quintero [10], respectively, and the existence results of solitons (finite energy travelling wave solutions) were obtained by Quintero and Montes [8] and Quintero [9], respectively.

As happens in water wave models, there is a Hamiltonian type structure which is clever to characterize solitary waves as critical points of the action functional and also provides relevant information for the study of the Cauchy problem. In our particular Boussinesq system (1), the Hamiltonian functional $\mathcal {H}$ is defined as

$$\begin{aligned} \mathcal {H}\begin{pmatrix}\eta \\ \Phi \end{pmatrix}=\frac{1}{2}\int _{\mathbb {R}} \left( \eta ^2+ d\mu (\eta _x)^2+(\Phi _x)^{2}+ b \mu (\Phi _{xx})^2 +\frac{2\epsilon }{p+1}\eta \left( \Phi _x\right) ^{p+1}\right) dx. \end{aligned}$$

and the Hamiltonian type structure is given by

$$\begin{aligned} \begin{pmatrix}\eta _t \\ \Phi _t \end{pmatrix}=\mathcal {J}\mathcal {H}'\begin{pmatrix}\eta \\ \Phi \end{pmatrix}, \quad \mathcal {J}= \begin{pmatrix}0 &{}\left( I- c\mu \partial ^2_x\right) ^{-1}\\ -\left( I- a\mu \partial ^2_x\right) ^{-1}&{}0 \end{pmatrix}. \end{aligned}$$

Note that for $a=c$ the operator $\mathcal J$ becomes skew symmetric

$$\begin{aligned} \mathcal {J}=(I- a\mu \partial ^2_x)^{-1} \begin{pmatrix}0 &{}\quad 1 \\ -1 &{}\quad 0 \end{pmatrix}, \end{aligned}$$

We see directly that the functional $\mathcal {H}$ is well defined when for t in some interval we have that $\eta (\cdot ,t), \Phi _x(\cdot ,t)\in H^1(\mathbb R)$. These conditions already characterize the natural space (energy space) in which we consider the well-posedness of the Cauchy problem and the existence of travelling wave solutions. Another special characteristic on the system (1) is that some well known water wave models as the one-dimensional Benney–Luke equation (see [12–14]) and the Korteweg–de Vries equation emerge from this Boussinesq type system (up to some order with respect to ${\epsilon }$ and $\mu$), making the system (1) very interesting from the physical and numerical view points.

In this paper, we will establish the local well-posedness for the Cauchy problem associated with the system (1) in the space $H^s\times {\mathcal {V}}^{s+1}$, where $H^s=H^s(\mathbb R)$ is the usual Sobolev space of order s and ${\mathcal {V}}^s$ is defined by the norm $\Vert \psi \Vert _{{\mathcal {V}}^s}=\Vert \psi '\Vert _{H^{s-1}}$. We also show global well-posedness for the Cauchy problem in the energy space $H^1\times {\mathcal {V}}^{2}$ when the initial date is small enough. We will see as usual that local well-posedness for the Cauchy problem associated with the system (1) follows by the Banach fixed point theorem and appropriate linear and nonlinear estimates using different results as a key ingredient in the case of spatial dimension one:

• (a) For $a, c>0$, we will use a bilinear estimative obtained by Bona and Tzvetkov [2].

• (b) For $a=c=0$, we will use the well known estimates for Kato’s commutator used successfully in the KdV model (see works by Kato [3–5]).

On the other hand, global existence for $a=c$ follows from the local existence, the conservation in time of the Hamiltonian, a Sobolev type inequality and the use of energy estimates. Existence of solitons involve the use of the mountain pass theorem and the existence of an appropriate local compact embedding from the space $H^1({\mathbb R}) \times {\mathcal {V}}$ to a special $L^q({\mathbb R})$ type space for $q\ge 2$.

The paper is organized as follows. In “Local Existence”, using semigroup estimates and nonlinear estimates, we show a local existence and uniqueness result for the Boussinesq system (1), via a standard fixed point argument. In “Global Existence for $a=c$”, from a variational approach which involves the characterization of invariant sets under the flow for the Boussinesq system (1) we obtain the global existence result for initial data small enough, in the case $a=c$. In “Existence of Solitons”, we prove the existence of solitons for the system (1) for $0<|w|<w_0$. We will see that solitons are characterized as critical points of a functional of action. Throughout this work, if not specified, we denote by K a generic constant varying line by line.

Local Existence

In this section we consider the Cauchy problem associated to the system (1) with the initial condition

$$\begin{aligned} \eta (0,\cdot )=\eta _0,\quad \Phi (0,\cdot )=\Phi _0. \end{aligned}$$

(2)

The main objective is to show that the Cauchy problem for the system (1) is locally well-posed. The notion of well-posedness to be used here is in the sense of Kato: consider an abstract Cauchy problem

$$\begin{aligned} \frac{du}{dt}=f(u), \quad u(0)=u_0. \end{aligned}$$

(3)

Suppose that there are two Banach spaces $Y\hookrightarrow X,$ with the embedding continuous, such that f is continuous from Y to X. We say that the problem (3) is locally well-posed in Y, if for each $u_0\in Y$ there are a real number $T=T(u_0)>0$ and a unique function $u\in C\left( [0, T],Y\right)$ satisfying the integral equation associated to (3), depending continuously on the initial data in the sense that the solution map $u_0 \mapsto u$ is continuous: if $u_n \rightarrow u$ en Y and $T'\in (0, T)$, then for n large enough $u_n \in C([0, T'],Y)$ and,

$$\begin{aligned} \lim _{n\rightarrow \infty }\sup _{[0, T']}\Vert u_n(t)-u(t)\Vert _Y=0. \end{aligned}$$

We say that the problem is globally well-posed in Y, if for every $u_0\in Y$ the number T can be taken arbitrarily large,. We recall that if E is a Banach space then C([0, T], E) denote the space of continuous functions defined in [0, T] with values in E.

The natural space in which we consider the well-posedness of the Cauchy problem associated with the Boussinesq system (1) is dictated by the definition of the Hamiltonian. Remember that the Hamiltonian is well defined when for t in some interval we have that $\eta (\cdot ,t), \Phi _x(\cdot ,t)\in H^1(\mathbb R)$, then we consider the following spaces. For $s\in {\mathbb R}$, the Sobolev space $H^s(\mathbb R)$ is defined as the completion of the Schwartz space $\mathcal {S}(\mathbb R)$ with respect to the norm given by

$$\begin{aligned} \Vert f\Vert ^2_{H^s}=\Vert (1+|\xi |^2)^{\frac{s}{2}}\widehat{f}\,\Vert ^2_{L^2(\mathbb R)} =\int _{\mathbb R}(1+|\xi |^2)^{s}|\widehat{f}(\xi )|^2d\xi , \end{aligned}$$

where the Fourier transform of a function w defined on ${\mathbb R}$ is given by

$$\begin{aligned} (\mathcal {F}w)(\xi )=\widehat{w}(\xi )= \int _{\mathbb R}e^{-ix\cdot \xi }w(x)\,dx. \end{aligned}$$

The space ${\mathcal {V}}^s$ denote the completion of $\mathcal {S}(\mathbb R)$ with respect to the norm given by

$$\begin{aligned} \Vert f\Vert ^2_{{\mathcal {V}}^s}=\Vert f'\Vert ^2_{H^{s-1}}. \end{aligned}$$

Note that ${\mathcal {V}}^s$ is a Hilbert space with inner product

$$\begin{aligned} (f,g)_{{\mathcal {V}}^s}=(f',g')_{H^{s-1}}. \end{aligned}$$

Moreover,

$$\begin{aligned} \Vert f\Vert ^2_{{\mathcal {V}}^s} =\int _{{\mathbb R}}(1+|\xi |^2)^{s-1}|\xi |^2|\widehat{f}(\xi )|^2d\xi . \end{aligned}$$

We will show, under some conditions on a, c, p and s, the local well-posedness for the Boussinesq system (1) with the initial condition (2), in the space $H^{s}\times {\mathcal {V}}^{s+1}$. Hereafter, we assume $b,d>0$.

We note that if we formally derive the second equation of the Boussinesq system (1), we find that the system (1) is transformed in the following system

$$\begin{aligned} \left\{ \begin{array}{l} (I-a\mu \partial ^2_{x}) \eta _t +\partial _x(I-b \mu \partial _{x}^2 )u +\epsilon \partial _x(\eta u^p)= 0, \\ (I-c \mu \partial ^2_{x})u_t + \partial _x(I-d \mu \partial _{x}^2 )\eta +\frac{\epsilon }{p+1}\partial _x(u^{p+1})=0 \end{array}\right. \end{aligned}$$

(4)

in the variables $\eta$, $u=\Phi _x$. For this system, we see that the quantities

$$\begin{aligned} \int _{\mathbb R}u(t, x)\,dx, \quad \int _{\mathbb R}\eta (t, x)\,dx \end{aligned}$$

are conserved in time for classical solutions and even for mild solutions. So, if we consider the Cauchy problem associated with initial data in an appropriate Sobolev space such that

$$\begin{aligned} \widehat{u}_0(0)= \int _{\mathbb R}u_0(x)\,dx=0, \end{aligned}$$

(5)

then we have that

$$\begin{aligned} \widehat{u}(t, \xi )= \int _{\mathbb R}u(t, x)\,dx=0, \end{aligned}$$

for $t\in {\mathbb R}$, as long as the solution exists. Now, it is known that if

$$\begin{aligned} \dot{H}^{r}:=H^r \cap \{f\in H^r:\hat{f}(0)=0\}, \end{aligned}$$

then there is an onto linear map $\partial _x^{-1}: \dot{H}^{r} \rightarrow H^{r+1}$ defined via the Fourier transform by

$$\begin{aligned} \widehat{\partial _x^{-1}(f)}(\xi )= \frac{\widehat{f}(\xi )}{i\xi }. \end{aligned}$$

Moreover, for a given function $u \in \dot{H}^{r}$, the function $\Phi = \partial _x^{-1}u \in {\mathcal {V}}^{r+1}$ is such that $u=\Phi _x$. So, by solving the Cauchy problem associated for the system (4) with a initial condition satisfying (5), we are able to solve the Cauchy problem associated with the system (1).

