1 Introduction

Nonlinear science is the basic subject of studying various nonlinear phenomena, it not only has important scientific significance, but also has practical significance in engineering, social and economic fields such as the financial problems, and the human survival environment. With the rapid development of nonlinear science emerged a large number of nonlinear evolution equations, for these different equations, under the background of different physical has different role. Shallow water equation is one of the more important equation, is widely applied to describe the fluid dynamics and environmental science and other fields, e.g., the tsunami is a kind of shallow water. Therefore, exact solutions of these equations become the important problem of attention. There are many articles in recent decades show the specific solutions, such as \(\mathrm \exp \big (-\phi (\xi )\big )\)-expansion method [1,2,3,4], \(\big (\frac{G'}{G}\big )\)-expansion method [5,6,7], the Riccati equation method [8], the generalized \(\tanh \)-expansion method [9], the local symmetry method [10, 11], the Bäcklund transformation method [12], the inverse scattering method [13], the CK direct reduction method [14], the Lie group analysis method [15,16,17,18,19,20,21,22,23], the dynamical system method [24,25,26,27] and the power series method [28, 29], and so on.

In this paper, we consider the following nonlinear shallow water wave equation:

$$\begin{aligned} u_{t}-u_{xxt}+a_{1}uu_{x}+a_{2}u_{x}u_{xx}-uu_{xxx}=0, \end{aligned}$$
(1.1)

where \(u=u(x,t)\) denotes the unknown function, \(a_{1}\) and \(a_{2}\) are arbitrary constants.

In particular, if \(a_{1}=3 \), \( a_{2}=-2 \), then Eq. (1.1) is the famous Camassa–Holm (CH) equation [30]; If \(a_{1}=4 \), \( a_{2}=-3 \), then Eq. (1.1) is the Degasperis-Procesi (DP) equation [31].

This paper is organized as follows. In Sect. 2, the infinitesimal generators and Lie algebra of the nonlinear evolution equation (1.1) are obtained by using the Lie group analysis method. In Sect. 3, Eq. (1.1) is transformed into a dynamical system by using the traveling wave transform, and the first integral of the system is calculated. The singularity and bifurcations under different parameters are given with the help of Maple symbol calculation software, then the exact traveling wave solutions are provided accordingly. In Sect. 4, the symmetry reduction and exact analytic solution of the nonlinear equation are obtained by using the power series method. In Sect. 5, some conclusion and new findings are given.

2 Lie Symmetry Analysis of Eq. (1.1)

Recall that the geometric vector field of system (1.1) is as follows:

$$\begin{aligned} V=\xi (x,t,u)\frac{\partial }{\partial x}+\tau (x,t,u)\frac{\partial }{\partial t}+\phi (x,t,u)\frac{\partial }{\partial u}, \end{aligned}$$
(2.1)

where the coefficient functions \(\xi (x,t,u)\), \(\tau (x,t,u)\), \(\phi (x,t,u)\) are to be determined. (2.1) is also called the infinitesimal symmetry of (1.1). Then V must satisfy the Lie symmetry condition

$$\begin{aligned} \mathrm {pr}^{(3)}V(\Delta )|_{\Delta =0}=0, \end{aligned}$$
(2.2)

where \(\hbox {pr}^{(3)}V\) denotes the 3th prolongation of V, and \(\Delta =u_{t}-u_{xxt}+a_{1}uu_{x}+a_{2}u_{x}u_{xx}-uu_{xxx}\),

$$\begin{aligned} \mathrm {pr}^{(3)}V=V+\phi ^{t}\frac{\partial }{\partial u_{t}}-\phi ^{xxt}\frac{\partial }{\partial u_{xxt}}+\phi ^{x}\frac{\partial }{\partial u_{x}}+\phi ^{xx}\frac{\partial }{\partial u_{xx}}+\phi ^{xxx}\frac{\partial }{\partial u_{xxx}}. \end{aligned}$$
(2.3)

