Dynamics of Solitary Waves of the Rosenau-RLW Equation

Abstract

In this paper we study the solitary waves of the Rosenau-RLW equation. By using some trigonometric function methods, a family of stable solitary wave solutions are obtained, revealing an intrinsic relationship among the amplitude, frequency and wave speed, for what should be an equation relevant to modeling in a number of fields.

Introduction

This paper is concerned with the existence and stability of exact solitary wave solutions of the Rosenau-RLW equation

$$\begin{aligned} u_t+{\varepsilon }u_x+{\alpha }u_{txx}+\beta u_{txxxx}+uu_x=0, \end{aligned}$$

(1)

where ${\varepsilon }>0,$ $\beta >0$ and ${\alpha }\in \mathbb{R }$ are constants. Zuo et al. in [36] considered (1) as a generalization of the Rosenau equation (${\alpha }=0$) which is used to describe the dynamics of dense discrete systems [24, 25, 26]. He studied the initial-boundary value problem of the Rosenau-RLW equation by a Crank–Nicolson difference scheme. Equation (1) is a regularized counterpart of the Rosenau-KdV equation

$$\begin{aligned} u_t+{\varepsilon }u_x+{\alpha }u_{xxx}+\beta u_{txxxx}+uu_x=0, \end{aligned}$$

(2)

which has been studied in [17, 35]. When ${\alpha }<0$ and $\beta =0,$ (1) can be also considered as a generalization of the BBM equation [7, 23] which together with the KdV equation,

$$\begin{aligned} u_t+u_{xxx}+uu_x=0, \end{aligned}$$

(3)

arise as models for one-dimensional long wavelength surface waves propagating in weakly nonlinear dispersive media [1, 11, 19, 23, 31], as well as the evolution of weakly nonlinear ion acoustic waves in plasmas [29].

Our interest in the present paper is first to search for exact solutions of (1). Next we investigate the orbital stability of the obtained solitary waves. By a solitary wave solution of the Rosenau-RLW equation, we mean a traveling-wave solution of Eq. (1) of the form

$$\begin{aligned} u_s(x,t)={\varphi _c}(\xi )={\varphi _c}(x-ct) \end{aligned}$$

decaying at infinity, where $c\in \mathbb{R }$ is the speed of wave propagation. Alternatively, it is a solution ${\varphi _c}$ of the equation

$$\begin{aligned} (c-{\varepsilon }){\varphi _c}+{\alpha }c {\varphi _c}^{\prime \prime }+\beta c{\varphi _c}^{\prime \prime \prime \prime }-\frac{1}{2}{\varphi _c}^{2}=0, \end{aligned}$$

(4)

where “$^{\prime }=\mathrm{d}/\mathrm{d}\xi$”. To find solutions of (4), because of the well known solitary traveling-wave solution associated with the KdV equation, we use trigonometric methods [8, 9, 16], based on the sech-method, including two families of traveling wave solutions of (4). Various traveling wave solutions are obtained, revealing an intrinsic relationship among the amplitude, frequency and wave speed.

Following the methods in [6, 10, 12, 18, 28, 30], we employ the invariants

$$\begin{aligned} Q(u)=\frac{1}{2}\int \limits _\mathbb{R }\left( u^2-{\alpha }u_x^2+\beta u_{xx}^2\right) \mathrm{d}x \end{aligned}$$

(5)

and

$$\begin{aligned} E(u)=-\int \limits _\mathbb{R }\left( \frac{{\varepsilon }}{2} u^2+\frac{1}{6}u^{3}\right) \mathrm{d}x, \end{aligned}$$

(6)

as well as the fact that a solitary wave ${\varphi _c}$ of (1) is a critical point of energy $E(\cdot )+cQ(\cdot )$ subject to constant charge $Q,$ to show that the usual necessary and sufficient condition for the stability of solitary wave holds: defining $d(c)=E({\varphi _c})+cQ({\varphi _c}),$ the solitary wave ${\varphi _c}$ is stable if $d^{\prime \prime }(c)>0.$ A direct computation shows that $d^{\prime }(c)=Q({\varphi _c})$ (see [28]). One can also observe that (1) has the hamiltonian structure. Indeed, one has $u_t=\mathfrak J E^{\prime }(u),$ where

$$\begin{aligned} \mathfrak J ={\partial }_xM^{-1} \qquad {\mathrm{and}}\qquad M=I+{\alpha }{\partial }_{x}^2+\beta {\partial }_x^4. \end{aligned}$$

The main ingredient in stability theory presented in [6, 10, 12, 18, 28, 30] is to verify the following spectral condition holds ([4]).

Assumption 1

(Spectral structure) There exists $(\omega _1,\omega _2)\subset \mathbb{R },$ with ${\varepsilon }\le \omega \le \omega _2\le +\infty ,$ such that $c\mapsto {\varphi _c}$ is a nontrivial smooth curve, and for each $c\in (\omega _1,\omega _2)$ and a solitary wave ${\varphi _c},$ the linearized operator

$$\begin{aligned} L={\alpha }c\frac{\mathrm{d}^2}{\mathrm{d}\xi ^2}+\beta c\frac{\mathrm{d}^4}{\mathrm{d}\xi ^4}-{\varphi _c}+c-{\varepsilon }\end{aligned}$$

(7)

is self-adjoint closed unbounded on a dense subspace of $L^2(\mathbb{R })$ and enjoys the following spectral properties: it has a unique negative simple with eigenfunction $\chi _c,$ the zero eigenvalue is simple with eigenfunction ${\varphi _c}^{\prime }$ and the remainder of the spectrum of $L$ is positive and bounded away zero. Moreover the mapping $c\mapsto \chi _c$ is a continuous curve with values in $H^2(\mathbb{R })$ and $\chi _c(x)>0.$

It is clear that $L$ is self-adjoint closed unbounded linear operator from $H^2(\mathbb{R })$ into $H^{-2}(\mathbb{R }),$ $M^{1/2}LM^{1/2}$ is self-adjoint on $L^2(\mathbb{R }),$ and $L({\varphi _c}^{\prime })=0,$ where

$$\begin{aligned} M=I+{\alpha }\frac{\mathrm{d}^2}{\mathrm{d}\xi ^2}+\beta \frac{\mathrm{d}^4}{\mathrm{d}\xi ^4}. \end{aligned}$$

To verify the spectral properties of $L,$ we will apply the results of Albert [2], and Albert and Bona [3]. In [2, 3], the authors considered the following KdV-type equation

$$\begin{aligned} u_t+u^pu_x-\mathfrak M u_x=0, \end{aligned}$$

(8)

where $\mathfrak M$ is a Fourier multiplier $\widehat{\mathfrak{M g}}(\xi )=m(\xi )\widehat{g}(\xi )$ and $m$ is a measurable locally bounded even function on $\mathbb{R }$ satisfying

$$\begin{aligned} C_1|\xi |^{\nu _1}\le m(\xi )\le C_2(1+|\xi |)^{\nu _2}, \end{aligned}$$

for $\nu _1\le \nu _2,$ $|\xi |\ge |\xi _0|$ and $C_1,C_2>0,$ and $m(\xi )>\kappa >0,$ for all $\xi \in \mathbb{R }.$ It is shown that, for a solitary wave ${\varphi _c}$ of (8), the associated linearized operator $\mathfrak M +c-{\varphi _c}$ satisfies the spectral structure mention above provided that ${\varphi _c}$ is a positive solitary wave such that $\widehat{{\varphi _c}}>0$ and $\widehat{{\varphi _c}^p}$ belongs to the $PF(2)$-class defined by Karlin in [20] (see Definition 2). It is worth noticing that Souganidis and Strauss in [28] considered Assumption (1) and used the ideas of [12, 18, 30] to study the instability of solitary waves of the following BBM-type equation

$$\begin{aligned} u_t+\mathfrak M u_t+u_x+u^pu_x=0, \end{aligned}$$

(9)

where $\widehat{\mathfrak{M g}}(\xi )=m(\xi )\widehat{g}(\xi )$ is a Fourier multiplier with appropriate conditions, similar to (8), on $m.$ We will apply the ideas of [2, 3, 12, 18, 28] to show the stability of our explicit solitary wave solutions of (1). Rest of this paper is divided into three sections. The next section is devoted the local and global well-posedness of (1), by using the properties of the kernel $\mathbb H$ (see (12)). In the third section, some explicit solitary waves will be obtained. Finally in the last section, we prove the orbital stability of the obtained solutions. We end this section by introducing some notations that will be used throughout this article.