Now, we will focus in the local well posedness for the the Cauchy problem associated with the system (4). Note that by defining the operators $A=I-a \mu \partial _x^2$, $B=I-b \mu \partial _x^2$, $C=I-c \mu \partial _x^2$ and $D=I-d\mu \partial _x^2$ via the Fourier transform as

$$\begin{aligned} \widehat{Af}= (1+a \mu \xi ^2)\widehat{f}, \quad \widehat{Bf}= (1+{b\mu }\xi ^2)\widehat{f}, \quad \widehat{Cf}= (1+{c\mu }\xi ^2)\widehat{f}, \quad \widehat{Df}= (1+d \mu \xi ^2)\widehat{f}, \end{aligned}$$

we see that the system (4) can be written as

$$\begin{aligned} \begin{pmatrix} \eta \\ u \end{pmatrix}_t+M\begin{pmatrix} \eta \\ u \end{pmatrix}+F\begin{pmatrix} \eta \\ u \end{pmatrix}=0, \end{aligned}$$

(6)

where M is a linear operator and F corresponds to the nonlinear part,

$$\begin{aligned} M=\begin{pmatrix} 0 &{}\quad \partial _xA^{-1}B\\ \partial _xC^{-1}D &{}\quad 0 \end{pmatrix}, \quad F\begin{pmatrix} \eta \\ u \end{pmatrix} ={\epsilon }\begin{pmatrix} \partial _xA^{-1}\left( \eta u^p\right) \\ \frac{1}{p+1}\partial _xC^{-1} (u^{p+1}) \end{pmatrix}. \end{aligned}$$

In order to consider the Cauchy problem associated with the first order equation (6), we need to describe the semigroup S(t) associated with the linear problem

$$\begin{aligned} \begin{pmatrix} \eta \\ u \end{pmatrix}_t+M\begin{pmatrix} \eta \\ u \end{pmatrix}=0. \end{aligned}$$

(7)

If we consider the Sobolev type space $Y^s=H^{s}\times H^s$ with norm given by

$$\begin{aligned} \Vert (\eta ,u)\Vert ^2_{Y^s }=\Vert \eta \Vert ^2_{H^s}+\Vert u\Vert ^2_{H^s}. \end{aligned}$$

Then the unique solution of the linear problem (7) with the initial condition

$$\begin{aligned} (\eta (0,\cdot ),u(0,\cdot ))=(\eta _0,u_0) \in Y^s, \end{aligned}$$

(8)

is given by

$$\begin{aligned} (\eta (t), u(t))=S(t)(\eta _0, u_0), \end{aligned}$$

where S(t) is defined as

$$\begin{aligned} S(t)= \begin{pmatrix} \mathcal {F}^{-1} &{} 0\\ \\ 0 &{} \mathcal {F}^{-1} \end{pmatrix}\begin{pmatrix} \cos \left( \xi \Lambda (\xi )t\right) &{} -i\frac{\varphi _1(\xi )}{\Lambda (\xi )}\sin (\xi \Lambda (\xi )t)\\ \\ -i\frac{\varphi _2(\xi )}{\Lambda (\xi )}\sin \left( \xi \Lambda (\xi )t\right) &{} \cos \left( \xi \Lambda (\xi )t\right) \end{pmatrix}\begin{pmatrix} \mathcal {F} &{} &{} 0\\ \\ 0 &{} &{} \mathcal {F} \end{pmatrix}, \end{aligned}$$

and the functions $\varphi _i, \Lambda :{\mathbb R}\longrightarrow {\mathbb R}$ are given by

$$\begin{aligned} \varphi _1(\xi )=\sqrt{\frac{1+b\mu \xi ^2}{1+a\mu \xi ^2}}, \quad \varphi _2(\xi )=\sqrt{\frac{1+d\mu \xi ^2}{1+c\mu \xi ^2}}, \quad \Lambda ^2(\xi )=\frac{(1+b\mu \xi ^2)(1+d\mu \xi ^2)}{(1+a\mu \xi ^2)(1+c\mu \xi ^2)}. \end{aligned}$$

It is convenient to set

$$\begin{aligned} Q(t)(\widehat{\eta }, \widehat{u})=\bigl (Q_1(t), Q_2(t)\bigr )(\widehat{\eta }, \widehat{u}), \end{aligned}$$

where

$$\begin{aligned} Q_1(t)(\widehat{\eta }, \widehat{u})(\xi )&=\cos \left( \xi \Lambda (\xi )t\right) \widehat{\eta }(\xi ) -i\frac{\varphi _1(\xi )}{\Lambda (\xi )}\sin (\xi \Lambda (\xi )t)\widehat{u}(\xi ),\nonumber \\ Q_2(t)(\widehat{\eta }, \widehat{u})(\xi )&=-i\frac{\varphi _2(\xi )}{\Lambda (\xi )}\sin \left( \xi \Lambda (\xi )t\right) \widehat{\eta }(\xi )+ \cos \left( \xi \Lambda (\xi )t\right) \widehat{u}(\xi ). \end{aligned}$$

Then we have that

$$\begin{aligned} S(t)(\eta ,\Phi )=(\mathcal {F}^{-1}(Q_1(t)(\widehat{\eta },\widehat{\Phi })), \mathcal {F}^{-1}(Q_2(t)(\widehat{\eta },\widehat{\Phi }))). \end{aligned}$$

On the other hand, it is known that the Duhamel’s principle implies that if $(\eta , \Phi )$ is a solution of (6) with the initial condition (8), then this solution satisfies the integral equation

$$\begin{aligned} \begin{pmatrix} \eta \\ \Phi \end{pmatrix} (t)=S(t)\begin{pmatrix} \eta _0\\ u_0 \end{pmatrix}-\int _0^tS(t-\tau )\,F\begin{pmatrix} \eta \\ u \end{pmatrix}(\tau )\,d\tau . \end{aligned}$$

(9)

Hereafter, we refer a couple $(\eta ,u)\in C([0,T], Y^s)$ satisfying the integral equation (9) as a mild solution for the Cauchy problem associated with the system (7) with initial condition (8). Now, we will establish the existence of mild solutions. For this, we use some linear and nonlinear estimates. Let us start with the following result.

Lemma 2.1

Suppose $s\in {\mathbb R}$. Then for all $t\in {\mathbb R}$, S(t) is a bounded linear operator from $Y^s$ into $Y^s$. Moreover, there exists $K_1>0$ such that for all $t\in {\mathbb R}$,

$$\begin{aligned} \Vert S(t)(\eta ,u)\Vert _{Y^s}\le K_1\Vert (\eta ,u)\Vert _{Y^s}. \end{aligned}$$

Proof

First note that there is a constant $\beta >0$ such that $0<\frac{\varphi _i}{\Lambda }\le \beta$. Then we have that

$$\begin{aligned} \Vert \mathcal {F}^{-1}(Q_1(t)(\widehat{\eta }, \widehat{u})) \Vert ^2_{H^{s}}&\le \int _{{\mathbb R}} (1+\xi ^2)^{s}\left| \cos (\xi \Lambda (\xi )t)\right| ^2|\widehat{\eta }(\xi )|^2d\xi \\&\quad +\int _{{\mathbb R}} (1+\xi ^2)^{s}\frac{\varphi ^2_1(\xi )}{\Lambda ^2(\xi )}|\sin \left( \xi \Lambda (\xi )t\right) |^2| \widehat{u}(\xi )|^2d\xi \\&\le \int _{{\mathbb R}} (1+\xi ^2)^{s}|\widehat{\eta }(\xi )|^2d\xi +\beta ^2\int _{{\mathbb R}} (1+\xi ^2)^{s}|\widehat{u}(\xi )|^2d\xi \\&\le K(\beta )(\Vert \eta \Vert ^2_{H^{s}}+\Vert u\Vert ^2_{H^{s}}). \end{aligned}$$

In a similar fashion, we see that

$$\begin{aligned} \Vert \mathcal {F}^{-1} (Q_2(t)(\widehat{\eta }, \widehat{u})) \Vert ^2_{H^s}&=\int _{{\mathbb R}}(1+\xi ^2)^{s} \frac{\varphi ^2_2(\xi )}{\Lambda ^2(\xi )}|\sin \left( \xi \Lambda (\xi )t\right) |^2|\widehat{\eta }(\xi )|^2\,d\xi \\&\quad +\int _{{\mathbb R}} (1+|\xi |^2)^{s}|\cos (\xi \Lambda (\xi )t)|^2|\widehat{u}(\xi )|^2d\xi \\&\le \beta ^2\int _{{\mathbb R}} (1+\xi ^2)^{s}|\widehat{\eta }(\xi )|^2 +\int _{{\mathbb R}}(1+\xi ^2)^{s}|\widehat{u}(\xi )|^2d\xi \\&\le K(\beta )(\Vert \eta \Vert ^2_{H^{s}}+\Vert u\Vert ^2_{H^s}). \end{aligned}$$

Then we obtain that

$$\begin{aligned} \Vert S(t)(\eta ,u)\Vert ^2_{Y^s}&=\Vert \mathcal {F}^{-1}(Q_1(t)(\widehat{\eta }, \widehat{u}))\Vert ^2_{H^{s}} +\Vert \mathcal {F}^{-1}(Q_2(t)(\widehat{\eta }, \widehat{u})\Vert ^2_{H^{s}}\\&\le K\Vert (\eta ,u)\Vert ^2_{Y^s}, \end{aligned}$$

and S(t) have the required property.

Next, we want to perform the estimates for nonlinear terms of system (6) (Lemma 2.4), which will follow by an estimate obtained by J. Bona and N. Tzvetkov (see Lemma 1 in [2]) in the case $a, c>0$ and the well known estimates for the commutator of Kato in the case $a=c=0$. First, note that for $r>0$, the Fourier multiplier $\psi _r(\xi )=\frac{\xi }{1+r\xi ^2}$ is associated with the operator $R^{-1}\partial _x$ with $R=I-r\partial _x^2$, since we have that

$$\begin{aligned} \widehat{R^{-1}\partial _x u}(\xi )=\psi _r(\xi )\widehat{u}(\xi ). \end{aligned}$$

Lemma 2.2

(Bona and Tzvetkov [2]) Let $r>0$, $s\ge 0$ and $u,v\in H^s(\mathbb R)$. Then there exists a constant $K(r)>0$ such that

$$\begin{aligned} \Vert R^{-1}(\partial _x)uv\Vert _{H^{s}(\mathbb R)}\le K(r)\Vert u\Vert _{H^{s}(\mathbb R)}\Vert v\Vert _{H^{s}({\mathbb R})}. \end{aligned}$$

Now, let $J=\left( I-\partial _x^2\right) ^{1/2}$ be the operator defined by

$$\begin{aligned} \widehat{Jf}=(1+|\xi |^2)^{1/2}\widehat{f}, \end{aligned}$$

and let $[\, \, , \,]$ be the commutator defined by

$$\begin{aligned}{}[J^s, u ]v= J^s(uv)-uJ^sv. \end{aligned}$$

Lemma 2.3

(Kato [3–5]) Suppose $s>\frac{3}{2}$, $t>\frac{1}{2}$ and $u\in H^s(\mathbb R), w\in H^{s-1}(\mathbb R)$. Then there exists a constant $K>0$ such that

• (1) $\left\| [J^s, u]w\right\| _{L^2({\mathbb R})}\le K\Vert u\Vert _{H^s({\mathbb R})}\Vert w\Vert _{H^{s-1}({\mathbb R})}$.

• (2) $\left\| u\partial _xw\right\| _{L^2({\mathbb R})}\le K\Vert \partial _xu\Vert _{H^t({\mathbb R})}\Vert w\Vert _{L^2({\mathbb R})}$.

We now will establish the nonlinear estimates.

Lemma 2.4

Suppose a, c, p and s are such that

• (i) $a, c>0, \, p=1, s\ge 0$, or

• (ii) $a, c>0, \, p>1, \, s>\frac{1}{2}$, or

• (iii) $a=c=0, \, p\ge 1, \, s>\frac{3}{2}$.