Then the invariant condition reads as follows

$$\begin{aligned} \phi ^{t}-\phi ^{xxt}+a_{1}u_{x}\phi +a_{1}u\phi ^{x}+a_{2}u_{xx}\phi ^{x}+a_{2}u_{x}\phi ^{xx}-u_{xxx}\phi -u\phi ^{xxx}, \end{aligned}$$
(2.4)

where the coefficient functions are given by

$$\begin{aligned} \phi ^{x}= & {} D_{x}\phi -u_{x}D_{x}\xi -u_{t}D_{x}\tau ,\\ \phi ^{t}= & {} D_{t}\phi -u_{x}D_{t}\xi -u_{t}D_{t}\tau ,\\ \phi ^{xx}= & {} D^{2}_{x}\phi -u_{x}D^{2}_{x}\xi -2u_{xx}D_{x}\xi -u_{t}D^{2}_{x}\tau -2u_{xt}D_{x}\tau ,\\ \phi ^{xxx}= & {} D^{3}_{x}\phi -3u_{xxx}D_{x}\xi -3u_{xx}D^{2}_{x}\xi -u_{t}D^{3}_{x}\tau -3u_{xt}D^{2}_{x}\tau -3u_{xxt}D_{x}\tau ,\\ \phi ^{xxt}= & {} D_{t}D^{2}_{x}\phi -2u_{xxt}D_{x}\xi -2u_{xx}D_{x}D_{t}\xi -u_{xt}D^{2}_{x}\xi -u_{x}D^{2}_{x}D_{t}\xi -u_{xxx}D_{t}\xi \\&-2u_{xxt}D_{x}\tau - 2u_{tx}D_{x}D_{t}\tau -u_{tt}D^{2}_{x}\tau -u_{t}D^{2}_{x}D_{t}\tau -u_{xxt}D_{t}\tau , \end{aligned}$$

\(D_{x}\), \(D_{t}\) are total differential operators with respect to x and t, respectively. Then, Substituting the above coefficient function into Eq. (2.4), and solving the deterministic system of equations about \(\xi \), \(\tau \), \(\phi \), so we get the solution

$$\begin{aligned} \xi =C_{3},\quad \tau =C_{1}t+C_{2},\quad \phi =-C_{1}u, \end{aligned}$$
(2.5)

where \(C_{1}\), \(C_{2}\), \(C_{3}\) are arbitrary constants.

Then, in terms of the Lie symmetry analysis method, we can obtain all of the geometric vector fields of the system

$$\begin{aligned} V_{1}=\frac{\partial }{\partial x},\quad V_{2}=\frac{\partial }{\partial t},\quad V_{3}=t\frac{\partial }{\partial t}-u\frac{\partial }{\partial u}. \end{aligned}$$

It is necessary to show that the vector fields are closed under the Lie bracket. In fact, we have:

$$\begin{aligned} {[}V_{1},V_{1}] &= [V_{2},V_{2}]=[V_{3},V_{3}]=0,\\ {[}V_{1},V_{2}]= & {} -[V_{2},V_{1}]=[V_{1},V_{3}]=-[V_{3},V_{1}]=0,\\ {[}V_{2},V_{3}]= & {} -[V_{3},V_{2}]=V_{2}. \end{aligned}$$

Furthermore, for Eq. (1.1), the one-parameter group \(g_{i}(\varepsilon )\) generated by the \(V_{i}\)\((i=1,2,3)\) are given in following table:

$$\begin{aligned}&g_{1}: (x,t,u)\longrightarrow (x+\varepsilon ,t,u), \end{aligned}$$
(2.6)
$$\begin{aligned}&g_{2}: (x,t,u)\longrightarrow (x,t+\varepsilon ,u), \end{aligned}$$
(2.7)
$$\begin{aligned}&g_{3}: (x,t,u)\longrightarrow (x, te^{\varepsilon }, ue^{-\varepsilon }). \end{aligned}$$
(2.8)

Applying above group \(g_{1}\), \(g_{2}\) and \(g_{3}\), we can get the Lie symmetry theorem:

Theorem 2.1

If\(u = f(x, t)\)is a solution of Eq. (1.1), so are the following function expressions

$$\begin{aligned} u_{1}&= g_{1}(\varepsilon )f(x,t)=f(x-\varepsilon ,t),\nonumber \\ u_{2}&= g_{2}(\varepsilon )f(x,t)=f(x,t-\varepsilon ),\nonumber \\ u_{3}&= g_{3}(\varepsilon )f(x,t)=e^{-\varepsilon }f(x,te^{-\varepsilon }), \end{aligned}$$
(2.9)

where\(\varepsilon \)is any real number.

3 Traveling Wave Transformations, Bifurcations and Exact Solutions of Eq. (1.1)

3.1 Traveling Wave Transformation Derived from \(cV_{1}+V_{2}\)

For the linear combination \(cV_{1}+V_{2}\), we have

$$\begin{aligned} u=u(\xi ), \end{aligned}$$
(3.1)

where \(\xi =x-ct\) is the group-invariant and c is the propagating wave velocity. Substitution of (3.1) into Eq. (1.1), we obtain:

$$\begin{aligned} -cu'+cu'''+a_{1}uu'+a_{2}u'u''-uu'''=0, \end{aligned}$$
(3.2)

where \(u'=\frac{du}{d\xi }\). Integrating the above ordinary differential equation (ODE) (3.2) , we get

$$\begin{aligned} a_{1}u^{2}-2cu+(a_{2}+1)(u')^{2}+2(c-u)u''=0. \end{aligned}$$
(3.3)

Equation (3.3) is equivalent to the following singular system

$$\begin{aligned} \frac{du}{d\xi }=y, \quad \frac{dy}{d\xi }=\frac{-(a_{2}+1)y^{2}-a_{1}u^{2}+2cu}{2(c-u)}, \end{aligned}$$
(3.4)

which has the Hamiltonian function

$$\begin{aligned} H(u,y)=y^{2}+\frac{a_{1}u^{2}}{a_{2}-1}-\frac{2c(a_{1}a_{2}+a_{2}^{2}+a_{1}-1)u}{a_{2}(a_{2}^{2}-1)}+\frac{2c^{2}(a_{1}+a_{2}-1)}{a_{2}(a_{2}^{2}-1)}=h, \end{aligned}$$
(3.5)

where h is an arbitrary constant.

3.2 Bifurcation Sets and Phase Portraits

In this section, we shall study all possible bifurcations and phase portraits of the vector fields given by system (3.4) in the parametric space. First, making the timescale transformation \(d\xi =2(c-u)d\zeta \), singular system (3.4) can be reduced to the following regular planar system

$$\begin{aligned} \frac{du}{d\zeta }=2y(c-u), \frac{dy}{d\zeta }=-(a_{2}+1)y^{2}-a_{1}u^{2}+2cu, \end{aligned}$$
(3.6)

which has the same Hamiltonian function and topological phase portraits as Eq. (3.4) expect on the singular line \(u=c\). Now we discuss the bifurcation of the phase portraits of system (3.6) in the parameter space \((a_{1}\), \(a_{2})\). By qualitative analysis, we have the following results.

Firstly, note that there are two equilibrium points \(A_{1}(0,0)\) and \(A_{2}\big (\frac{2c}{a_{1}},0\big )\) of system (3.6), the determinant of the coefficient matrix for the linearized system of Eq. (3.4) about this point writes

$$\begin{aligned} J(u_{e}, y_{e})=4(a_{2}+1)y_{e}^{2}+4(c-u_{e})(a_{1}u_{e}-c), \end{aligned}$$
(3.7)

By the theory of planar dynamical systems, for an equilibrium point of a planar integrable system, if \(J>0\), then it is a center point; if \(J<0\), then then the equilibrium point is a saddle point; if \(J=0\) and the Poincaré index of the equilibrium point is 0, then the equilibrium point is a cusp.

3.2.1 The Case \(a_{2}>0\)

  1. 1.