Notation

We shall denote by $\widehat{\varphi }$ the Fourier transform of ${\varphi },$ defined as

$$\begin{aligned} \widehat{\varphi }(\zeta )=\int \limits _{\mathbb{R }}\;{\varphi }(\omega )\mathrm{e}^{-\mathrm{i}\omega \zeta }\;\mathrm{d}\omega . \end{aligned}$$

For $1\le p <\infty ,$ $L^p = L^p(\mathbb{R })$ connotes the $p$th-power Lebesgue-integrable functions with the usual modification for the case $p=\infty .$

For $s\in \mathbb{R },$ we denote by $H^s=H^s\left( \mathbb{R }\right) ,$ the nonhomogeneous Sobolev space defined by

$$\begin{aligned} H^s\left( \mathbb{R }\right) =\left\{ {\varphi }\in {\fancyscript{S}}^{\prime }\left( \mathbb{R }\right) \,:\,\Vert {\varphi }\Vert _{H^s\left( \mathbb{R }\right) }<\infty \right\} \!, \end{aligned}$$

where

$$\begin{aligned} \Vert {\varphi }\Vert _{H^s\left( \mathbb{R }\right) } =\left( \int \limits _\mathbb{R }\left( 1+|\zeta |^2\right) ^s|\widehat{{\varphi }}(\zeta )|^2\mathrm{d}\zeta \right) ^{1/2}, \end{aligned}$$

and ${\fancyscript{S}}^{\prime }\left( \mathbb{R }\right)$ is the space of tempered distributions.

If $\mathfrak{X }$ is any Banach space and $T > 0,$ $C( 0, T ;\mathfrak{X })$ is the class of continuous functions from $[0, T]$ into $\mathfrak{X }$ with its usual norm

$$\begin{aligned} \Vert u\Vert _{C( 0,T ;\mathfrak{X })}=\max _{t\in [0,T]}\Vert u(t)\Vert _{\mathfrak{X }}. \end{aligned}$$

If $\mathcal{Y }\subset \mathfrak{X }$ is a subset, then $C(0, T ; \mathcal{Y })$ is the collection of elements $u$ in $C(0, T ;\mathfrak{X })$ such that $u(t)\in \mathcal{Y }$ for $t\in [0,T].$ When $T =+\infty ,$ $C(0,+\infty ;\mathfrak{X })$ is a Fréchet space with defining set of semi-norms $\max _{t\in [0,N]}\Vert u(t)\Vert _{\mathfrak{X }},$ for $N\in \mathbb N .$ The Banach space $C^1(0, T ;\mathfrak{X })$ is the subspace of $C(0, T ;\mathfrak{X })$ for which the limit

$$\begin{aligned} u^{\prime }(t)=\lim _{h\rightarrow 0}\frac{u(t+h)-u(t)}{h} \end{aligned}$$

exists in $C(0, T ;\mathfrak{X }).$ It is equipped with the obvious norm. Inductively, one defines $C^k(0, T ;\mathfrak{X })$ and $C^k(0,+\infty ;\mathfrak{X }).$

Given a solitary wave ${\varphi }_c$ of (1), the orbit of $\mathcal O _{{\varphi }_c}$ is defined by the set $\mathcal O _{{\varphi }_c}=\{\tau _r{\varphi }_c;\;r\in \mathbb{R }\},$ where $\tau _r{\varphi }_c(\cdot )={\varphi }_c(\cdot +r).$ We also denote by

$$\begin{aligned} U_{\epsilon }=U_{{\varphi _c},{\epsilon }}=\left\{ u_0;\;\inf _{\psi \in \mathcal O _{{\varphi _c}}}\Vert u_0-\psi \Vert _{H^2}<{\epsilon }\right\} \end{aligned}$$

the ${\epsilon }$-neighborhood of the orbit $\mathcal O _{{\varphi _c}}.$

For any positive numbers $a$ and $b,$ the notation $a\lesssim b$ means that there exists a positive constant such that

Well-Posedness

In this section we are going to study the well-posedness issue for (1). In this section we assume that ${\alpha }<2\sqrt{\beta }.$

Definition 1

An evolution equation $u_t = \mathfrak A u,$ with $u(0) = u_0,$ is said to be locally (in time) well-posed in a Banach space $\mathfrak{X }$ if for any $u_0\in \mathfrak{X },$ there is a positive number $T$ such that the equation possesses a unique solution $u$ which lies in $C( 0, T ;\mathfrak{X }).$ Moreover, the solution $u$ must depend continuously on $u_0.$ The evolution equation is well-posed globally in time if $T$ can be chosen arbitrarily large.

We note that a formal integration in the temporal variable then leads to the Rosenau-RLW integral equation

$$\begin{aligned} u(t)=u_0(x)+\int \limits _0^t\int \limits _\mathbb{R }\mathbb{H }(x-y)\left( {\varepsilon }u(y,s)+\frac{1}{2}u^{2}(y,s)\right) \mathrm{d}y\;\mathrm{d}s, \end{aligned}$$

(10)

where $u_0(x) = u(x, 0)$ is the initial data and

$$\begin{aligned} \widehat{\mathbb{H }}(\xi )=\frac{\mathrm{i}\xi }{1-{\alpha }\xi ^2+\beta \xi ^4}. \end{aligned}$$

(11)

More precisely, by using the residue theorem, one can easily see that

$$\begin{aligned} \mathbb H (x)=\left\{ \begin{array}{ll} \frac{\pi \mathrm{sgn}(x)}{\beta ({\lambda }_1^2-{\lambda }_2^2)}\left( \mathrm{e}^{-{\lambda }_1|x|}-\mathrm{e}^{-{\lambda }_2|x|}\right) ,&{} \quad {\alpha }<-{\alpha }_*,\\ -\frac{\pi \mathrm{sgn}(x)}{2\root 4 \of {\beta ^3}}|x|\mathrm{e}^{-\beta ^{-1/4}|x|},&{} \quad {\alpha }=-{\alpha }_*,\\ \frac{\pi \mathrm{sgn}(x)\mathrm{e}^{-\sigma |x|}}{2\beta \sigma \omega (\sigma ^2+\omega ^2)} \left( \sigma ^2H_1(x)-\omega H_2(x)\right) ,&{} \quad {\alpha }\in (-{\alpha }_*,{\alpha }_*), \end{array}\right. \end{aligned}$$

(12)

where

$$\begin{aligned} \left. \begin{array}{l} {\alpha }_*=2\beta ^{1/2},\\ {\lambda }_1=\sqrt{-\frac{1}{2\beta }({\alpha }+\sqrt{{\alpha }^2-4\beta })},\\ {\lambda }_2=\sqrt{-\frac{1}{2\beta }({\alpha }-\sqrt{{\alpha }^2-4\beta })},\\ \sigma =\frac{1}{2}\sqrt{2\beta ^{-1/2}-{\alpha }\beta ^{-1}},\\ \omega =\frac{1}{2}\sqrt{2\beta ^{-1/2}+{\alpha }\beta ^{-1}},\\ H_1(x)=\cos (\sigma x)-\sin (\sigma |x|),\\ H_2(x)=\sigma \cos (\omega x)-\omega \sin (\omega |x|). \end{array}\right\} \end{aligned}$$

(13)

Figure 1 illustrates the shape of kernel of (12) for $\beta =1,$ and ${\alpha }=-3,$ ${\alpha }=-2$ and ${\alpha }=1$ respectively.