Then there are constants $K_2,K_3>0$ such that

$$\begin{aligned} \Vert F(\eta ,u)\Vert _{Y^s}&\le K_2\Vert (\eta ,u)\Vert ^{p+1}_{Y^s}, \end{aligned}$$

(10)

$$\begin{aligned} \Vert F(\eta ,u)-F(\eta _1,u_1)\Vert _{Y^s}&\le K_3\Vert (\eta ,u)-(\eta _1,u_1)\Vert _{Y^s}\left( \Vert (\eta , u)\Vert _{Y^s}+\Vert (\eta _1,u_1)\Vert _{Y^s}\right) ^p. \end{aligned}$$

(11)

Proof

We write $F=\epsilon \left( F_1,\frac{1}{p+1}F_2\right)$ where

$$\begin{aligned} F_1(\eta ,u)=A^{-1}\partial _x(\eta u^p), \quad F_2(\eta ,u)=C^{-1}\partial _x(u^{p+1}). \end{aligned}$$

First we assume that $a,c>0$, $p=1$ and $s\ge 0$. Note that the Lemma 2.2 holds for $\psi _{a}(\xi )=\frac{\xi }{1+a\xi ^2}$. Then we have that

$$\begin{aligned} \Vert F_1(\eta ,u)\Vert _{H^{s}}&= \Vert A^{-1}\partial _x\left( \eta u\right) \Vert _{H^{s}}\\&\le K(a)\Vert \eta \Vert _{H^{s}}\Vert u\Vert _{H^{s}}\\&\le K(a)(\Vert \eta \Vert ^2_{H^{s}}+\Vert u\Vert ^2_{H^{s}})\\&= K(a)\Vert (\eta ,u)\Vert _{Y^s}^2. \end{aligned}$$

Similarly we have that

$$\begin{aligned} \Vert F_2(\eta ,u)\Vert _{H^{s}}&=\Vert C^{-1}\partial _x(u^2)\Vert _{H^s}\le K(c)\Vert u\Vert ^2_{H^{s}}\le K(c)\Vert (\eta ,u)\Vert _{Y^s}^2. \end{aligned}$$

In other words, we have established estimate (1). Now we prove estimate (2). In fact,

$$\begin{aligned} \Vert F_1(\eta ,u)-F_1(\eta _1,u_1)\Vert _{H^{s}}&\le \Vert A^{-1}\partial _x\left( \eta u -\eta _1 u_1\right) \Vert _{H^{s}}\\&\le \Vert A^{-1}\partial _x(\eta (u-u_1))\Vert _{H^{s}}+\Vert A^{-1}\partial _x(\eta -\eta _1) u_1\Vert _{H^{s}} \\&\le K(a)\left( \Vert \eta \Vert _{H^{s}}\Vert u-u_1\Vert _{H^{s}}+ \Vert \eta -\eta _1\Vert _{H^{s}}\Vert u_1\Vert _{H^{s}}\right) \\&\le K(a) \left( \Vert \eta \Vert _{H^{s}}+\Vert u_1\Vert _{H^{s}}\right) \left( \Vert \eta -\eta _1\Vert _{H^{s}}+\Vert u-u_1\Vert _{H^{s}}\right) \\&\le K(a)\left( \Vert (\eta , u)\Vert _{Y^s}+\Vert (\eta _1,u_1)\Vert _{Y^s}\right) \Vert (\eta ,u)-(\eta _1,u_1)\Vert _{Y^s}. \end{aligned}$$

In a similar fashion we have that

$$\begin{aligned} \Vert F_2(\eta ,u)-F_2(\eta _1,u_1)\Vert _{H^{s}}&=\Vert C^{-1}\partial _x(u^2-u_1^2)\Vert _{H^{s}}\\&=\Vert C^{-1}\partial _x(u+u_1)(u-u_1)\Vert _{H^{s}} \\&\le K(c)\Vert u+u_1\Vert _{H^{s}}\Vert u-u_1\Vert _{H^{s}}\\&\le K(c)\left( \Vert (\eta ,u)\Vert _{Y^s}+\Vert (\eta _1,u_1)\Vert _{Y^s}\right) \Vert (\eta ,u)-(\eta _1,u_1)\Vert _{Y^s}. \end{aligned}$$

Then we conclude that

$$\begin{aligned} \Vert F(\eta ,u)-F(\eta _1,u_1)\Vert _{Y^s}&\le K(a, c)( \Vert F_1(\eta ,u)-F_1(\eta _1,u_1)\Vert _{H^s} +\Vert F_2(\eta ,u)\\&\quad -F_2(\eta _1,u_1)\Vert _{H^{s}})\le K(a, c)(\Vert (\eta ,u)\Vert _{Y^s}\\&\quad +\Vert (\eta _1,u_1)\Vert _{Y^s}) \Vert (\eta ,u)-(\eta _1,u_1)\Vert _{Y^s}. \end{aligned}$$

Now we suppose that $s>\frac{1}{2}$ and $p>1$. Using the Lemma 2.2 and that $H^s(\mathbb R)$ is an algebra we obtain that

$$\begin{aligned} \Vert F_1(\eta ,u)\Vert _{H^{s}}&= \Vert A^{-1}\partial _x(\eta u^p)\Vert _{H^{s}} \\&\le K(a)\Vert \eta u^p\Vert _{H^s}\\&\le K(a)\Vert \eta \Vert _{H^s}\Vert u\Vert ^p_{H^s}\\&\le K(a)\Vert (\eta ,u)\Vert _{Y^s}^{p+1}, \end{aligned}$$

and also that

$$\begin{aligned} \Vert F_2(\eta ,u)\Vert _{H^{s}}&=\Vert C^{-1}\partial _x(u^{p+1})\Vert _{H^s}\le K(c)\Vert u\Vert ^{p+1}_{H^{s}}\le K(c)\Vert (\eta ,u)\Vert _{Y^s}^{p+1}. \end{aligned}$$

Moreover, we see that

$$\begin{aligned} \Vert F_1(\eta ,u)-F_1(\eta _1,u_1)\Vert _{H^s}&\le \Vert A^{-1}\partial _x(\eta (u^p-u_1^p))\Vert _{H^{s}}+\Vert A^{-1}\partial _x((\eta -\eta _1) u_1^p)\Vert _{H^{s}}\\&\le K(a)\Vert \eta \Vert _{H^{s}}\Vert u^p-u_1^p\Vert _{H^{s}} +\Vert \eta -\eta _1\Vert _{H^{s}}\Vert u_1\Vert ^p_{H^s}. \end{aligned}$$

But a simple calculation shows that

$$\begin{aligned} \Vert u^p-u_1^p\Vert _{H^{s}}\le K(p)\Vert u-u_1\Vert _{H^{s}}\left( \Vert u\Vert _{H^{s}}+\Vert u_1\Vert _{H^{s}}\right) ^{p-1}. \end{aligned}$$

Then we have that

$$\begin{aligned} \Vert F_1(\eta ,u)\!-\!F_1(\eta _1,u_1)\Vert _{Y^s} \!\le \!K(a, c, p)\Vert (\eta ,u)\!-\!(\eta _1,u_1)\Vert _{Y^s}\left( \Vert (\eta ,u)\Vert _{Y^s}\!+\!\Vert (\eta _1,u_1)\Vert _{Y^s}\right) ^p. \end{aligned}$$

In a similar fashion we obtain the same estimate for $\Vert F_2(\eta ,u)-F_2(\eta _1,u_1)\Vert _{Y^s}$ and then (1) and (2) hold.

We assume now that $a=c=0$, $p\ge 1$ and $s>\frac{3}{2}$. First we will that if $v,\partial _x w\in H^s$ then there exists $K>0$ such that

$$\begin{aligned} \Vert v\partial _xw\Vert _{H^s}\le K\Vert v\Vert _{H^s}\Vert w\Vert _{H^s}. \end{aligned}$$

(12)

In fact, from Lemma 2.3 we see that

$$\begin{aligned} \Vert v\partial _xw\Vert _{H^s}&=\Vert J^s(v\partial _xw)\Vert _{L^2}\\&\le \Vert \,[J^s, v]\partial _xw\,\Vert _{L^2}+\Vert v\partial _x J^sw\Vert _{L^2}\\&\le K(\Vert v\Vert _{H^s}\Vert \partial _xw\Vert _{H^{s-1}}+\Vert \partial _x v\Vert _{H^{s-1}}\Vert J^sw\Vert _{L^2})\\&\le K\Vert v\Vert _{H^s}\Vert w\Vert _{H^s}. \end{aligned}$$

Then, using (12) and that $H^s({\mathbb R})$ is an algebra we have that

$$\begin{aligned} \Vert F_1(\eta , u)\Vert _{H^s}&\le K(p)(\Vert \partial _x\eta u^p\Vert _{H^s}+ \Vert \eta u^{p-1}\partial _xu\Vert _{H^s})\\&\le K(p)\Vert \eta \Vert _{H^s}\Vert u\Vert ^p_{H^s}\\&\le K(p)\Vert (\eta , u)\Vert ^{p+1}_{Y^s}, \end{aligned}$$

and also that

$$\begin{aligned} \Vert F_2(\eta , u)\Vert _{H^s}&\le K(p)\Vert u^p\partial _xu\Vert _{H^s}\le K(p)\Vert u\Vert ^{p+1}_{H^s}\le K(p)\Vert (\eta , u)\Vert ^{p+1}_{Y^s}. \end{aligned}$$

Thus, we conclude that there exists $K>0$ such that

$$\begin{aligned} \Vert F(\eta ,u)\Vert _{Y^s}\le K\Vert (\eta ,u)\Vert _{Y^s}^{p+1}. \end{aligned}$$

In a similar way we obtain the part (2) and then the theorem follows.

Next, we establish the local well-posedness for the system (4) in the space $Y^s=H^{s} \times H^s$. For this we will show the existence of a mild solution for the integral equation (9) for a, c, p and s as in Lemma 2.4, using the Banach fixed point theorem. Moreover, if a, c, p and s are as in Lemma 2.4, with $s>\frac{1}{2}$ in the case $a,c>0$ and $p=1$, we already have classical solutions.

Theorem 2.1

Let a, c, p and s be as in Lemma 2.4. Then for all $(\eta _0,u_0)\in Y^s$ there exists a time $T>0$ which depends only on $\Vert (\eta _0,u_0)\Vert _{Y^s}$ such that the problem (4) with initial condition (8) has a unique solution $(\eta ,u)$ satisfying that $(\eta ,u)\in C\left( [0,T], Y^s\right)$. Moreover, as in Lemma 2.4 with $s>\frac{1}{2}$ for $a,c>0$, $p=1$, we have

$$\begin{aligned} (\eta ,u)\in C([0,T], Y^s)\cap C^1([0,T],Y^{s-1}). \end{aligned}$$

On the other hand, for all $0\,<T'<T$ there exists a neighborhood $\,\mathbb {V}$ of $\,(\eta _0,u_0)$ in $Y^s$ such that the correspondence $(\tilde{\eta }_0,\tilde{u}_0)\longrightarrow (\tilde{\eta }(\cdot ),\tilde{u}(\cdot ))$, that associates to $(\tilde{\eta }_0,\tilde{u}_0)$ the solution $(\tilde{\eta }(\cdot ), \tilde{u}(\cdot ))$ of the problem (4) with initial condition $(\tilde{\eta }_0,\tilde{u}_0)$ is a Lipschitz mapping from $\mathbb {V}$ in $C([0,T'],Y^s)$.

Proof

Given $T>0$ we define the space $X^s(T)=C([0,T],Y^s)$, equipped with the norm defined by

$$\begin{aligned} \Vert U\Vert _{X^s(T)}=\max _{t\in [0,T]}\Vert U(\cdot ,t)\Vert _{Y^s}. \end{aligned}$$

It is easy to see that $X^s(T)$ is a Banach space. Let $B_R(T)$ be the closed ball of radius R centered at the origin in $X^s(T)$, i.e.

$$\begin{aligned} B_R(T)=\{U\in X^s(T):\Vert U\Vert _{X^s(T)}\le R\}. \end{aligned}$$

For fixed $U_0=(\eta _0,u_0)\in Y$, we define the map

$$\begin{aligned} \Psi (U(t))=S(t)U_0-\int _0^tS(t-\tau )F(U(\tau ))\,d\tau , \end{aligned}$$

where $U=(\eta ,u)\in X(T)$. We will show that the correspondence $\,U(t)\mapsto \Psi (U(t))\,$ maps $B_R(T)$ into itself and is a contraction if R and T are well chosen. In fact, if $t\in [0,T]$ and $U\in B_R(T)$, then using Lemma 2.1 and statement (1) of Lemma 2.4 we have that

$$\begin{aligned} \Vert \Psi (U(t))\Vert _{Y^s}&\le K_1\left( \Vert U_0\Vert _{Y^s}+K_2\int _0^t\Vert U(\tau )\Vert _{Y^s}^{p+1}d\tau \right) \\&\le K_1(\Vert U_0\Vert _{Y^s}+K_2R^{p+1}T). \end{aligned}$$

Choosing $R=2K_1\Vert U_0\Vert _{Y^s}$ and $T>0$ such that

$$\begin{aligned} (2K_1)^{p+1}K_2\Vert U_0\Vert ^p_{Y^s}\,T\le 1, \end{aligned}$$

we obtain that

$$\begin{aligned} \Vert \Psi (U(t))\Vert _{Y^s}&\le K_1\,\Vert U_0\Vert _{Y^s}(1+(2K_1)^{p+1}C_2\Vert U_0\Vert ^p_{Y^s}\,T)\le 2\,K_1\Vert U_0\Vert _{Y^s}= R. \end{aligned}$$