    For \(c>0\), \(a_{1}>1\), as shown in Fig. 1a, system (3.6) always has two equilibrium points, where \(A_{1}(0,0)\) is a saddle point, \(A_{2}\big (\frac{2c}{a_{1}},0\big )\) is a center point.

  2. 2.

    For \(c>0\), \(a_{1}=1\), as shown in Fig. 1b, system (3.6) always has two equilibrium points, where \(A_{1}(0,0)\) is a saddle point, \(A_{2}\big (\frac{2c}{a_{1}},0\big )\) is a cusp.

  3. 3.

    For \(c>0\), \(0<a_{1}<1\), as shown in Fig. 1c, system (3.6) always has two equilibrium points, where \(A_{1}(0,0)\) and \(A_{2}\big (\frac{2c}{a_{1}},0\big )\) are a saddle point.

  4. 4.

    For \(c>0\), \(a_{1}<0\), as shown in Fig. 1d, e, system (3.6) always has two equilibrium points, where \(A_{1}(0,0)\) is a saddle point, \(A_{2}\big (\frac{2c}{a_{1}},0\big )\) is a center point.

  5. 5.

    For \(c>0\), \(a_{1}=0\), as shown in Fig. 1f, system (3.6) always has two equilibrium points, where \(A_{1}(0,0)\) is a saddle point, \(A_{2}\big (\frac{2c}{a_{1}},0\big )\) is a center point.

  6. 6.

    For \(c<0\), \(a_{1}>0\), as shown in Fig. 1g, h and j, system (3.6) always has two equilibrium points, where \(A_{1}(0,0)\) and \(A_{2}\big (\frac{2c}{a_{1}},0\big )\) are a saddle point.

  7. 7.

    For \(c<0\), \(a_{1}<0\), as shown in Fig. 1k–m, system (3.6) always has two equilibrium points, where \(A_{1}(0,0)\) is a saddle point, \(A_{2}\big (\frac{2c}{a_{1}},0\big )\) is a center point.

  8. 8.

    For \(c<0\), \(a_{1}=0\), as shown in Fig. 1n, system (3.6) always has two equilibrium points, where \(A_{1}(0,0)\) is a saddle point, \(A_{2}\big (\frac{2c}{a_{1}},0\big )\) is a center point.

Fig. 1
figure 1

Bifurcations of phase portraits of system for \(a_{2}>0\)

Full size image

3.2.2 The Case \(a_{2}<0\)

  1. 1.

    For \(c>0\), \(a_{1}=1\), as shown in Fig. 2a, system (3.6) always has two equilibrium points, where \(A_{1}(0,0)\) and \(A_{2}\big (\frac{2c}{a_{1}},0\big )\) are a saddle point.

  2. 2.

    For \(c>0\), \(a_{1}>1\), as shown in Fig. 2b, system (3.6) always has two equilibrium points, where \(A_{1}(0,0)\) is a saddle point, \(A_{2}\big (\frac{2c}{a_{1}},0\big )\) is a cusp point.

  3. 3.

    For \(c>0\), \(0<a_{1}<1\), as shown in Fig. 2c, system (3.6) always has two equilibrium points, where \(A_{1}(0,0)\) and \(A_{2}\big (\frac{2c}{a_{1}},0\big )\) are a saddle point.

  4. 4.

    For \(c>0\), \(a_{1}<0\), as shown in Fig. 2d and e, system (3.6) always has two equilibrium points, where \(A_{1}(0,0)\) is a saddle point, \(A_{2}\big (\frac{2c}{a_{1}},0\big )\) is a center point.

  5. 5.

    For \(c>0\), \(a_{1}=0\), as shown in Fig. 2f, system (3.6) always has two equilibrium points, where \(A_{1}(0,0)\) is a saddle point, \(A_{2}\big (\frac{2c}{a_{1}},0\big )\) is a center point.

  6. 6.