Fig. 1

Kernel $\mathbb H$ in (12) for $\beta =1.$ Figures correspond to ${\alpha }=-3,$ ${\alpha }=-2$ and ${\alpha }=1$ respectively from left to right and then up to down

First we study the local well-posedness, based on Definition 1, in $L^q(\mathbb{R })$-spaces. See [23] for similar results.

Theorem 2

Let $q\ge 2.$ Then (10) is well-posed in $L^q(\mathbb{R })$ in the sense of Definition 1. Moreover, the flow map $\mathfrak{G }:u_0\mapsto u,$ that associates to the initial data $u_0$ the unique solution $u,$ is real analytic.

Proof

First one can observe from (11) that $\mathbb{H }\in L^\ell (\mathbb{R }),$ for any $1\le \ell \le \infty .$ Thus by the Young inequality,

$$\begin{aligned} \left\| \mathbb{H }*\left( {\varepsilon }u+\frac{1}{2}u^{2}\right) \right\| _{L^q(\mathbb{R })}\le C\left( \Vert \mathbb{H }\Vert _{L^1(\mathbb{R })}\Vert u\Vert _{L^q(\mathbb{R })}+\Vert \mathbb{H }\Vert _{L^{q/(q-1)}(\mathbb{R })}\Vert u\Vert _{L^q(\mathbb{R })}^{2}\right) , \end{aligned}$$

which is to say,

$$\begin{aligned} \mathbb{H }*\left( {\varepsilon }u+\frac{1}{2}u^{2}\right) \in L^q(\mathbb{R }). \end{aligned}$$

Now we define the constants $r=2\Vert u_0\Vert _{L^q(\mathbb{R })}$ and

$$\begin{aligned} T=\frac{1}{2C\left( \Vert \mathbb{H }\Vert _{L^1(\mathbb{R })}+\Vert \mathbb{H }\Vert _{L^{q/(q-1)}(\mathbb{R })}r\right) } \end{aligned}$$

and define ${\fancyscript{X}}={\fancyscript{X}}_{T,r}=C( 0,T ;B_r(u_0)),$ where $B_r(u_0)$ is the closed ball in $L^q(\mathbb{R })$ of the radius $r$ centered at $u_0.$ The set ${\fancyscript{X}}$ is a complete metric space with the distance $d$ induced by the norm on $C( 0, T ; L^q(\mathbb{R })).$ We show that the operator

$$\begin{aligned} \varPhi (u)=u_0(x)+\int \limits _0^t\int \limits _\mathbb{R }\mathbb H (x-y)\left( {\varepsilon }u(y,s)+\frac{1}{2}u^{2}(y,s)\right) \mathrm{d}y\;\mathrm{d}s \end{aligned}$$

is contractive from ${\fancyscript{X}}$ to itself.

For any $u\in {\fancyscript{X}},$ we have

$$\begin{aligned} d(\varPhi (u),0)=\Vert \varPhi (u)\Vert _{\fancyscript{X}}\le \Vert u\Vert _{L^q(\mathbb{R })}+TC \left( \Vert \mathbb{H }\Vert _{L^1(\mathbb{R })}r+\Vert \mathbb{H }\Vert _{L^{q/(q-1)}(\mathbb{R })}r^{2}\right) \le r, \end{aligned}$$

(14)

so that $\varPhi$ maps ${\fancyscript{X}}$ to ${\fancyscript{X}}.$ Moreover the Cauchy–Schwarz inequality implies for $u,v\in {\fancyscript{X}}$ that

$$\begin{aligned}&\left\| \mathbb{H }*\left( {\varepsilon }(u-v)+\frac{1}{2}\left( u^{2}-v^{2}\right) \right) \right\| _{L^q(\mathbb{R })}\nonumber \\&\quad \le C_0\left( \Vert \mathbb{H }\Vert _{L^1(\mathbb{R })}+\Vert \mathbb{H }\Vert _{L^{q/(q-1)}(\mathbb{R })}r \right) \Vert u-v\Vert _{L^q(\mathbb{R })}, \end{aligned}$$

(15)

Consequently, it holds that

$$\begin{aligned} \Vert \varPhi (u)-\varPhi (v)\Vert _{L^q(\mathbb{R })}\le C\int \limits _0^t\left[ \Vert \mathbb{H }\Vert _{L^1(\mathbb{R })}+\Vert \mathbb{H }\Vert _{L^{q/(q-1)}(\mathbb{R })}r \right] \Vert u(\tau )-v(\tau )\Vert _{L^q(\mathbb{R })}\mathrm{d}\tau . \end{aligned}$$

Therefore, we obtain that

$$\begin{aligned} d(\varPhi (u),\varPhi (v))&\le CT\left[ \Vert \mathbb{H }\Vert _{L^1(\mathbb{R })}+\Vert \mathbb{H }\Vert _{L^{q/(q-1)}(\mathbb{R })}r \right] \Vert u-v\Vert _{\fancyscript{X}}\\&\le \frac{1}{2}\Vert u-v\Vert _{\fancyscript{X}}=\frac{1}{2}d(u,v). \end{aligned}$$

It shows that $\varPhi$ is contractive which implies the desired local well-posedness result.

To prove the second part of theorem, we will use an argument analogous to [5], [14, Theorem 3], [27, Theorem 3.3] and [32, 33, 34] (see also [13, 21, 22]). Define an operator $\varPsi$ as

$$\begin{aligned} \varPsi (u)=\int \limits _0^t\mathbb{H }*({\varepsilon }u+g(u))\mathrm{d}s, \end{aligned}$$

where $g(u)=\frac{1}{2}u^{2}.$ It is straightforward to see that $\varPsi$ is Fréchet differentiable and for $v,z\in C( 0,T ;L^q(\mathbb{R }))$ we have

$$\begin{aligned} \varPsi ^{\prime }(v)z=\int \limits _0^t\int \limits _\mathbb{R }\mathbb{H }(x-y)(1+g^{\prime }(v))z\;\mathrm{d}y\mathrm{d}\tau . \end{aligned}$$

Now we define, for $a,b\in C( 0,T ;L^q(\mathbb{R })),$ that

$$\begin{aligned} \varLambda (a,b)=b-a-\varPsi (b); \end{aligned}$$

so that when $a=u_0$ and $b=u,$ where $u$ is the fixed point of the operator $\varPsi$ corresponding to initial data $u_0,$ then $\varLambda (u_0,u)=0$ and

$$\begin{aligned} D_b\varLambda (u_0,u)z=z-\varPsi ^{\prime }(u)z. \end{aligned}$$