So that $\Psi$ maps $B_R(T)$ to itself. Let us prove that $\Psi$ is a contraction. If $U,V\in B_R(T)$, then by the definition of $\Psi$ we have that

$$\begin{aligned} \Psi (U(t))-\Psi (V(t))=-\int _0^tS(t-\tau )[F(U(\tau ))-F(V(\tau ))]d\tau . \end{aligned}$$

Then using the statement (2) of Lemma 2.4 we see that for $t\in [0,T]$,

$$\begin{aligned} \left\| \Psi (U(t))-\Psi (V(t))\right\| _{Y^s}&\le K_1K_3\int _0^t\left( \Vert U(\tau )\Vert _{Y^s} +\Vert V(\tau )\Vert _{Y^s}\right) ^p\Vert U(\tau )-V(\tau )\Vert _{Y^s}\,d\tau \\&\le K_1K_3(2R)^pT\Vert U-V\Vert _{X^s(T)}\\&\le 4^pK_1^{p+1}K_3\Vert U_0\Vert ^p_{Y^s}\,T\Vert U-V\Vert _{X^s(T)}. \end{aligned}$$

We choose T enough small so that (2) holds and

$$\begin{aligned} \alpha = 4^pK_1^{p+1}K_3\Vert U_0\Vert ^p_{Y^s}\,T\le \frac{1}{2}. \end{aligned}$$

So, we conclude that

$$\begin{aligned} \left\| \Psi (U)-\Psi (V)\right\| _{X^s(T)}&\le \alpha \Vert U-V\Vert _{X^s(T)}. \end{aligned}$$

Therefore $\Psi$ is a contraction. Thus, there exists a unique fixed point of $\Psi$ in $B_R(T)$, which is a solution of the integral equation (9). Now, if $\left( \eta (t), u(t)\right) \in C([0,T], Y^s)$ is a integral or mild solution, obviously $\left( \eta (0), u(0)\right) =(\eta _0, u_0)$.

Now assume that a, c, p and s are as in Lemma 2.4, with $s>\frac{1}{2}$ in the case $a,c>0$ and $p=1$. We define the function $H\in C([0, T]: Y^s)$ by $H(t)=F(\eta (t), u(t))$. From Lemma 2.4, we have that $H\in L^1([0, T]:Y^s)$ since from inequality (10) for $s>\frac{1}{2}$,

$$\begin{aligned} \Vert F(\eta (t),u(t))\Vert _{C_b(\mathbb R)}\le K_3 \Vert F(\eta (t),u(t))\Vert _{Y^s}\le K_3K_2\Vert (\eta (t),u(t))\Vert ^{p+1}_{Y^s}, \end{aligned}$$

where $C_b({\mathbb R})$ denotes the space of bounded continuous functions defined on ${\mathbb R}$. From this fact and the smoothness properties of the semigroup S, we conclude that the function defined on [0, T] by

$$\begin{aligned} W(t)=\int _0^t S(t-\tau )F(\eta (\tau ), u(\tau ))\,d\tau \end{aligned}$$

is such that $W\in C([0, T]:Y^s)$. On the other hand, we also have that

$$\begin{aligned} \frac{1}{h}\left( W(t+h)-W(t)\right)= & {} \frac{1}{h}\int _t^{t+h} S(t-\tau +h)F(\eta (\tau ), u(\tau ))\,d\tau \nonumber \\&+\left( \frac{ S(h)-I}{h}\right) \int _0^{t} S(t-\tau )F(\eta (\tau ), u(\tau ))\,d\tau . \end{aligned}$$

(13)

Taking limit as $h\rightarrow 0$ and using the continuity of H, we have that

$$\begin{aligned} W'(t)= F(\eta (t), u(t)) -M(\eta (t), u(t)), \end{aligned}$$

and so $U(t)= S(t)U_0 - W(t)$ is such that $U\in C([0, T]:Y^{s})\cap C^1([0, T]:Y^{s-1})$ is a local classical solution of (7). In other words, $\left( \eta (t), u(t)\right)$ is a local classical solution for the Cauchy problem associated with the system (6) and initial condition (8). The uniqueness and continuous dependence of the solution are obtained by standard arguments.

Our main result in this section related with the existence and uniqueness of mild and classical solutions for the Cauchy problem associated with (1) is a direct consequence of the Theorem 2.1. For system (1), we have existence of integral or mild solutions for a, c, p and s as in Lemma 2.4, and for a, c, p and s as in Lemma 2.4, with $s>\frac{1}{2}$ in the case $a,c>0$ and $p=1$, we already have classical solutions.

Theorem 2.2

Let a, c, p and s be as in Lemma 2.4. Then for all $(\eta _0,\Phi _0)\in H^{s}\times {\mathcal {V}}^{s+1}$ there exists a time $T>0$ which depends only on $\Vert (\eta _0,\Phi _0)\Vert _{H^{s}\times {\mathcal {V}}^{s+1}}$ such that the Cauchy problem associated with the Boussinesq system (1) and the initial condition $(\eta _0,\Phi _0)$ has a unique solution $(\eta ,\Phi )$ satisfying that $(\eta ,u)\in C\left( [0,T], Y^s\right) \cap C^1([0,T],Y^{s-1})$. Moreover, as in Lemma 2.4 with $s>\frac{1}{2}$ for $a, c>0$, $p=1$, we have

$$\begin{aligned} (\eta ,\Phi )\in C([0,T],H^{s}\times {\mathcal {V}}^{s+1})\cap C^1([0,T],H^{s-1}\times {\mathcal {V}}^{s}). \end{aligned}$$

Moreover, for all $0\,<T'<T$ there exists a neighborhood $\,\mathbb {V}$ of $\,(\eta _0,\Phi _0)$ in $H^{s}\times {\mathcal {V}}^{s+1}$ such that the correspondence $(\tilde{\eta }_0,\tilde{\Phi }_0)\longrightarrow (\tilde{\eta }(\cdot ),\tilde{\Phi }(\cdot ))$, that associates to $(\tilde{\eta }_0,\tilde{\Phi }_0)$ the solution $(\tilde{\eta }(\cdot ), \tilde{\Phi }(\cdot ))$ of the problem (1) with initial condition $(\tilde{\eta }_0,\tilde{\Phi }_0)$ is a Lipschitz mapping from $\mathbb {V}$ in $C([0,T'],H^{s}\times {\mathcal {V}}^{s+1})$.

Proof

By hypothesis, if $u_0=\partial _x \Phi _0$, then we have that $(\eta _0, u_0)\in Y^s$, and also that $\widehat{u}_0(0)=0$. Now, from Theorem 2.1, there exist $T=T(\Vert (\eta _0,u_0)\Vert _{Y^s})>0$ and a unique solution $(\eta ,u)$ of the problem (4) with initial condition $\,(\eta _0,\Phi _0)$ satisfying that

$$\begin{aligned} (\eta ,u)\in C([0,T], Y^s)\cap C^1([0,T],Y^{s-1}), \end{aligned}$$

and also that

$$\begin{aligned} \widehat{u}_0(0)= \int _{\mathbb R}u_0(x)\,dx= \int _{\mathbb R}u(t, x)\,dx = \widehat{u}(t, \xi )=0. \end{aligned}$$

So, the couple $(\eta , \Phi )$ where $\Phi (t, x)= \partial ^{-1}_x u(t, x)$ is a mild solution of the Cauchy problem (1) with initial condition $\,(\eta _0,\Phi _0)$ satisfying that

$$\begin{aligned} (\eta ,\Phi )\in C([0,T],H^{s}\times {\mathcal {V}}^{s+1})\cap C^1([0,T],H^{s-1}\times {\mathcal {V}}^{s}). \end{aligned}$$

The last part follows by noting that if $\Phi (t, x)= \partial ^{-1}_x u(t, x)$, then we have that

$$\begin{aligned} \Vert (\eta (t, \cdot ),u(t, \cdot ))\Vert _{Y^s}= \Vert (\eta (t, \cdot ),\Phi (t, \cdot )) \Vert _{H^{s}\times {\mathcal {V}}^{s+1}}. \end{aligned}$$

Global Existence for $a=c$

In this section for $a=c$ we will establish that any local solution in time of the system (1) can be extended for any $t>0$. The result will depends strongly on fact that the Hamiltonian $\mathcal H$ is conserved in time on classical and mild solutions. Before we go further, for solutions of the system (1) a direct computation shows that

$$\begin{aligned} \partial _t\mathcal H\left( \eta (t,x),\Phi (t,x)\right) =\mu (c-a)\int _{\mathbb R}\partial _x^2\Phi _t\eta _t\,dx, \end{aligned}$$

meaning that the Hamiltonian $\mathcal H$ is conserved in time on classical and mild solutions if and only if $a=c$. Now note that

$$\begin{aligned} \mathcal H(\eta ,\Phi )&=\frac{1}{2} \int _{\mathbb {R}}\left( \eta ^2+d\mu \left( \eta _x\right) ^2 +\left( \Phi _x\right) ^2 +b\mu \left( \Phi _{xx}\right) ^2+\frac{2{\epsilon }}{p+1}\eta \left( \Phi _x\right) ^{p+1} \right) dx \nonumber \\&= \frac{1}{2}\left( \mathcal E(\eta , \Phi )+ G(\eta , \Phi )\right) , \end{aligned}$$

(14)

where functional $\mathcal E$ (energy) and G are given by

$$\begin{aligned} \mathcal {E}(\eta ,\Phi )&=\int _{\mathbb {R}}(\eta ^2+d\mu \left( \eta _x\right) ^2 +\left( \Phi _x\right) ^2 +b\mu \left( \Phi _{xx}\right) ^2)dx, \\ G(\eta ,\Phi )&=\frac{2{\epsilon }}{p+1}\int _{{\mathbb R}}\eta \left( \Phi _x\right) ^{p+1}dx. \end{aligned}$$

We will see that the global well-posedness follows by using a variational approach and the fact that the energy $\sqrt{\mathcal {E}}$ is a norm in the space $H^1({\mathbb R}) \times {\mathcal {V}}^2$, since for some constant $K(b, d, \mu )>1$,

$$\begin{aligned} K(b, d, \mu )^{-1}\Vert (\eta , \Phi )\Vert ^2_{H^1\times {\mathcal {V}}^2}\le \mathcal E(\eta ,\Phi )\le K(b, d, \mu )\Vert (\eta , \Phi )\Vert ^2_{H^1\times {\mathcal {V}}^2}. \end{aligned}$$

(15)

A key ingredient in our analysis depends upon the variational characterization of the number $\delta _0$ defined by

$$\begin{aligned} \delta _0&=\inf \left\{ \sup _{\lambda \ge 0}\mathcal H(\lambda (\eta ,\Phi )):(\eta ,\Phi ) \in H^1\times {\mathcal {V}}^2 {\setminus } \{0\}\right\} \\&=\inf \left\{ \sup _{\lambda \ge 0} \mathcal H(\lambda (\eta ,\Phi )):(\eta ,\Phi ) \in H^1\times {\mathcal {V}}^2, \ G(\eta ,\Phi ) < 0\right\} . \end{aligned}$$

Note that for $G(\eta ,\Phi ) \ge 0$, we have that $\sup _{\lambda \ge 0} \mathcal H(\lambda (\eta ,\Phi ))=\infty$. It is straightforward to see that

$$\begin{aligned} \delta _0= \frac{p}{2(p+2)}\left( \frac{2}{p+2}\right) ^{\frac{2}{p}}K_p^{-\frac{p+2}{p}}, \end{aligned}$$

(16)

where $K_p$ is defined as

$$\begin{aligned} K_p= \sup \left\{ \frac{G^{\frac{2}{p+2}}(\eta ,\Phi )}{\mathcal E(\eta ,\Phi )} :(\eta ,\Phi ) \in H^1\times {\mathcal {V}}^2{\setminus } \{0\}\right\} . \end{aligned}$$

(17)