    For \(c<0\), \(a_{1}>0\), as shown in Fig. 2g and h, system (3.6) always has two equilibrium points, where \(A_{1}(0,0)\) and \(A_{2}\big (\frac{2c}{a_{1}},0\big )\) are a saddle point.

  7. 7.

    For \(c<0\), \(a_{1}<0\), as shown in Fig. 2j and k, system (3.6) always has two equilibrium points, where \(A_{1}(0,0)\) is a saddle point, \(A_{2}\big (\frac{2c}{a_{1}},0\big )\) is a center point.

  8. 8.

    For \(c<0\), \(a_{1}=0\), as shown in Fig. 2l, system (3.6) always has two equilibrium points, where \(A_{1}(0,0)\) is a saddle point, \(A_{2}\big (\frac{2c}{a_{1}},0\big )\) is a center point.

Fig. 2
figure 2

Bifurcations of phase portraits of system for \(a_{2}<0\)

Full size image

3.3 Exact Explicit Traveling Wave Solutions to Eq. (1.1)

In this section, by using the planar dynamical system (3.4) and the first integral H(u, y) to do calculations, we will give all explicit traveling wave solutions to Eq. (1.1) under the given parameter conditions shown in Figs. 1 and 2.

3.3.1 The Case for \(a_{2}>0\)

  1. (i)

    Suppose that \(a_{1}>1\), \(c>0\), we consider the case of Fig. 1a.

    Corresponding to \(H(u,y)=h\), the function (3.5) can be written as

    $$\begin{aligned} y^{2}= & {} -\frac{a_{1}u^{2}}{a_{2}-1}+\frac{2c(a_{1}a_{2}+a_{2}^{2}+a_{1}-1)u}{a_{2}(a_{2}^{2}-1)}-\frac{2c^{2}(a_{1}+a_{2}-1)}{a_{2}(a_{2}^{2}-1)}+h\nonumber \\= & {} -\frac{a_{1}}{a_{2}-1}(u-u_{1})(u_{2}-u) \end{aligned}$$
    (3.8)

    By using above formula Eq. (3.8) and the first equation of system Eq. (3.4), we obtain

    $$\begin{aligned} \frac{du}{\sqrt{(u-u_{1})(u_{2}-u)}}=\sqrt{-\frac{a_{1}}{a_{2}-1}}d\xi . \end{aligned}$$
    (3.9)

    Integrating Eq. (3.9), we have traveling wave solution of Eq. (1.1) as follows

    $$\begin{aligned} u(\xi )=u_{2}-\frac{u_{2}-u_{1}}{2(m+\sqrt{m^{2}+m}+1)}, \end{aligned}$$
    (3.10)

    where \(m=\tan ^{2}\left( \sqrt{\frac{-a_{1}}{a_{2}-1}}\xi +C_{1}\right) \), \(C_{1}\) is a constant, and \(u_{2}<0<u_{1}\).

  2. (ii)

    Suppose that \(0<a_{1}<1\), \(c>0\), we consider the case of Fig. 1c.

    Corresponding to \(H(u,y)=h\), the function (3.5) can be written as

    $$\begin{aligned} y^{2}= & {} -\frac{a_{1}u^{2}}{a_{2}-1}+\frac{2c(a_{1}a_{2}+a_{2}^{2}+a_{1}-1)u}{a_{2}(a_{2}^{2}-1)}-\frac{2c^{2}(a_{1}+a_{2}-1)}{a_{2}(a_{2}^{2}-1)}+h\nonumber \\= & {} -\frac{a_{1}}{a_{2}-1}(u_{1}-u)(u-u_{2}). \end{aligned}$$
    (3.11)

    By using above formula Eq. (3.11) and the first equation of system Eq. (3.4), we obtain

    $$\begin{aligned} \frac{du}{\sqrt{(u_{1}-u)(u-u_{2})}}=\sqrt{-\frac{a_{1}}{a_{2}-1}}d\xi . \end{aligned}$$
    (3.12)

    Integrating Eq. (3.12), we have the explicit representation of smooth periodic wave solutions of Eq. (1.1) as follows

    $$\begin{aligned} u(\xi )=u_{2}+\frac{u_{1}-u_{2}}{2(m+\sqrt{m(m+1)}+1)}, \end{aligned}$$
    (3.13)

    where \(m=\tan ^{2}\left( \sqrt{\frac{-a_{1}}{a_{2}-1}}\xi +C_{2}\right) \), \(C_{2}\) is a constant, and \(0<u_{2}<u_{1}\).