Furthermore, it is seen from the definition that

$$\begin{aligned} \Vert \varPsi ^{\prime }(u)z\Vert _{L^q(\mathbb{R })}&\le T\sup _{0\le t\le T}\Vert \mathbb{H }*(({\varepsilon }+g^{\prime }(u))z)\Vert _{L^q(\mathbb{R })}\\&\le CT\left( \Vert \mathbb{H }\Vert _{L^1(\mathbb{R })}+\Vert \mathbb{H }\Vert _{L^{q/(q-1)}(\mathbb{R })}r \right) \Vert z\Vert _{\fancyscript{X}}\\&= \frac{1}{2}\Vert z\Vert _{\fancyscript{X}}. \end{aligned}$$

Hence,

$$\begin{aligned} D_v\varLambda (u_0, u) = I - \varPsi ^{\prime }(u) \end{aligned}$$

is invertible and therefore, by Implicit Function Theorem [15], the flow map $\mathfrak{G }(u_0) = u$ is a $C^1$ map, and

$$\begin{aligned} D_{u_0}u=-\left( I-\varPsi ^{\prime }(u)\right) ^{-1}D_a\varLambda (u_0,u); \end{aligned}$$

and the second assertion of Theorem 2 follows. $\square$

Theorem 3

Let $s\ge 0.$ Then for any $u_0\in H^s(\mathbb{R }),$ there is a number $T>0$ and a unique solution $u\in C( 0, T ;H^s(\mathbb{R }))$ of (10) with $u(0)=u_0.$ Moreover, the flow map $\mathfrak{G }:u_0\mapsto u,$ that associates to the initial data $u_0$ the unique solution $u,$ is real analytic. In addition, $u(t)$ satisfies $E(u(t))=E(u_0)$ and $Q(u(t))=Q(u_0)$ for all $t\in [0,T).$

Proof

Take the Fourier transform in (10) with respect to the spatial variable, we obtain

$$\begin{aligned} \widehat{u}(\xi ,t) =\widehat{u}_0(\xi )+\int \limits _0^t\frac{\mathrm{i}\xi }{1-{\alpha }\xi ^2+\beta \xi ^4}\left( {\varepsilon }\widehat{u}+\frac{1}{2}\widehat{u^{2}}\right) (\xi ,\tau )\mathrm{d}\tau . \end{aligned}$$

Now we define, for any $T > 0,$ an operator $\mathcal A :C( 0,T ;H^s)\rightarrow C( 0,T ;H^s)$ by

$$\begin{aligned} \widehat{\mathcal{A u}}(\xi ,t) =\widehat{u}_0(\xi )+\int \limits _0^t\frac{\mathrm{i}\xi }{1-{\alpha }\xi ^2+\beta \xi ^4}\left( {\varepsilon }\widehat{u}+\frac{1}{2}\widehat{u^{2}}\right) (\xi ,\tau )\mathrm{d}\tau . \end{aligned}$$

When $s\ge 0,$ if $u\in H^s,$ then for any $\xi \in \mathbb{R },$

$$\begin{aligned} (1+|\xi |)^s|\widehat{u^2}(\xi )|&\le \left( (1+|\cdot |)^s|\widehat{u}(\cdot )|\right) *\left( (1+|\cdot |)^s|\widehat{u}(\cdot )|\right) (\xi )\nonumber \\&\le \int \limits _\mathbb{R }(1+|\xi |)^{2s}|\widehat{u}(\xi )|^2\mathrm{d}\xi =\Vert u\Vert _{H^s}^2. \end{aligned}$$

(16)

Consequently,

$$\begin{aligned} \int \limits _\mathbb{R }(1+|\xi |)^{2s}\frac{\xi ^2}{(1-{\alpha }\xi ^2+\beta \xi ^4)^2}|\widehat{u^{2}}(\xi )|^2\mathrm{d}\xi \lesssim \Vert u\Vert _{H^s}^{4}; \end{aligned}$$

and it is concluded that $\mathcal A u\in C( 0,+\infty ;H^s(\mathbb{R })),$ if $u\in C( 0,+\infty ;H^s(\mathbb{R })).$ Following the steps laid out in the proof of Theorem 2, it can be shown that in all cases, when $T > 0$ is chosen sufficiently small, the operator $\mathcal A$ is contractive in $C\left( [0, T) ;B_{2\Vert u_0\Vert _s}(0)\right) ,$ where the ball $B_{2\Vert u_0\Vert _s}(0)$ is in $H^s(\mathbb{R }).$ The contraction mapping principle completes the proof. Invariance of $E$ and $Q$ follows by a standard argument. $\square$

Theorem 4

Let $s\!\!\ge \!\!2.$ Then for any $u_0\in H^s(\mathbb{R }),$ there is a unique solution $u\in C( 0, +\infty ;H^s(\mathbb{R }))$ of (10) with $u(0)=u_0.$

Proof

By Theorem 3, there exists a $T>0$ and a unique solution $u$ of (1) with $u(0)=u_0$ such that $u\in C( 0,T ;H^s(\mathbb{R })).$ It remains to show that $T$ can be taken arbitrarily large. First we note for $s=2$ that the invariant (6) implies that the solution can be extended from $C( 0, T ; H^2(\mathbb{R }))$ to $C( 0,+\infty ;H^2(\mathbb{R })).$ Next, when $s>2,$ we multiply both sides of (1) by $2(I+D)^{2s-4}u(x,t)$ and integrate over $\mathbb{R }$ with respect to $x$ to obtain at least for smooth solutions that

$$\begin{aligned}&2\int \limits _\mathbb{R }\left( (I+D)^{2s-4}u(x,t)\right) Mu_t(x,t)\mathrm{d}x\\&\quad =-2\int \limits _\mathbb{R }\left( (I+D)^{2s-4}u(x,t)\right) \left( {\varepsilon }u(x,t)+\frac{1}{2}u^2(x,t)\right) _x\mathrm{d}x\\&\quad =-\int \limits _\mathbb{R }\mathrm{i}\xi (1+|\xi |)^{2s-4}\overline{\widehat{u}}(\xi ,t)\;\widehat{u^2}(\xi ,t)\mathrm{d}\xi , \end{aligned}$$

where $D=(-\partial _x^2)^{1/2}$ and $M=I+{\alpha }{\partial }_x^2+\beta {\partial }_x^4.$ Using the fact

$$\begin{aligned} 1-{\alpha }\xi ^2+\beta \xi ^4\sim (1+|\xi |)^4, \end{aligned}$$

(17)

it follows that

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\int \limits _\mathbb{R }(1-{\alpha }\xi ^2+\beta \xi ^4)(1+|\xi |)^{2s-4}|\widehat{u}(\xi ,t)|^2\mathrm{d}\xi \nonumber \\&\qquad \le \int \limits _\mathbb{R }(1+|\xi |)^{2s-3}|\widehat{u}(\xi ,t)|\left| \widehat{u^2}(\xi ,t)\right| \mathrm{d}\xi \nonumber \\&\qquad \le \Vert u(t)\Vert _{H^s}\Vert u^{2}(t)\Vert _{H^{s-3}}\nonumber \\&\qquad \le \Vert u(t)\Vert _{H^s}\Vert u^{2}(t)\Vert _{H^s}. \end{aligned}$$

(18)