In fact, from the Young inequality and that the embedding $H^1({\mathbb R})\hookrightarrow L^q({\mathbb R})$ is continuous for $q\ge 2$, we see that there is $K_1=K_1({\epsilon }, p)>0$ such that for all $(\eta , \Phi )\in H^1({\mathbb R})\times {\mathcal {V}}^2$,

$$\begin{aligned} \left| G(\eta ,\Phi )\right| \le K_1(\Vert \eta \Vert ^{p+2}_{H^1}+\Vert \Phi \Vert ^{p+2}_{{\mathcal {V}}^2})\le K_1\Vert (\eta , \Phi )\Vert ^{\frac{p+2}{2}}_{H^1\times {\mathcal {V}}^2}. \end{aligned}$$

Thus, from (15), we obtain that

$$\begin{aligned} |G(\eta ,\Phi )|^\frac{2}{p+2}\le K_1({\epsilon }, p)K(b, d, \mu )\mathcal E(\eta ,\Phi ), \end{aligned}$$

meaning that $K_p$ is finite. Now, for $\lambda \ge 0$, we define the function

$$\begin{aligned} V(\lambda )= \mathcal H(\lambda (\eta ,\Phi ))= \frac{\lambda ^{2}}{2}\mathcal E(\eta ,\Phi )+\frac{\lambda ^{p+2}}{2}G(\eta ,\Phi ). \end{aligned}$$

Then we have that $V'(\lambda )=0$ if and only if $\lambda _0=0$ or $\lambda _1^pG(\eta ,\Phi )=-\frac{2}{p+2}\mathcal E(\eta ,\Phi )$. Since $V(0)=0$, then

$$\begin{aligned} \sup _{\lambda \ge 0}V(\lambda )=V(\lambda _1)=\frac{p}{2(p+2)}\left( \frac{2}{p+2}\right) ^{\frac{2}{p}} \left( \frac{G^{\frac{2}{p+2}}(\eta ,\Phi ) }{\mathcal E(\eta ,\Phi )} \right) ^{-\frac{p+2}{p}}. \end{aligned}$$

This formula implies the desired equality (16). We note that the constant $K_p$ establishes a Sobolev type inequality, since we have that

$$\begin{aligned} |G(\eta , \Phi )|^{\frac{1}{p+2}} \le K_p^{\frac{1}{2}} \sqrt{\mathcal E(\eta , \Phi )} \le K(b, d, \mu )^{\frac{1}{2}}K_p^{\frac{1}{2}} \Vert (\eta , \Phi )\Vert _{H^1\times {\mathcal {V}}^2}. \end{aligned}$$

(18)

Before we go further, we consider the auxiliary functional $\mathcal H_1(U)= \mathcal H'(U)(U)$, which has can be expressed as

$$\begin{aligned} \mathcal H_1(\eta ,\Phi )=\mathcal E(\eta , \Phi ) +\frac{p+2}{2}G(\eta , \Phi ). \end{aligned}$$

(19)

In particular, we have that

$$\begin{aligned} \mathcal H(\eta ,\Phi )= \frac{p}{2(p+2)}\mathcal {E}(\eta , \Phi ) + \frac{1}{p+2}\mathcal H_1(\eta ,\Phi ). \end{aligned}$$

(20)

We have the following result related with invariance of quantities under the flow of solutions for the Cauchy problem associated with the system (1).

Lemma 3.1

Let $(\eta , \Phi )$ be a local solution of (1) with initial condition $(\eta _0,\Phi _0) \in H^1({\mathbb R})\times {\mathcal {V}}^2$ on $[0, T_0)$ such that $\mathcal {H}(\eta _0, \Phi _0)<\delta _0$ and $\mathcal H_1(\eta _0, \Phi _0) >0$. Then for $t \in [0, T_0)$ we have that $\mathcal H(\eta (t),\Phi (t)) < \delta _0$, $\mathcal {H}_1(\eta (t),\Phi (t))>0$ and

$$\begin{aligned} e(t)= \sup _{r\in [0, t]} \mathcal {E}(\eta (r),\Phi (r))<\frac{2(p+2)}{p}\delta _0. \end{aligned}$$

Proof

First we observe that the Hamiltonian $\mathcal H$ is conserved in time on solutions. In fact, after integration by parts, we obtain that

$$\begin{aligned} \frac{d}{dt}\mathcal {H}(\eta (t),\Phi (t))&=\int _{\mathbb {R}}\left( \left( \eta -d\mu \partial _x^2\eta +\frac{{\epsilon }}{p+1}(\partial _x\Phi )^{p+1} \right) \eta _t \right. \\&\quad \left. + (-\partial _x^2\Phi +b\mu \partial _x^4\Phi -{\epsilon }\partial _x[\eta \left( \partial _x\Phi \right) ^p])\Phi _t\right) \,dx \\&=\int _{\mathbb {R}}((I-a\mu \partial _x^2)\eta _t\Phi _t -(I-a\mu \partial _x^2)\Phi _t\eta _t)\,dx\\&=0, \end{aligned}$$

since the operator $I-a\mu \partial _x^2$ is self adjoint on $L^2({\mathbb R})$. In other words, we have on classical solutions that

$$\begin{aligned} \mathcal H(\eta (t),\Phi (t))=\mathcal H(\eta _0,\Phi _0)<\delta _0, \end{aligned}$$

(21)

as long as the solution exist for $0\le t <T_0$.

Now, assume that there is $t_2\in (0, T_0)$ such that $\mathcal H_1(\eta (t_2),\Phi (t_2))<0$, then by continuity, there is $0<t_1<t_2$ such that

$$\begin{aligned} \mathcal H_1(\eta (t_1),\Phi (t_1))=0, \quad (\eta (t_1),\nabla \Phi (t_1))\ne 0. \end{aligned}$$

(22)

Then, from (20), we have that

$$\begin{aligned} 0<\mathcal E(\eta (t_1),\Phi (t_1))=\frac{2(p+2)}{p}\mathcal H(\eta (t_1),\Phi (t_1))-\frac{2}{p}\mathcal H_1(\eta (t_1),\Phi (t_1))<\frac{2(p+2)}{p}\delta _0. \end{aligned}$$

(23)

But from the Sobolev type inequality (18) we conclude that

$$\begin{aligned} \left| G(\eta (t_1),\quad \Phi (t_1))\right|&\le K_p^{\frac{p+2}{2}}\left[ \mathcal E(\eta (t_1),\Phi (t_1))\right] ^{\frac{p}{2}}\mathcal E(\eta (t_1),\Phi (t_1))\\&< K_p^{\frac{p+2}{2}}\left[ \frac{2(p+2)}{p}\delta _0\right] ^{\frac{p}{2}}\mathcal E(\eta (t_1),\Phi (t_1))\\&< \left( \frac{2}{p+2}\right) \mathcal E(\eta (t_1),\Phi (t_1)), \end{aligned}$$

which implies, by using (19), that we already have $\mathcal H_1(\eta (t_1),\Phi (t_1))>0$, but this is a contradiction. In other words, we have shown that $\mathcal H_1(\eta (t),\Phi (t))>0$, since the case $\mathcal H_1(\eta (t_2),\Phi (t_2))=0$ also provides a contradiction.

Now, as a consequence of invariance of the Hamiltonian given by (21), we have for $t\in [0,T_0)$ and the Sobolev type inequality (18) that

$$\begin{aligned} \mathcal {E}(\eta (t),\Phi (t)) \le 2\mathcal {H}(\eta (0),\Phi (0))+|{G}(\eta (t),\Phi (t))| \le 2\delta _0+K_p^{\frac{p+2}{2}}(\mathcal {E}(\eta (t),\Phi (t)))^{\frac{p+2}{2}}. \end{aligned}$$

(24)

Then from this inequality we conclude that

$$\begin{aligned} e(t) \le 2\delta _0+K_p^{\frac{p+2}{2}}(e(t))^{\frac{p+2}{2}}. \end{aligned}$$

Now consider the function f defined for $x>0$ as $f(x)=x- K_p^{\frac{p+2}{2}}x^{\frac{p+2}{2}}-2\delta _0$. Note that $f(0)=-2\delta _0<0$ and that there is a unique $x_0>0$ such that $f'(x_0)=0$. In fact,

$$\begin{aligned} f'(x_0)=0 \Leftrightarrow x_0=\left( \frac{2}{p+2}\right) ^{\frac{2}{p}}K_p^{-\frac{p+2}{p}}=\frac{2(p+2)}{p}\delta _0, \end{aligned}$$

and so, we also have that $f(x_0)=0$ and that $f(x)<0$ for $x\ne x_0$. We want to show that in fact $e(t) < x_0$ for $t\in [0, T_0)$. So, assume that for some $0<t_1<T_0$ we have that $\mathcal E(\eta (t_1), \Phi (t_1)) \ge x_0$. Then from Eq. (20) we have that

$$\begin{aligned} \delta _0&>\mathcal H(\eta (t_1),\Phi (t_1))= \frac{p}{2(p+2)}\mathcal {E}(\eta (t_1), \Phi (t_1)) + \frac{1}{p+2}\mathcal H_1(\eta (t_1),\Phi (t_1)) \\&\ge \delta _0 +\frac{1}{p+2}\mathcal H_1(\eta (t_1),\Phi (t_1)), \end{aligned}$$

meaning that $\mathcal H_1(\eta (t_1),\Phi (t_1))\le 0$, but we have that $\mathcal H_1(\eta (t),\Phi (t))>0$ para $t\in [0, T_0)$. In other words, $\mathcal E(\eta (t), \Phi (t)) < x_0$, and so $e(t) \le x_0$, for $t\in [0, T_0)$ as claimed.

The proof of the existence of global solutions for the system (1) is based on the Lemma 3.1.

Theorem 3.1

Assume $a=c\ge 0$ and $p\ge 1$. Let $(\eta _0,\Phi _0) \in H^1({\mathbb R})\times {\mathcal {V}}^2$ be such that $\mathcal {H}(\eta _0,\Phi _0)<\delta _0$ and $\mathcal H_1(\eta _0,\Phi _0)>0$. Then there exists a unique global solution $(\eta ,\Phi ) \in C([0, \infty ), H^1({\mathbb R})\times {\mathcal {V}}^2)$ of the Boussinesq system (1) satisfying the initial condition

$$\begin{aligned} (\eta (0, \cdot ) ,\Phi (0, \cdot ))=(\eta _{0},\Phi _{0}). \end{aligned}$$

Proof

First we assume $a=c>0$. Then, if $(\eta _0,\Phi _0) \in H^1({\mathbb R})\times {\mathcal {V}}^2$, by the local existence result, there is a maximal existence time $T_0>0$ and a unique solution $(\eta ,\Phi )\in C([0, T_0), H^1({\mathbb R})\times {\mathcal {V}}^2)$ of the Cauchy problem associated with the system (1) with initial condition $(\eta (0, \cdot ) ,\Phi (0, \cdot ))=(\eta _0,\Phi _0)$. From the conservation in time of the Hamiltonian and the hypothesis we see that

$$\begin{aligned} \mathcal {H}(\eta (t),\quad \Phi (t)) =\frac{p}{2(p+2)}\mathcal E\left( \eta (t),\quad \Phi (t)\right) + \frac{1}{p+2}\mathcal H_1(\eta (t),\Phi (t))=\mathcal {H}(\eta _{0},\Phi _{0}) <\delta _0. \end{aligned}$$

Hence, using Lemma 3.1 we have that $\mathcal H_1(\eta (t),\Phi (t))> 0$ . Then we also have that

$$\begin{aligned} \mathcal E(\eta (t),\quad \Phi (t)) \le \frac{2(p+2)}{p}\mathcal {H}(\eta _{0},\Phi _{0})<\frac{2(p+2)}{p}\delta _0. \end{aligned}$$

But from (15) we obtain that for $t\in [0,T_0)$,

$$\begin{aligned} \left\| (\eta (t),\Phi (t))\right\| ^2_{H^1\times {\mathcal {V}}^2} \le K(b, d, \mu )\mathcal E(\eta (t),\Phi (t)) < \frac{2(p+2)}{p}K(b, d, \mu )\delta _0. \end{aligned}$$

This fact implies that the solution $(\eta , \Phi )$ is bounded in time on the space $H^1({\mathbb R})\times {\mathcal {V}}^2$ and that for any finite $T_0 < \infty$ we are able to conclude that

$$\begin{aligned} \lim _{t \rightarrow T^{-}_0}\Vert (\eta (t), \Phi (t))\Vert ^2_{H^1\times {\mathcal {V}}^2} < \infty . \end{aligned}$$

In other words, we have that $(\eta , \Phi )$ can be extended in time.