  3. (iii)

    Suppose that \(a_{1}<0\), \(c>0\), we consider the case of Fig. 1e.

    Corresponding to \(H(u,y)=h\), the function (3.5) can be written as

    $$\begin{aligned} y^{2}= & {} -\frac{a_{1}u^{2}}{a_{2}-1}+\frac{2c(a_{1}a_{2}+a_{2}^{2}+a_{1}-1)u}{a_{2}(a_{2}^{2}-1)}-\frac{2c^{2}(a_{1}+a_{2}-1)}{a_{2}(a_{2}^{2}-1)}+h\nonumber \\= & {} -\frac{a_{1}}{a_{2}-1}u(u_{1}-u). \end{aligned}$$
    (3.14)

    By using above formula Eq. (3.14) and the first equation of system Eq. (3.4), we obtain

    $$\begin{aligned} \frac{du}{\sqrt{u(u_{1}-u)}}=\sqrt{-\frac{a_{1}}{a_{2}-1}}d\xi . \end{aligned}$$
    (3.15)

    Integrating Eq. (3.15), we have the explicit representation of smooth periodic wave solutions of Eq. (1.1) as follows

    $$\begin{aligned} u(\xi )=u_{1}-\frac{u_{1}}{2(m+\sqrt{m(m+1)}+1)}, \end{aligned}$$
    (3.16)

    where \(m=\tan ^{2}\left( \sqrt{\frac{-a_{1}}{a_{2}-1}}\xi +C_{3}\right) \), \(C_{3}\) is a constant, and \(u_{1}>0\).

  4. (iv)

    Suppose that \(a_{1}=0\), \(c>0\), we consider the case of Fig. 1f.

    Corresponding to \(H(u,y)=h\), the function (3.5) can be written as

    $$\begin{aligned} y^{2}=\frac{2cu}{a_{2}}-\frac{2c^{2}}{a_{2}(a_{2}^2+1)}+h=\frac{2c}{a_{2}}(u_{1}-u). \end{aligned}$$
    (3.17)

    By using above formula Eq. (3.17) and the first equation of system Eq. (3.4), we obtain

    $$\begin{aligned} \frac{du}{\sqrt{(u_{1}-u)}}=\sqrt{\frac{2c}{a_{2}}}d\xi . \end{aligned}$$
    (3.18)

    Integrating Eq. (3.18), we have traveling wave solution of Eq. (1.1) as follows

    $$\begin{aligned} u(\xi )=\frac{1}{4}\left( \sqrt{\frac{2c}{a_{2}}}\xi +C_{4}\right) ^{2}+u_{1}, \end{aligned}$$
    (3.19)

where \(C_{4}\) is a constant, and \(u_{1}<0\).

3.3.2 The Case for \(a_{2}<0\)

  1. (i)

    Suppose that \(a_{1}<0\), \(c>0\), we consider the case of Fig. 2e.

    Corresponding to \(H(u,y)=h\), the function (3.5) can be written as

    $$\begin{aligned} y^{2}= & {} -\frac{a_{1}u^{2}}{a_{2}-1}+\frac{2c(a_{1}a_{2}+a_{2}^{2}+a_{1}-1)u}{a_{2}(a_{2}^{2}-1)}-\frac{2c^{2}(a_{1}+a_{2}-1)}{a_{2}(a_{2}^{2}-1)}+h\nonumber \\= & {} -\frac{a_{1}}{a_{2}-1}u(u-u_{1}). \end{aligned}$$
    (3.20)