Now by (17) and the invariance $E,$ we have

$$\begin{aligned} \Vert \widehat{u}(t)\Vert _{L^1}^2&\lesssim \Vert u\Vert _{H^2}^2 \lesssim \int \limits _\mathbb{R }u^2-{\alpha }u_x^2+\beta u_{xx}^2\mathrm{d}x \\&= \int \limits _\mathbb{R }u_0^2-{\alpha }(\partial _x u_0)^2+\beta (\partial _x^2 u_0)^2\mathrm{d}x \lesssim \Vert u_0\Vert _{H^2}^2. \end{aligned}$$

This implies that

$$\begin{aligned} \Vert u^2(t)\Vert _{H^s}^2&= \int \limits _\mathbb{R }(1+|\xi |)^{2s}|\widehat{u^2}(\xi ,t)|\mathrm{d}\xi \nonumber \\&\lesssim \int \limits _\mathbb{R }\int \limits _\mathbb{R }\left( 1+|\xi -\eta |^{2s}+|\eta |^{2s}\right) |\widehat{u}(\xi -\eta ,t)\widehat{u}(\eta ,t)|^2\mathrm{d}\eta \mathrm{d}\xi \nonumber \\&\lesssim \Vert \widehat{u}(t)*\widehat{ u}(t)\Vert _{L^2}^2+\Vert \widehat{D^s u}(t)*\widehat{u}(t)\Vert _{L^2}^2 \lesssim \Vert \widehat{u}(t)\Vert _{L^1}^2\Vert u(t)\Vert _{H^s}^2\nonumber \\&\le C \Vert u(t)\Vert _{H^s}^2, \end{aligned}$$

(19)

where $C=C(\Vert u_0\Vert _{H^2}).$ Combining (18) and (19) leads to

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\int \limits _\mathbb{R }\left( 1-{\alpha }\xi ^2+\beta \xi ^4 \right) (1+|\xi |)^{2s-4}|\widehat{u}(\xi ,t)|^2\mathrm{d}\xi \lesssim \Vert u(t)\Vert _{H^s}^2. \end{aligned}$$

(20)

Integrating the last inequality with respect to $t$ and using (17) yields

$$\begin{aligned} \Vert u(t)\Vert _{H^s}^2\le C_1\Vert u_0\Vert _{H^s}^2+C_2\int \limits _0^t\Vert u(\tau )\Vert _{H^s}^2\mathrm{d}\tau . \end{aligned}$$

By the Gronwall lemma, there are two constants $C_1$ and $C_2$ in which $C_1$ is dependent only on $\Vert u_0\Vert _{H^s}$ and $C_2$ only on $\Vert u_0\Vert _{H^2}$ such that $\Vert u(t)\Vert _{H^s}\le C_1\exp (C_2t).$ This a priori bound allows us to iterate the local theory and achieve a globally defined solution. $\square$

Solitary Waves

In this section, we establish the existence of solitary waves of (1). Here, we propose two types of $L^1$-solutions of (4). We first consider the sech-ansatz; actually our hypothesis is ${\varphi _c}(\xi )=A\mathrm{sech}^q(b\xi ),$ where $A$ is the amplitude of the solitary wave and $b$ is the inverse width of the solitary wave. One can see after balancing ${\varphi _c}^{\prime \prime \prime \prime }$ with ${\varphi _c}^2$ that $q=4.$ Hence plugging $A\mathrm{sech}^4(b\xi )$ into (4), collecting the coefficients $\mathrm{sech}^j(b\xi )$ and equating these coefficients to zero, there obtains

$$\begin{aligned} \left\{ \begin{array}{ll} -2 {\varepsilon }+2 c+512 \beta c{b}^{4}+32 c\alpha {b}^{2}=0,\\ -40 \alpha -208 \beta b^2=0,\\ -A+1680\beta c b^4=0. \end{array}\right. \end{aligned}$$

(21)

After some calculations, we obtain from system (21) that

$$\begin{aligned} A=A(c)= {\frac{35}{12}}( c-{\varepsilon })\quad {\mathrm{and}}\quad b=b(c)= \frac{1}{12} \sqrt{ \frac{13({\varepsilon }-c)}{c{\alpha }}}, \end{aligned}$$

(22)

such that ${\alpha }<0,$ $c>{\varepsilon }$ and

$$\begin{aligned} \beta =\frac{36c{\alpha }^2}{169(c-{\varepsilon })}. \end{aligned}$$

(23)

One can also easily check that $\frac{\mathrm{d}A}{\mathrm{d}c}>0$ and $\frac{\mathrm{d}b}{\mathrm{d}c}>0,$ for all $c>{\varepsilon };$ so that the mapping $c\rightarrow {\varphi _c}$ is smooth from $({\varepsilon },+\infty )$ into $H^s(\mathbb{R }),$ for all $s\in \mathbb{N }.$ Therefore, we obtain

$$\begin{aligned} {\varphi _c}(\xi )&= {\varphi _c}(x-ct) \nonumber \\&= {\frac{35}{12}}( c-{\varepsilon })\mathrm{sech}^4\left( \sqrt{ \frac{13({\varepsilon }-c)}{144c{\alpha }}}(x-ct)\right) . \end{aligned}$$

(24)

By using the idea of the sech-ansatz method above, we propose the second type of solution which is

$$\begin{aligned} {\varphi _c}(\xi )=\sum \limits _{j=1}^4A_j\mathrm{sech}^j(b\xi ), \end{aligned}$$

where $A_j,b\in \mathbb{R }.$ It is straightforward to see after balancing ${\varphi _c}^{\prime \prime \prime \prime }$ with ${\varphi _c}^2$ that $A_1=A_3=0.$ Hence plugging this form into (4), collecting the coefficients $\mathrm{sech}^j(b\xi )$ and equating these coefficients to zero, there obtains

$$\begin{aligned} \left\{ \begin{array}{ll} 32 \beta c{b}^{4}+8c\alpha {b}^{2}+2 c-2 {\varepsilon }=0,\\ -A_2^2-240 \beta c{b}^{4}A_2 -12 c\alpha {b}^{2}A_2+2 A_4\left( c-{\varepsilon }+256 \beta c{b}^{4}+16 c\alpha {b}^{2}\right) =0,\\ -40 c\alpha {b}^{2}A_4+240\beta c{b}^{4}A_2-2 A_2A_4-2080 \beta c{b}^{4}A_4=0,\\ -A_4+1680 \beta c{b}^{4}=0. \end{array}\right. \end{aligned}$$

(25)

After some calculations, we obtain from system (25) that

$$\begin{aligned} A_2={\frac{910}{293}}(c-{\varepsilon }),\quad A_4={\frac{1085}{293}}(c-{\varepsilon })\quad {\mathrm{and}}\quad b=\frac{1}{1758}{ {\sqrt{\frac{799890 \left( {\varepsilon }-c \right) }{{\alpha }c} }}{}}, \end{aligned}$$

(26)

such that $c>{\varepsilon },$ ${\alpha }<0$ and

$$\begin{aligned} \beta =\frac{27249c{\alpha }^2}{828100(c-{\varepsilon })}. \end{aligned}$$

(27)

One can also observe that $\frac{\mathrm{d}A_j}{\mathrm{d}c}>0,$ $j=2,4,$ and $\frac{\mathrm{d}b}{\mathrm{d}c}>0,$ for all $c>{\varepsilon };$ so that the mapping $c\rightarrow {\varphi _c}$ is smooth from $({\varepsilon },+\infty )$ into $H^s(\mathbb{R }),$ for all $s\in \mathbb{N }.$ Therefore, we obtain

$$\begin{aligned} {\varphi _c}(\xi )&= {\varphi _c}(x-ct) \nonumber \\&= {\frac{910}{293}}(c-{\varepsilon })\mathrm{sech}^2\left( b(x-ct)\right) +{\frac{1085}{293}}(c-{\varepsilon })\mathrm{sech}^4(b(x-ct)),\qquad \end{aligned}$$

(28)

where $b$ is as above.