Now, we assume $a=c=0$. Let $s_0>\frac{3}{2}$ be fixed, then by density there exists $(\eta _{0,k} \Phi _{0,k})\in H^{s_0}\times {\mathcal {V}}^{s_0+1}$ such that

$$\begin{aligned} (\eta _{0,k}, \Phi _{0,k})\rightarrow (\eta _{0}, \Phi _{0}) \quad \text {in } H^1\times {\mathcal {V}}^2, \quad \text {as } k\rightarrow \infty . \end{aligned}$$

From the local existence result, for each $k\in \mathbb {Z}^+$ there is $T_{0,k}>0$ and a unique solution $(\eta _k, \Phi _k)$ of the Cauchy problem for the Boussinesq system (1) with initial condition $(\eta _{k}(0, \cdot ), \Phi _{k}(0, \cdot ))=(\eta _{0,k}, \Phi _{0,k})$. On the other hand, there exists $k_0\in \mathbb {Z}^+$ such that $\mathcal {H}(\eta _{0,k},\Phi _{0,k})<\delta _0$ and $\mathcal {H}_1(\eta _{0,k},\Phi _{0,k})>0$ for $k\ge k_0$. Now, for $k\ge k_0$ we have that

$$\begin{aligned} \mathcal {H}(\eta _k,\Phi _k) =\frac{p}{2(p+2)}\mathcal E\left( \eta _k,\Phi _k\right) + \frac{1}{p+2}\mathcal H_1(\eta _k,\Phi _k)=\mathcal {H}(\eta _{0,k},\Phi _{0,k}) <\delta _0. \end{aligned}$$

From Lemma 3.1 we have that $\mathcal H_1(\eta _k,\Phi _k)> 0$ for $k\ge k_0$ . Then we also have that

$$\begin{aligned} \mathcal E(\eta _k,\Phi _k) \le \frac{2(p+2)}{p}\mathcal {H}(\eta _{0,k},\Phi _{0,k})<\frac{2(p+2)}{p}\delta _0. \end{aligned}$$

But from (15) we obtain for $k\ge k_0$ and $t\in [0,T_0)$ that

$$\begin{aligned} \left\| (\eta _k,\Phi _k)\right\| ^2_{H^1\times {\mathcal {V}}^2} \le K\mathcal E(\eta _k,\Phi _k) < \frac{2(p+2)}{p}K\delta _0. \end{aligned}$$

This fact implies that $\{(\eta _k, \Phi _k)\}_{k}$ is bounded sequence in the space $H^1({\mathbb R})\times {\mathcal {V}}^2$ and that for any finite $T_0 < \infty$ and $k\ge k_0$ we are able to conclude that

$$\begin{aligned} \lim _{t \rightarrow T^{-}_0}\Vert (\eta _k, \Phi _k)\Vert ^2_{H^1\times {\mathcal {V}}^2} < \infty . \end{aligned}$$

In other words, for $k\ge k_0$ we have that $(\eta _k, \Phi _k)$ can be extended in time. Since $\{(\eta _k, \Phi _k)\}_{k}$ is bounded sequence in $H^1({\mathbb R})\times {\mathcal {V}}^2$, then there is a subsequence, denoted the same, and $(\eta , \Phi )\in H^1({\mathbb R})\times {\mathcal {V}}^2$ such that

$$\begin{aligned} (\eta _k, \Phi _k)\rightharpoonup (\eta , \Phi ) \quad \text {(weakly) } \text {in } H^1({\mathbb R})\times {\mathcal {V}}^2, \quad \text {as } k\rightarrow \infty . \end{aligned}$$

It is no hard to prove that $(\eta , \Phi )\in C([0,\infty ),\, H^1({\mathbb R})\times {\mathcal {V}}^2)$ is a weak solution of the Cauchy problem for the system (1) satisfying $(\eta (0, \cdot ), \Phi (0, \cdot ))=(\eta _0, \Phi _0)$.

As a consequence of the previous result, we are able to establish that the Cauchy problem associated with the Boussinesq system (1) has global solution in time for initial data $(\eta _0, \Phi _0) \in H^1({\mathbb R})\times {\mathcal {V}}^2$ small enough such that $(\eta _0, \Phi _0)\not = 0$.

Theorem 3.2

Let $p\ge 1$. Then there exists $\delta >0$ such that for any $(\eta _0, \Phi _0) \in H^1({\mathbb R})\times {\mathcal {V}}^2$ with $\Vert (\eta _0, \Phi _0)\Vert _{H^1\times {\mathcal {V}}^2} \le \delta$, the Cauchy problem (1)–(2) has a unique global solution

$$\begin{aligned} (\eta ,\Phi ) \in C([0, \infty ),H^1({\mathbb R})\times {\mathcal {V}}^2 ) \cap C^1([0, \infty ),L^2({\mathbb R}) \times H^1({\mathbb R})) . \end{aligned}$$

Proof

If $G(\eta _0, \Phi _0)\ge 0$, then using (19) we have directly that

$$\begin{aligned} \mathcal H_1(\eta _0, \Phi _0)=\mathcal E(\eta _0, \Phi _0)+\frac{p+2}{2}G(\eta _0, \Phi _0)>0. \end{aligned}$$

Now, If $G(\eta _0, \Phi _0)<0$, then we see from (15) that

$$\begin{aligned} \mathcal H_1(\eta _0, \Phi _0)&=\mathcal E(\eta _0, \Phi _0)+\frac{p+2}{2}G(\eta _0, \Phi _0)\\&\ge K^{-1}(b, d, \mu )\left( \Vert (\eta _0, \Phi _0)\Vert ^2_{H^1\times {\mathcal {V}}^2} +\frac{p+2}{2}K(b, d, \mu )G(\eta _0, \Phi _0)\right) . \end{aligned}$$

Thus, for $\Vert (\eta _0, \Phi _0)\Vert ^2_{H^1\times {\mathcal {V}}^2}$ sufficiently small we would have $\mathcal H_1(\eta _0, \Phi _0)>0$, since

$$\begin{aligned} G(\eta _0, \Phi _0) =O( \mathcal E(\eta _0, \Phi _0)^{\frac{p+2}{2}})=O(\Vert (\eta _0, \Phi _0)\Vert ^{p+2}_{H^1\times {\mathcal {V}}^2}). \end{aligned}$$

From (15), (14) and (18) we see that there exists $K_1(b, d, \mu ,{\epsilon },p)>0$ such that

$$\begin{aligned} \mathcal H(\eta _0, \Phi _0) \le K_1(b, d, \mu ,{\epsilon },p)(1+\Vert (\eta _0, \Phi _0)\Vert _{H^1\times {\mathcal {V}}^2}^p) \Vert (\eta _0, \Phi _0)\Vert ^2_{H^1\times {\mathcal {V}}^2}, \end{aligned}$$

and from (15) we have that

$$\begin{aligned} \mathcal E(\eta _0, \Phi _0)\le K(b, d, \mu )\Vert (\eta _0, \Phi _0)\Vert _{H^1\times {\mathcal {V}}^2}^2. \end{aligned}$$

Hence, we choose $\delta >0$ in a such way that

$$\begin{aligned} K_1(b, d, \mu ,{\epsilon },p) (1+\delta ^p) \delta ^2< \delta _0 \quad \text{ and }\quad K(b, d, \mu )\delta ^2 <\frac{2(p+2)}{p}\delta _0. \end{aligned}$$

Let $(\eta _0, \Phi _0) \in H^1\times {\mathcal {V}}^2$ be such that $\Vert (\eta _0, \Phi _0)\Vert _{H^1\times {\mathcal {V}}^2} \le \delta$, then we see that $\mathcal H(\eta _0, \Phi _0)<\delta _0$. Moreover, from the Sobolev type inequality (18) we obtain that

$$\begin{aligned} |G(\eta _0, \Phi _0)|&\le K_p^{\frac{p+2}{2}} (\mathcal E(\eta _0, \Phi _0))^{\frac{p}{2}}\mathcal E(\eta _0, \Phi _0) \nonumber \\&<K_p^{\frac{p+2}{2}}\left( \frac{2(p+2)}{p}\delta _0\right) ^{\frac{p}{2}} \mathcal E(\eta _0, \Phi _0) \nonumber \\&<\frac{p+2}{2}\mathcal E(\eta _0, \Phi _0). \end{aligned}$$

Then from (19) we have that $\mathcal H_1(\eta _0, \Phi _0)>0$ and the conclusion follows from the previous lemma.

Existence of Solitons

In this section we will establish the existence of finite energy travelling wave solutions or solitons for the 1D-Boussinesq system with $a=c\ge 0$, $b, d>0$ and wave speed $\omega$ satisfying $0<|\omega |<\omega _0$, where $\omega _0=\min \left\{ 1, \frac{d}{a}, \frac{b}{a}\right\}$ for $a\ne 0$ and $\omega _0=1$ for $a=0$. We will see that the solitary waves are characterized as critical points of some functional, for which the existence of critical points follows as a consequence of the mountain pass theorem without the Palais–Smale condition and the existence of a local compact embedding result (see Lemma 4.1).

By a solitary wave solution we shall mean a solution $(\eta , \Phi )$ of (1) of the form

$$\begin{aligned} \eta (t,x)= \frac{1}{\epsilon }u\left( \frac{x-\omega t}{\sqrt{\mu }}\right) , \quad \Phi (t,x)= \frac{\mu }{\epsilon }v\left( \frac{x-\omega t}{\sqrt{\mu }}\right) . \end{aligned}$$

Then we have that the travelling wave profile (u, v) should satisfy the system

$$\begin{aligned} \left\{ \begin{array}{l} bv''''-v^{''}+\omega (u'-au''') -[u (v')^p\,]' =0, \\ u-du''-\omega (v'-cv''') +\frac{1}{p+1}(v')^{p+1}=0. \end{array}\right. \end{aligned}$$

(25)

Next, we define the appropriate spaces. The usual space $H^1(U), \,U\subset \mathbb R$, is the Hilbert space defined as the closure of $C^\infty (U)$ with respect to the norm

$$\begin{aligned} \Vert \phi \Vert ^2_{H^1(U)}=\int _{U}(\phi ^2+(\phi ')^2)dx. \end{aligned}$$

We denote by $\mathcal {V}$ the closure of $C^\infty _0(\mathbb R)$ with respect to the norm given by

$$\begin{aligned} \Vert \psi \Vert ^2_{\mathcal V}&= \int _{\mathbb R} ((\psi ')^2 + (\psi '')^2 )dx = \Vert \psi '\Vert ^2_{H^1(\mathbb R)}. \end{aligned}$$

Note that $(\mathcal {V}, \Vert \cdot \Vert _{{\mathcal {V}}})$ is a Hilbert space with inner product

$$\begin{aligned} (\phi , \psi )_{\mathcal {V}}= (\phi ', \psi ')_{H^{1}({\mathbb R})}. \end{aligned}$$

Also we define the Hilbert space $\mathcal {X}=H^1({\mathbb R})\times {\mathcal {V}}$ with respect to norm

$$\begin{aligned} \Vert (\phi , \psi )\Vert ^2_{\mathcal {X}}=\Vert \phi \Vert ^2_{H^1(\mathbb R)}+\Vert \psi \Vert ^2_{{\mathcal {V}}}. \end{aligned}$$