Finally, we propose the ansatz

$$\begin{aligned} {\varphi _c}(\xi )=\sum \limits _{j=1}^2\frac{A_j}{(B_j+\cosh (b\xi ))^j}. \end{aligned}$$

One can see after balancing again ${\varphi _c}^{\prime \prime \prime \prime }$ with ${\varphi _c}^2$ that $A_1=0.$ Hence plugging

$$\begin{aligned} {\varphi _c}(\xi )=\frac{A}{(B+\cosh (b\xi ))^2} \end{aligned}$$

into (4), collecting the coefficients $\mathrm{sech}^j(b\xi )$ and equating these coefficients to zero, there obtains

$$\begin{aligned} \left\{ \begin{array}{ll} 16 \beta c{b}^{4}+4 c\alpha {b}^{2}- {\varepsilon }+ c=0,\\ -33\beta c{b}^{4}+3 c\alpha {b}^{2}-2 {\varepsilon }+2 c=0,\\ 72 \beta c{b}^{4}{B}^{2}-240 \beta c{b}^{4}-A+12 c{B}^{2}-12 c\alpha {b}^{2}-12 {\varepsilon }{B}^{2}=0,\\ - A+96 \beta c{b}^{4}+4 c{B}^{2}-4 {\varepsilon }{B}^{2}-2 \beta c{b}^{4}{B}^{2}-2 c\alpha {b}^{2}{B}^{2}-12 c\alpha {b}^{2}=0,\\ 2c{B}^{4}-12 c\alpha {b}^{2}{B}^{2}+240 \beta c{b}^{4}-48 \beta c{b}^{4}{B}^{2}-2 {\varepsilon }{B}^{4}-A{B}^{2}=0. \end{array}\right. \end{aligned}$$

(29)

After some computations, we obtain from system (29) that

$$\begin{aligned} A={\frac{35}{3}}(c-{\varepsilon }),\quad B=\pm 1\quad {\mathrm{and}}\quad b= \sqrt{\frac{13({\varepsilon }-c) }{36{\alpha }c}}, \end{aligned}$$

(30)

such that $c>{\varepsilon },$ ${\alpha }<0$ and

$$\begin{aligned} \beta =\frac{36c{\alpha }^2}{169(c-{\varepsilon })}. \end{aligned}$$

(31)

Therefore, we obtain

$$\begin{aligned} {\varphi _c}^\pm (\xi )={\varphi _c}^\pm (x-ct)= \frac{35(c-{\varepsilon })}{3\left( 1\pm \cosh \left( \sqrt{\frac{13({\varepsilon }-c) }{36{\alpha }c}}(x-ct)\right) \right) ^2}. \end{aligned}$$

(32)

One can observe that ${\varphi _c}^+$ is actually identical with solution (24), while ${\varphi _c}^-$ has a singularity in $\xi =0$ (see Fig. 3).

Figure 2 shows the wave profiles of (24) and (28) and their Fourier transform.

Fig. 2

Up is solitary waves of (24) and (28) at $t=0,$ and down is their Fourier transform. The circle-curves correspond to (28)

Fig. 3

The graph of solution ${\varphi _c}^-$ given by (32)

Stability

In this section, we are going to study the orbital stability of the solitary waves obtained in the previous section. Hereafter we assume that ${\alpha }<0$ and $c>{\varepsilon }.$ First we recall the definition of orbital stability.

Definition 2

We say that ${\varphi _c}$ is orbitally stable in $H^2(\mathbb{R })$ by the flow generated by the Rosenau-RLW equation (1) if the initial value problem associated to (1) is globally well-posed in $H^2(\mathbb{R }),$ and for every ${\epsilon },$ there is $\delta >0$ such that for all $u_0\in U_\delta$ the solution $u(t)$ of (1) with $u(0)=u_0$ satisfies $u(t)\in U_{\epsilon }$ for all $t>0.$

Definition 3

A function $w:\mathbb{R }\rightarrow \mathbb{R }$ is said to be in the class $PF(2)$ if for all $x\in \mathbb{R },$ $w(x)>0,$

$$\begin{aligned} w(x_1-y_1)w(x_2-y_2)\ge w(x_1-y_2)w(x_2-y_1),\quad {\mathrm{for}}\quad x_1<x_2\quad {\mathrm{and}}\quad y_1<y_2 \quad \end{aligned}$$

(33)

and the strict inequality holds in (33) whenever the intervals $(x_1,x_2)$ and $(y_1,y_2)$ intersect.

The following result is proved in [3].

Theorem 5

If $f$ a twice-differentiable positive function on $\mathbb{R }$ satisfying

$$\begin{aligned} \frac{\mathrm{d}^2}{\mathrm{d}x^2}\log f(x)<0, \end{aligned}$$

for $x\ne 0,$ then $f\in PF(2).$

The following theorem [2] gives some spectral structure of the linearized operator $L$ about a solitary wave ${\varphi _c}.$

Theorem 6

Let ${\varphi _c}$ be an even positive solitary wave of (1). Suppose that $\widehat{{\varphi _c}}\in PF(2),$ then the operator $L$ satisfies Assumption 1.

By using Theorem 6, the proof of the following stability theorem can be obtained by using the arguments given in [6, 10, 18, 30].

Theorem 7

Let ${\varphi _c}$ be a positive solitary wave of (1). Suppose that Assumption 1 holds for the linearized operator $L.$ Then ${\varphi _c}$ is orbitally stable, if $d^{\prime \prime }(c)>0.$

Theorem 8

Let $c>{\varepsilon }$ and ${\alpha }<0.$ Then the solitary wave ${\varphi _c}$ of (1) obtained in (24) is orbitally stable by the flow of the Rosenau-RLW equation.

The proof of Theorem 7 is a special case of [18, Theorem 3.5], and we will give it for the sake of completeness.

Lemma 1

Let $d^{\prime \prime }(c)>0.$ Then $\langle Ly,y\rangle >0,$ if $y\in H^2(\mathbb{R })$ and $\langle y,Q^{\prime }({\varphi _c})\rangle =\langle y,{\varphi _c}^{\prime }\rangle =0.$

Proof

First by using $d^{\prime }(c)=Q({\varphi _c}),$ we have

$$\begin{aligned} 0<d^{\prime \prime }(c)=\left\langle M{\varphi _c},\mathrm{d}{\varphi _c}/\mathrm{d}c \right\rangle =-\left\langle L\mathrm{d}{\varphi _c}/\mathrm{d}c,\mathrm{d}{\varphi _c}/\mathrm{d}c \right\rangle . \end{aligned}$$