The existence of solitons for the system (1) is a consequence of a variational approach which apply a minimax type result, since solutions (u, v) of the system (25) are critical points of the functional $J_{\omega }$ given by

$$\begin{aligned} J_{\omega }=I_\omega (u, v)+ G(u, v), \end{aligned}$$

where the functionals $I_\omega$ and G are defined on the space $\mathcal {X}$ by

$$\begin{aligned} I_\omega (u, v)= & {} \int _{\mathbb R}[u^2 + d(u')^{2} + (v')^2+b( v'')^2-2\omega uv'-\omega \left( a+c \right) u' v'' ]dx\\= & {} \int _{\mathbb R}[u^2 + d(u')^{2} + (v')^2+b( v'')^2-2\omega ( uv'+ au' v'' )]dx,\\ G(u, v)= & {} \frac{2}{p+1}\int _{\mathbb R}u(v')^{p+1}dx. \end{aligned}$$

First we have that $I_\omega , G, J_{\omega } \in C^2(\mathcal {X}, {\mathbb R})$ and its derivatives in (u, v) in the direction of (z, w) are given by

$$\begin{aligned} \langle I_\omega '(u, v), (z, w)\rangle&= 2 \int _{\mathbb R} ( uz+ d u'z' + v'w' + bv''w'' )dx \\&\quad -2\omega \int _{\mathbb R}(uw'+ v'z+a(u' w''+ v''z'))dx \\ \langle G'(u, v), (z, w)\rangle&=\frac{2}{p+1}\int _{\mathbb R}( (v')^{p+1}z+(p+1)u (v')^pw')dx. \end{aligned}$$

As a consequence of this, after integration by parts, we conclude that

$$\begin{aligned} J_{\omega }'(u, v)= 2\begin{pmatrix}u-d u'' -\omega (v'-av''')+\frac{1}{p+1}( v')^{p+1}\\ \\ b v''''- v'' +\omega (u'-au''')-[u (v')^p]' \end{pmatrix}, \end{aligned}$$

meaning that critical points of the functional $J_{\omega }$ satisfy the travelling wave Eq. (25). Hereafter, we will say that weak solutions for (25) are critical points of the functional $J_{\omega }$. In particular, we have that

$$\begin{aligned} \left<J_{\omega }'(u, v), (u, v)\right>&= 2I_\omega (u, v)+(p+2)G(u, v) \nonumber \\&= 2J_{\omega }(u, v) + pG(u, v) . \end{aligned}$$

(26)

Thus on any critical point (u, v), we have that

$$\begin{aligned} J_{\omega }(u, v)&= \frac{p}{p+2}I_\omega (u, v), \end{aligned}$$

(27)

$$\begin{aligned} J_{\omega }(u, v)&= -\frac{p}{2}G(u, v), \end{aligned}$$

(28)

$$\begin{aligned} I_\omega (u, v)&= -\frac{p+2}{2}G(u, v). \end{aligned}$$

(29)

One can see easily that the functional G is well-defined on $\mathcal {X}$. Note that $u, v' \in H^{1}(\mathbb R) \hookrightarrow L^{q}(\mathbb R)$ for all $q\ge 2$, therefore by applying Young’s inequality we obtain that

$$\begin{aligned} |G(u,v)|\le K(\Vert u\Vert ^{p+2}_{L^{p+2}(\mathbb R)}+\Vert v'\Vert ^{p+2}_{L^{p+2}(\mathbb R)})\le K\Vert (u, v)\Vert _{\mathcal {X}}^{p+2}. \end{aligned}$$

(30)

Moreover, for $0<|\omega |<\omega _0$ there are some positive constants $K_1(a,b,d, \omega )<K_2(a,b,d, \omega )$ such that

$$\begin{aligned} K_1 \Vert (u, v)\Vert _{\mathcal {X}}^{2} \le I_\omega (u, v)\le K_2 \Vert (u, v)\Vert _{\mathcal {X}}^{2}. \end{aligned}$$

(31)

In fact, using the definition of $I_\omega$ and Young inequality we obtain that

$$\begin{aligned} I_\omega (u,v)&\le \int _{\mathbb R} [(1+|\omega |)u^2+(1+|\omega |)(v')^2+\left( b+|\omega | a\right) (v'')^2+\left( d+|\omega | a\right) (u')^2 ]dx \nonumber \\&\le \max \left( 1+|\omega |,b+|\omega | a, d+|\omega a| \right) \Vert (u,v)\Vert ^2_{\mathcal {X}}. \end{aligned}$$

Additionally,

$$\begin{aligned} I_\omega (u,v)&\ge \int _{\mathbb R} [(1-|\omega |)u^2+(1-|\omega |)(v')^2+\left( b-|\omega | a\right) (v'')^2+\left( d-|\omega | a\right) (u')^2 ]dx \nonumber \\&\ge \min \left( 1-|\omega |,b-|\omega | a, d-|\omega |a \right) \Vert (u,v)\Vert ^2_{\mathcal {X}}, \end{aligned}$$

showing that the inequality (31) holds.

Our approach to show the existence of a non trivial critical point for $J_{\omega }$ is to use the mountain pass theorem without the Palais–Smale condition (see Ambrosetti et. al. [1], Willem [15]) to build a Palais–Smale sequence for $J_{\omega }$ for a minimax value and use a local embedding result to obtain a critical point for $J_{\omega }$ as a weak limit of such Palais–Smale sequence.

Theorem 4.1

Let X be a Hilbert space, $\varphi \in C^1(X, {\mathbb R})$, $e\in X$ and $r>0$ such that $\Vert e\Vert _X > r$ and

$$\begin{aligned} \beta =\inf _{\Vert u\Vert _X=r} \varphi (u) > \varphi (0) \ge \varphi (e). \end{aligned}$$

Then, given $n\in \mathbb N$, there is $u_n \in X$ such that

$$\begin{aligned} \varphi (u_n) \rightarrow \delta , \quad \text{ and } \quad \varphi '(u_n) \rightarrow 0 \quad \text{ in }\, X', \end{aligned}$$

(PS)

where

$$\begin{aligned} \delta =\inf _{\gamma \in \Gamma } \max _{t\in [0, 1]}\varphi (\gamma (t)), \quad \text{ and } \quad \Gamma =\left\{ \gamma \in C([0, 1], X):\gamma (0)=0, \ \ \gamma (1)= e \right\} . \end{aligned}$$

Before we go further, we establish an important result for our analysis, which is related with the characterization of “vanishing sequences” in $\mathcal {X}$. Define $\varrho$ on $\mathcal {X}$ as

$$\begin{aligned} \varrho (u, v)&= u^2 + (v')^2, \end{aligned}$$

and for $\zeta \in {\mathbb R}$ and $r>0$ we will denote by $B_r(\zeta )$ the ball in $\mathbb R$ of center $\zeta$ and radius r.

Theorem 4.2

Let $q\ge 2$. If $\{(u_n, v_n)\}_n$ is a bounded sequence in $\mathcal {X}$ and there is a positive constant $r>0$ such that

$$\begin{aligned} \lim _{n\rightarrow \infty }\sup _{\zeta \in {\mathbb R}}\int _{B_r(\zeta )} \varrho (u_n, v_n) \,dx=0. \end{aligned}$$

(32)

Then we have that

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert v_n\Vert _{{\mathcal {M}}^{q}(\mathbb R)}=\lim _{n\rightarrow \infty }\Vert u_n\Vert _{L^q(\mathbb R)}=0. \end{aligned}$$

Proof

First suppose that $\{w_n\}_n$ is a bounded sequence in $H^1(\mathbb R)$ and assume there is a positive constant $r>0$ such that

$$\begin{aligned} \lim _{n\rightarrow \infty }\sup _{\zeta \in {\mathbb R}}\int _{B_r(\zeta )} w_n^2 \,dx=0. \end{aligned}$$

(33)

We will see that $\lim _{n\rightarrow \infty }\Vert w_n\Vert _{L^q(\mathbb R)}=0$. In fact, let $\{w_n\}_n$ be a bounded sequence in $H^1(\mathbb R)$ satisfying the limit (33). Then we have for $q\ge 2$ that

$$\begin{aligned} \Vert w_n\Vert ^q_{L^q(B_r(\zeta ))} \le \Vert w_n\Vert _{L^2(B_r(\zeta ))}\Vert w_n \Vert ^{q-1}_{L^{2(q-1)}(B_r(\zeta ))} \le \Vert w_n\Vert _{L^2(B_r(\zeta ))}\Vert w_n\Vert ^{q-1}_{H^1(\mathbb R)}. \end{aligned}$$

Covering $\mathbb R$ by a countable number of balls of radius r in a such way that every point in $\mathbb R$ is contained in at most two balls $B_r(\zeta )$, we obtain that

$$\begin{aligned} \Vert w_n\Vert ^q_{L^q(\mathbb R)} \le 2 \sup _{\zeta \in {\mathbb R}}\Vert w_n\Vert _{L^2(B_r(\zeta ))}\Vert w_n\Vert ^{q-1}_{H^1(\mathbb R)}. \end{aligned}$$

We conclude using the hypothesis and that $\{w_n\}_n$ is a bounded sequence in $H^1(\mathbb R)$ that

$$\begin{aligned} \lim _{n\rightarrow \infty } \Vert w_n\Vert _{L^q(\mathbb R)}=0. \end{aligned}$$

Now suppose that $\{(u_n, v_n)\}_n$ is a bounded sequence in $\mathcal {X}$ and that it satisfies (32). Then $u_n, v'_n \in H^1(\mathbb R)$. Hence, for $w_n$ being defined as either $u_n$ or $v'_n$ we see that $w_n$ satisfies in each case the condition (33). By the previous observation, we conclude for $q\ge 2$ that $\lim _{n\rightarrow \infty }\Vert w_n\Vert _{L^q(\mathbb R)}=0$. In other words, we have for $q\ge 2$ that

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert u_n\Vert _{L^q(\mathbb R)}=\lim _{n\rightarrow \infty }\Vert v\Vert _{{\mathcal {M}}^{q}(\mathbb R)}=0. \end{aligned}$$

Now, we want to verify the mountain pass theorem hypotheses given in Theorem 4.1 and to build a Palais–Smale sequence for $J_{\omega }$.

Theorem 4.3

Let $\,0<|\omega |< \omega _0$. Then

• (1) There exists $\rho >0$ small enough such that $\beta (\omega ):=\inf _{\Vert z \Vert _{\mathcal {X}}=\rho } J_{\omega }(z)> 0$.

• (2) There is $e \in \mathcal {X}$ with $\Vert e\Vert _{\mathcal {X}}\ge \rho$ such that $J_{\omega }(e)\le 0$.

• (3) If $\delta (\omega )$ is defined as

  $$\begin{aligned} \delta (\omega )= \inf _{\gamma \in \Gamma }\max _{t \in [0, 1]}J_{\omega }(\gamma (t)), \quad \Gamma = \{ \gamma \in C([0, 1], {\mathcal {X}})| \ \gamma (0)=0, \ \gamma (1)= e\}, \end{aligned}$$

  then $\delta (\omega ) \ge \beta (\omega )$ and there is a sequence $(U_n)_n \in {\mathcal {X}}$ such that

  $$\begin{aligned} J_{\omega }(U_n) \rightarrow \delta , \quad J_{\omega }'(U_n) \rightarrow 0 \quad \text{ in } \mathcal {X}'. \end{aligned}$$

Proof

From inequalities (30)–(31), we have for any $(u,v)\in \mathcal {X}$ that

$$\begin{aligned} J_{\omega }(u, v)&\ge K_1\Vert (u, v)\Vert _{\mathcal {X}}^{2}- K_2\Vert (u, v)\Vert _{\mathcal {X}}^{p+2}\\&\ge (K_1-K_2\Vert (u, v)\Vert ^p_{\mathcal {X}})\Vert (u, v)\Vert _{\mathcal {X}}^{2}. \end{aligned}$$

Then for $\rho >0$ small enough such that

$$\begin{aligned} K_1-\rho ^p K_2>0, \end{aligned}$$

we conclude for $\rho =\Vert (u, v)\Vert _{\mathcal {X}}$ that

$$\begin{aligned} J_{\omega }(u, v) \ge ( K_1- \rho ^p K_2)\rho ^2:=\alpha >0. \end{aligned}$$

In particular, we also have that

$$\begin{aligned} \beta (\omega )=\inf _{\Vert z \Vert _{\mathcal {X}}=\rho } J_{\omega }(z)\ge \alpha >0. \end{aligned}$$

(34)

Now, it is not hard to prove that there exist $u_0, v_0 \in C^{\infty }_0(\mathbb R)$ such that $G(u_0, v_0)<0$. Then for any $t\in {\mathbb R}$ we have that

$$\begin{aligned} J_{\omega }(tu_0, tv_0)&= t^2 I_\omega (u_0, v_0) + t^{p+2}G(u_0, v_0)\\&= t^2 (I_\omega (u_0, v_0) + t^pG(u_0, v_0)). \end{aligned}$$

As a consequence of this, we have that

$$\begin{aligned} \lim _{t \rightarrow \infty }J_{\omega }(tu_0, tv_0)= - \infty . \end{aligned}$$

So, there is $t_0>0$ such that $e= t_0(u_0, v_0)\in \mathcal {X}$ satisfies that $t_0\Vert (u_0, v_0)\Vert _{\mathcal {X}}=\Vert e \Vert _{{\mathcal {X}}} >\rho$ and that $J_{\omega }(e) \le J_{\omega }(0)=0$. The third part follows by applying Theorem 4.1.

Now we are in position to establish the main result in this section, in which we use the existence of a local embedding result obtained by J. Quintero in the case two dimensional (see [11]). First of all, we know for $q\ge 2$ that the embedding $H^1({\mathbb R})\hookrightarrow L^q(\mathbb R)$ is continuous and the embedding $H^1({\mathbb R})\hookrightarrow L_{loc}^q(\mathbb R)$ is compact. Now, if we set for $q \ge 2$ and $Q \subset \mathbb R$ the Banach space

$$\begin{aligned} \mathcal {M}^{q}(Q) :\overline{\left\langle C^{\infty }_0(Q), \Vert \cdot \Vert _{(q)}\right\rangle }, \quad \Vert \psi \Vert ^q_{(q)}=\Vert \psi '\Vert ^q_{L^q(Q)}. \end{aligned}$$

Then the following embedding result holds (see [11]).

Lemma 4.1

For $q \ge 2$ we have that

• (1) The embedding ${\mathcal {V}}\hookrightarrow \mathcal {M}^q(\mathbb R)$ is continuous and the embedding ${\mathcal {V}}\hookrightarrow \mathcal {M}^q_{loc}(\mathbb R)$ is compact.

• (2) The embedding $\mathcal {X}\hookrightarrow L^q(\mathbb R)\times \mathcal {M}^q(\mathbb R)$ is continuous and the embedding $\mathcal {X}\hookrightarrow L_{loc}^q(\mathbb R)\times \mathcal {M}_{loc}^{q}(\mathbb R)$ is compact.

Using previous local embedding, we have the following existence result.

Theorem 4.4

Let $\,0<|\omega |<\omega _0$. Then the system (25) has a nontrivial solution in ${\mathcal {X}}$.

Proof

We will see that $\delta (\omega )$ is in fact a critical value of $J_{\omega }$. Let $\{(u_n, v_n)\} \subset {\mathcal {X}}$ be the sequence given by previous lemma. First note from (34) that $\delta (\omega )\ge \beta (\omega )\ge \alpha$. Using the definition of $J_{\omega }$ and (26) we have that

$$\begin{aligned} I_\omega (u_n, v_n) =\frac{p+2}{p}J_{\omega }(u_n, v_n) - \frac{1}{p}\langle J_{\omega }'(u_n, v_n), (u_n, v_n)\rangle . \end{aligned}$$

But from (31) we conclude for n large enough that

$$\begin{aligned} K_1 \Vert (u_n, v_n)\Vert _{{\mathcal {X}}}^{2} \le I_\omega (u_n, v_n) \le \frac{p+2}{p}(\delta (\omega )+1)+ \Vert (u_n, v_n)\Vert _{\mathcal {X}}. \end{aligned}$$

Then we have shown that $\{(u_n, v_n)\}_n$ is a bounded sequence in ${\mathcal {X}}$. We claim that

$$\begin{aligned} \alpha ^*=\overline{\lim _{n\rightarrow \infty }}\sup _{\zeta \in {\mathbb R}} \int _{B_1(\zeta )} \varrho (u_n, v_n)\,dx >0. \end{aligned}$$

If we suppose that

$$\begin{aligned} \overline{\lim _{n\rightarrow \infty }}\sup _{\zeta \in {\mathbb R}}\int _{B_1(\zeta )} \varrho (u_n, v_n)\,dx=0. \end{aligned}$$

Hence from Lemma 4.2 we conclude for $q\ge 2$ that

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert u_n\Vert _{L^q(\mathbb R)}=0, \quad \lim _{n\rightarrow \infty }\Vert v\Vert _{{\mathcal {M}}^{q}(\mathbb R)}=0. \end{aligned}$$

Now, we have from (34), (26) and (30) that

$$\begin{aligned} 0< \alpha \le \delta (\omega )&=J_{\omega }(u_n, v_n) - \frac{1}{2} \langle J_{\omega }'(u_n, v_n), (u_n, v_n)\rangle + o(1)\\&=\frac{p}{2}G(u_n, v_n) +o(1)\\&\le C[\Vert u_n\Vert _{L^{p+2}(\mathbb R)}^{p+2} + \Vert v_n\Vert ^{p+2}_{{\mathcal {M}}^{p+2}(\mathbb R)}]+o(1)\\&\le o(1). \end{aligned}$$

But this is a contradiction. Thus, there is a subsequence of $\{(u_n, v_n)\}_n$, denoted the same, and a sequence $\zeta _n \in \mathbb R$ such that

$$\begin{aligned} \int _{B_1(\zeta _n)} \varrho (u_n, v_n)\,dx\ge \frac{\alpha ^*}{2}. \end{aligned}$$

(35)

Now we define the sequence $(\tilde{u}_n(x), \tilde{v}_n(x)) =(u_n(x+\zeta _n), v_n(x+\zeta _n))$. For this sequence we also have that

$$\begin{aligned} \Vert (\tilde{u}_n, \tilde{v}_n)\Vert _{\mathcal {X}} = \Vert ({u}_n, {v}_n)\Vert _{\mathcal {X}}, \quad J_{\omega }(\tilde{u}_n, \tilde{v}_n) \rightarrow d, \quad J_{\omega }'(\tilde{u}_n, \tilde{v}_n) \rightarrow 0 \quad \text{ in } {\mathcal {X}}'. \end{aligned}$$

Then $\{(\tilde{u}_n, \tilde{v}_n)\}_n$ is a bounded sequence in ${\mathcal {X}}$. Thus, for some subsequence of $\left\{ (\tilde{u}_n, \tilde{v}_n)\right\} _n$, denoted the same, and for some $(u, v) \in {\mathcal {X}}$ we have that

$$\begin{aligned} (\tilde{u}_n, \tilde{v}_n) \rightharpoonup (u, v), \quad \text{ as }\, n\rightarrow \infty \quad (\hbox {weakly in}\, {\mathcal {X}}). \end{aligned}$$

Since the embedding ${\mathcal {X}} \hookrightarrow L_{loc}^q(\mathbb R)\times \mathcal {M}_{loc}^{q}(\mathbb R)$ is locally compact for $q \ge 2$ we see that

$$\begin{aligned} (\tilde{u}_n, \tilde{v}_n) \rightarrow (u, v) \quad \text {in } L_{loc}^q(\mathbb R)\times \mathcal {M}_{loc}^{q}(\mathbb R). \end{aligned}$$

Then $(u, v)\not = 0$ because using (35) we have that

$$\begin{aligned} \int _{B_{1}(0)} \varrho (u, v)\,dxdy=\lim _{n\rightarrow \infty } \int _{B_{1}(0)} \varrho (\tilde{u}_n, \tilde{v}_n)\,dxdy\ge \frac{\alpha ^*}{2}. \end{aligned}$$

Moreover, if $(z, w) \in C^{\infty }_0(\mathbb R)\times C^{\infty }_0(\mathbb R)$ with $supp\,z, w\subset \Omega$ we have that

$$\begin{aligned} \langle I_\omega '(u, v), (z, w)\rangle&= 2 \int _{\Omega } ( uz+d u' z' + v' w' + b v''w'' )dx \\&\quad -2\omega \int _{\Omega }(uw'+ v'z+a(u'w''+v''z'))dx\\&=2\lim _{n\rightarrow \infty } \int _{\Omega } ( \tilde{u}_nz+ d\tilde{u}_n' z' + \tilde{v}_n' w' + b \tilde{v}_n'' w'' )dx \\&\quad -2\omega \lim _{n\rightarrow \infty }\int _{\Omega }(\tilde{u}_nw'+ \tilde{v}_n'z+a(\tilde{u}_n' w''+\tilde{v}_n'' z'))dx\\&=\lim _{n\rightarrow \infty }\langle I_\omega '(\tilde{u}_n,\tilde{v}_n), (z, w)\rangle . \end{aligned}$$

Now noting that the sequences $\{(\tilde{v}_n')^{p+1}\}_n$ and $\{\tilde{u}_n \left( \tilde{v}_n'\right) ^p\}_n$ are bounded in $L^2(\mathbb R)$, then (taking a subsequence, if necessary), we have that

$$\begin{aligned} (\tilde{v}_n')^{p+1} \rightharpoonup (v')^{p+1}, \quad \tilde{u}_n \left( \tilde{v}_n'\right) ^p \rightharpoonup u\left( v'\right) ^p \quad \text{ in } L^2(\mathbb R). \end{aligned}$$

As a consequence of this, we have that

$$\begin{aligned} \int _{\Omega }\left( \tilde{v}_n' \right) ^{p+1}z dx \rightarrow \int _{\Omega }\left( v'\right) ^{p+1}z\,dx, \quad \int _{\Omega }\tilde{u}_n\left( \tilde{v}_n'\right) ^pw'dx \rightarrow \int _\Omega u\left( v'\right) ^pw'dx. \end{aligned}$$

In other words, we have shown that

$$\begin{aligned} \left<G'(u,v), (z, w)\right>= \lim _{n\rightarrow \infty } \langle G'(\tilde{u}_n, \tilde{v}_n), (z, w)\rangle \end{aligned}$$

and also that

$$\begin{aligned} \langle J_{\omega }'(u, v), (z, w)\rangle = \lim _{n\rightarrow \infty }\langle J_{\omega }'(\tilde{u}_n, \tilde{v}_n), (z, w) \rangle =0. \end{aligned}$$

Now, let $(z, w) \in {\mathcal {X}}$. By density, there is $(z_k, w_k )\in C^{\infty }_0(\mathbb R)\times C^{\infty }_0(\mathbb R)$ such that $(z_k, w_k) \rightarrow (z, w)$ in ${\mathcal {X}}$. Then

$$\begin{aligned} | \langle J_{\omega }'(u,v), (z, w) \rangle |&\le | \langle J_{\omega }'(u,v), (z-z_k, w-w_k) \rangle |+ | \langle J_{c}'(u,v), (z_k, w_k ) \rangle |\le \Vert J_{\omega }'(u,v) \\&\quad \times \Vert _{\mathcal {X}'}\Vert (z-z_k, w-w_k)\Vert _{{\mathcal {X}}} + | \langle J_{\omega }'(u,v), (z-z_k, w-w_k) \rangle | \rightarrow 0. \end{aligned}$$

Thus we have already established that $J_{\omega }'(u, v)=0$. In other words, (u, v) is a nontrivial solution for the system (25).