Write

$$\begin{aligned} \frac{\mathrm{d}{\varphi _c}}{\mathrm{d}c}=a_0\chi _c+b_0{\varphi _c}^{\prime }+p_0, \end{aligned}$$

where $p_0$ is in the positive subspace of $L.$ Recall that $L\chi _c=-{\lambda }^2\chi _c$ with ${\lambda }>0$ and $L({\varphi _c}^{\prime })=0.$ It follows that $\langle Lp_0,p_0\rangle <0.$ Now suppose that

$$\begin{aligned} \langle y,{\varphi _c}^{\prime }\rangle =\langle y,Q^{\prime }({\varphi _c})\rangle =0 \end{aligned}$$

and decompose $y$ into the sum $a\chi _c+p$ with $p$ in the positive subspace of $L.$ Because

$$\begin{aligned} 0=\left\langle L\mathrm{d}{\varphi _c}/\mathrm{d}c,y \right\rangle =-a_0a{\lambda }^2+\langle Lp_0,p\rangle , \end{aligned}$$

it is inferred that

$$\begin{aligned} \langle Ly,y\rangle \ge -a^2{\lambda }^2+\frac{\langle Lp,p_0\rangle ^2}{\langle Lp_0,p_0\rangle } >-a^2{\lambda }^2+\frac{(a_0a{\lambda })^2}{a_0^2{\lambda }^2}=0, \end{aligned}$$

as required. $\square$

It can be proved exactly as in the analogous case of [12] that there exists ${\epsilon }>0$ and a unique $C^1$-map $\varrho :U_{\epsilon }\rightarrow \mathbb{R }$ such that for every $u\in U_{\epsilon }$ and $r\in \mathbb{R },$ $\langle u(\cdot +\varrho (u)),{\varphi _c}^{\prime }\rangle =0,$ $\varrho (u(\cdot +r))=\varrho (u)-r$ and

$$\begin{aligned} \varrho ^{\prime }(u)=\frac{{\varphi _c}^{\prime }(\cdot -\varrho (u))}{{\int \nolimits _\mathbb{R }u(x){\varphi _c}^{\prime \prime }(x-\varrho (u))\;\mathrm{d}x}}. \end{aligned}$$

Lemma 2

Let $d^{\prime \prime }(c)>0.$ Then there is $C>0$ and ${\epsilon }>0$ such that

$$\begin{aligned} E(u)-E({\varphi _c})\ge C\Vert u(\cdot +\varrho (u))-{\varphi _c}\Vert _{H^2}^2, \end{aligned}$$

for all

$$\begin{aligned} u\in \widetilde{U}_{\epsilon }=\left\{ u\in U_{\epsilon }:\;Q(u)=Q({\varphi _c})\right\} . \end{aligned}$$

Proof

Write $u$ in the form

$$\begin{aligned} u(\cdot +\varrho (u))=(1+a){\varphi _c}+y, \end{aligned}$$

where $\langle {\varphi _c},y\rangle =0$ and $a$ is a scalar. Then by the translation invariance $Q$ and Taylor’s theorem,

$$\begin{aligned} Q({\varphi _c})=Q(u)=Q({\varphi _c})+\langle {\varphi _c},u(\cdot +\varrho (u))-{\varphi _c}\rangle +a, \end{aligned}$$

where

$$\begin{aligned} a=O\left( \Vert u(\cdot +\varrho (u))-{\varphi _c}\Vert ^2_{H^2}\right) \!. \end{aligned}$$

Hence

$$\begin{aligned} \mathcal S (u)=\mathcal S (u(\cdot +\varrho (u)))=\mathcal S ({\varphi _c})+\frac{1}{2}\langle Lv,v\rangle +o\left( \Vert v\Vert _{H^2}^2 \right) \!, \end{aligned}$$

where

$$\begin{aligned} v=u(\cdot +\varrho (u))-{\varphi _c}=a{\varphi _c}+y. \end{aligned}$$

Thus

$$\begin{aligned} E(u)-E({\varphi _c})=\frac{1}{2}\langle Lv,v\rangle +o(\Vert v\Vert _{H^2}^2) = \frac{1}{2}\langle Ly,y\rangle +o\left( \Vert v\Vert _{H^2}^2 \right) \!. \end{aligned}$$

Since $y$ is orthogonal to both ${\varphi _c}$ and ${\varphi _c}^{\prime },$ it follows from Lemma 1 that

$$\begin{aligned} E(u)-E({\varphi _c})\ge 2C\Vert y\Vert ^2_{H^2}+o(\Vert v\Vert _{H^2}^2), \end{aligned}$$

for some constant $C.$ It follows that

$$\begin{aligned} E(u)-E({\varphi _c})\ge \Vert v\Vert _{H^2}^2, \end{aligned}$$

by using the fact

$$\begin{aligned} \Vert y\Vert _{H^2}=\Vert v-a{\varphi _c}\Vert _{H^2}\ge \Vert v\Vert _{H^2}-O\left( \Vert v\Vert ^2_{H^2}\right) \!, \end{aligned}$$

for $\Vert v\Vert _{H^2}$ small. The proof of lemma is now complete. $\square$

Proof of Theorem 7

Assume that $d^{\prime \prime }(c)>0.$ Let $u_{n,0}\in H^2(\mathbb{R })$ be any sequence such that

$$\begin{aligned} \inf _{r}\Vert u_{n,0}-{\varphi _c}(\cdot +r)\Vert _{H^2}\rightarrow 0, \end{aligned}$$

as $n\rightarrow \infty .$ If $u_n$ is the unique solution (1) with initial data $u_n(0)=u_{n,0},$ let $t_n$ be an arbitrary sequence of times such that, for each $n,$ $u_n(\cdot ,t_n)\in {\partial }U_{{\epsilon }/2}.$ Since $E$ and $Q$ are continuous on $H^2(\mathbb{R })$ and translation invariant,

$$\begin{aligned} E(u_n(\cdot ,t_n))=E(u_{n,0})\rightarrow E({\varphi _c}) \end{aligned}$$

and

$$\begin{aligned} Q(u_n(\cdot ,t_n))=Q(u_{n,0})\rightarrow Q({\varphi _c}). \end{aligned}$$

Next choose $w_n\in U_{\epsilon }$ so that $Q(w_n)=Q({\varphi _c})$ and

$$\begin{aligned} \Vert w_n-u_n(\cdot ,t_n)\Vert _{H^2}\rightarrow 0. \end{aligned}$$

By Lemma 2,

$$\begin{aligned} 0\leftarrow E(w_n)-E({\varphi _c})\ge C\Vert w_n(\cdot +\varrho (w_n))-{\varphi _c}\Vert ^2_{H^2}=C\Vert w_n-{\varphi _c}(\cdot -\varrho (w_n))\Vert _{H^2}, \end{aligned}$$

and therefore

$$\begin{aligned} \Vert u_n(\cdot ,t_n)-{\varphi _c}(\cdot -\varrho (w_n))\Vert _{H^2}\rightarrow 0. \end{aligned}$$

This means that $u_n(\cdot ,t_n)$ tends to $\mathcal O _{{\varphi }_c}.$ This contradiction completes the proof of Theorem 7. $\square$

Now we are in position to prove Theorem 8.

Proof of Theorem 8

By Theorem 7, first of all we should study the behavior of the first two eigenvalues associated with the operator

$$\begin{aligned} L= {\alpha }c \frac{\mathrm{d}^2}{\mathrm{d}\xi ^2}+\beta c\frac{\mathrm{d}^4}{\mathrm{d}\xi ^4}-{\varphi _c}+c-{\varepsilon }. \end{aligned}$$

By applying Theorem 6, it suffices to show that $\widehat{{\varphi _c}}\in PF(2).$ But a straightforward calculation reveals from (24) that

$$\begin{aligned} \widehat{{\varphi }}(\xi )=A\frac{\pi \xi }{3b^2}\left( 1+\frac{\xi ^2}{4b^2}\right) \mathrm{csch}\left( \frac{\pi \xi }{2b}\right) \end{aligned}$$

and

$$\begin{aligned}&\frac{\mathrm{d}^2}{\mathrm{d}\xi ^2}\log \widehat{{\varphi _c}}(\xi )\\&\quad \!=\! -C_{(24)} \frac{ (64{b}^{6}\!+\!12 {\xi }^{4}{b}^{2}) \cosh ^2 \left( \frac{\pi \xi }{2b} \right) \!-\!12\xi ^{4}{b}^{2}\!-\!{\xi }^{6}{\pi }^{2}\!-\!16 {\xi }^{2}{\pi }^{2}{b}^{4}\!-\!8 {\xi }^{4}{\pi }^{2}{b}^{2}\!-\!64 {b}^{6} }{ \left( \cosh ^2 \left( \frac{\pi \xi }{2b} \right) \!-\!1 \right) {\xi }^{2} \left( 4 {b }^{2}\!+\!{\xi }^{2} \right) ^{2}}, \end{aligned}$$

where $C_{(24)}=\frac{A\pi }{12b^4},$ where $A$ and $b$ is are as in (22). By using the Taylor expansion

$$\begin{aligned} \cosh ^2\left( \frac{\pi \xi }{2b} \right) =\frac{1}{2}+\sum _{n=0}^\infty \frac{1}{2(2n)!}\left( \frac{\pi \xi }{b}\right) ^{2n}\!, \end{aligned}$$

(34)

it is readily seen that

$$\begin{aligned} \frac{\mathrm{d}^2}{\mathrm{d}\xi ^2}\log \widehat{{\varphi _c}}(\xi )<0, \end{aligned}$$

for $\xi \ne 0.$ Then by Theorem 7, we need to calculate $\frac{\mathrm{d}}{\mathrm{d}c}Q({\varphi _c}).$ Actually, we have from (24) that

$$\begin{aligned} Q({\varphi _c})=\left( {\frac{1 }{ b}} -{\frac{16}{9}}{\alpha }b +{\frac{1024}{99}} \beta b^3 \right) \frac{16A^2}{35}, \end{aligned}$$

and

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}c}Q({\varphi _c})=K_{(24)} \frac{\left( 1400c^3-1856{\varepsilon }c^2+403c{\varepsilon }^2+53{\varepsilon }^3\right) }{c\sqrt{\frac{({\varepsilon }-c)c}{{\alpha }}}}>0. \end{aligned}$$

where

$$\begin{aligned} K_{(24)}=\frac{70\sqrt{13}}{11583}. \end{aligned}$$

This completes the proof of Theorem 8. $\square$

Theorem 9

Let $c>{\varepsilon }$ and ${\alpha }<0.$ Then the solitary wave ${\varphi _c}$ of (1) obtained in (28) is orbitally stable by the flow of the Rosenau-RLW equation.

Proof

First we note that

$$\begin{aligned} \widehat{{\varphi }}_c(\xi )=\frac{\pi \xi }{b^2}\left( \frac{A_2}{2}+\frac{A_4}{3}\left( 1+\frac{\xi ^2}{4b^2}\right) \right) \mathrm{csch}\left( \frac{\pi \xi }{2b}\right) \end{aligned}$$

and

$$\begin{aligned} \frac{\mathrm{d}^2}{\mathrm{d}\xi ^2}\log (\widehat{{\varphi _c}})=-C_{(28)}\frac{\eta (\xi )}{\left( \cosh \left( \frac{\pi \xi }{2b}\right) ^2-1\right) \xi ^2(6 A_2 b^2+4 A_4 b^2+A_4 \xi ^2)^2}, \end{aligned}$$

where $C_{(28)}=\frac{\pi }{4b^4}$ and

$$\begin{aligned} \eta (\xi )&= (12A_4^2\xi ^4b^2+192A_2b^6A_4+144A_2^2b^6+64A_4^2b^6)\cosh \left( \frac{\pi \xi }{2b}\right) ^2 \\&\quad - 36\xi ^2\pi ^2A_2^2b^4-16\xi ^2\pi ^2A_4^2b^4-64A_4^2b^6 \\&\quad - 144A_2^2b^6-48\xi ^2\pi ^2A_2b^4A_4 \\&\quad - 12\xi ^4\pi ^2A_2b^2A_4-12A_4^2\xi ^4b^2-\xi ^6\pi ^2A_4^2-192A_2b^6A_4-8\xi ^4\pi ^2A_4^2b^2 \end{aligned}$$

By a straightforward calculation, it is readily seen from (34) that

$$\begin{aligned} \frac{\mathrm{d}^2}{\mathrm{d}\xi ^2}\log (\widehat{{\varphi _c}})<0 \end{aligned}$$

for all $\xi \ne 0.$ By the proof of Theorem 8, it is enough to calculate $\frac{\mathrm{d}}{\mathrm{d}c}Q({\varphi _c}).$ Indeed, we have from (28) that

$$\begin{aligned} Q({\varphi _c})&= -\frac{8\alpha b}{15} \left( A^2_2+{\frac{16}{7}}A_4A_2+{\frac{32}{21}}A^2_4 \right) +\frac{32\beta b^3}{21} \left( A^2_2+{\frac{512}{165}}A^2_4+{\frac{16}{5}}A_4A_2 \right) \\&\quad +\frac{2}{b}\left( \frac{A^2_2}{3}+{\frac{8}{15}} A_4A_2+{\frac{8}{35}} A_4^2 \right) . \end{aligned}$$

and

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}c}Q({\varphi _c})=K_{(28)}\frac{\left( 235578848 c^3-323443550 c^2{\varepsilon }+73379707 c{\varepsilon }^2+14484995{\varepsilon }^3\right) }{c\sqrt{\frac{({\varepsilon }-c)c}{{\alpha }}}}>0, \end{aligned}$$

where

$$\begin{aligned} K_{(28)}=\frac{20\sqrt{799890}}{32372885259}, \end{aligned}$$

and $A_2,$ $A_4$ and $b$ is are as in (26); and the proof of Theorem 9 is now complete. $\square$

Figure 1 illustrates the shape of kernel of (12) for $\beta =1,$ and ${\alpha }=-3,$ ${\alpha }=-2$ and ${\alpha }=1$ respectively. Finally, we observed in Theorem 3 that the solutions of (1) satisfies the conservation laws $Q$ and $E$ in (5) and (6). We calculate these conserved quantities by using the solitary waves given by (24), (28) and (32). Actually, we obtain

$$\begin{aligned} E_{(24)}({\varphi _c})=-{\frac{16{\varepsilon }A^2}{35b}}- \frac{256A^3}{2079b}, \end{aligned}$$

where $A$ and $b$ is are as in (22),

$$\begin{aligned} E_{(28)}({\varphi _c})=-{\frac{8A^3_2}{45b}} - \left( \frac{{\varepsilon }}{3}+{\frac{8A_4}{35}} \right) \frac{2A^2_2}{b} - \left( {\varepsilon }+{\frac{8A_4}{21}} \right) \frac{16 A_2A_4}{15b} -\left( {\frac{{\varepsilon }}{5}} +{\frac{16A_4}{297}} \right) \frac{16A^2_4}{7b} \end{aligned}$$

where $A_2,$ $A_4$ and $b$ is are as in (26).

Figures 4 and 5 illustrate the graphs of the invariants $E$ and $Q$ in terms of $c,$ when ${\alpha }=-1$ and ${\varepsilon }=1,$ for solutions (24) and (28), respectively.

Fig. 4

Invariants of $Q$ and $E$ of solution (24) as the functions in terms of $c.$ The dash-curve corresponds to $E$

Fig. 5

Invariants of $E$ in down and $Q$ in up of solution (28) as the functions in terms of $c$