    By using above formula Eq. (3.20) and the first equation of system Eq. (3.4), we obtain

    $$\begin{aligned} \frac{du}{\sqrt{u(u-u_{1})}}=\sqrt{-\frac{a_{1}}{a_{2}-1}}d\xi . \end{aligned}$$
    (3.21)

    Integrating Eq. (3.21), we have the explicit representation of smooth periodic wave solutions of Eq. (1.1) as follows

    $$\begin{aligned} u(\xi )=\frac{(u_{1}+2m)^2}{8m}, \end{aligned}$$
    (3.22)

where \(m=Ce^{\sqrt{-\frac{a_{1}}{a_{2}-1}}\xi }\), C is a constant, and \(u_{1}<0\).

In other cases, the exact traveling wave solutions are similar to the above, and we will not solve them one by one here.

4 Symmetry Reductions and Exact Analytic Solutions of Eq. (1.1)

In general, we cannot obtain the exact and explicit solutions for the nonlinear PDEs such as (1.1) by using the elementary functions. However, we know that the power series method can be used to solve nonlinear differential equations, including many complicated differential equations with nonconstant coefficients. In this section, we will consider the exact analytic solutions of the reduced Eq. (1.1) by using the power series method.

For the linear combination \(V_{1}+V_{3}\), we have

$$\begin{aligned} u=t^{-1}f(x-\ln t), \end{aligned}$$
(4.1)

where \(\xi =x-\ln t\) and \(w=ut\) are group-invariants. Substituting (4.1) into Eq. (1.1), we reduce this equation to the following ODE

$$\begin{aligned} a_{1}ff'+a_{2}f'f''-ff'''+f'''+f''-f'-f=0, \end{aligned}$$
(4.2)

where \(f'=\frac{df}{d\xi }\). Now, we seek a solution of Eq. (4.2) in a power series of the form

$$\begin{aligned} f(\xi )={\sum _{n=0}^\infty \alpha _{n}\xi ^{n}}. \end{aligned}$$
(4.3)

Substituting (4.3) into Eq. (4.2), we get

$$\begin{aligned}&a_{1}\alpha _{0}\alpha _{1}+a_{1}{\sum _{n=1}^\infty \bigg [{\sum _{k=0}^n(n-k+1)\alpha _{k}}}\alpha _{n-k+1}\bigg ]\xi ^{n}+2a_{2}\alpha _{1}\alpha _{2}\nonumber \\&\quad +a_{2}{\sum _{n=1}^\infty \bigg [{\sum _{k=0}^n(k+1)(n-k+1)}}(n-k+2)\alpha _{k+1} \alpha _{n-k+2}\bigg ]\xi ^{n}\nonumber \\&\quad -6\alpha _{0}\alpha _{3}-{\sum _{n=1}^\infty \bigg [{\sum _{k=0}^n(n-k+1)(n-k+2)(n-k+3)\alpha _{k}}}\alpha _{n-k+3}\bigg ]\xi ^{n}\nonumber \\&\quad -\alpha _{0}-\sum _{n=1}^{\infty }\alpha _{n}\xi ^{n}-\alpha _{1}-\sum _{n=1}^{\infty }(n+1)\alpha _{n+1}\xi ^{n}+2\alpha _{2}\nonumber \\&\quad +{\sum _{n=1}^\infty (n+1)(n+2)\alpha _{n+2}\xi ^{n}}+6\alpha _{3}+{\sum _{n=1}^\infty (n+1)(n+2)(n+3)\alpha _{n+3}\xi ^{n}}=0. \end{aligned}$$
(4.4)

From (4.4), comparing coefficients, for \(n=0\), one can get

$$\begin{aligned} \alpha _{3}=\frac{1}{6(\alpha _{0}-1)}(a_{1}\alpha _{0}\alpha _{1}+2a_{2}\alpha _{1}\alpha _{2}-\alpha _{0}-\alpha _{1}+2\alpha _{2}), \end{aligned}$$
(4.5)

where and in what follows \(\alpha _{0}-1\ne 0\).

Generally, for \(n\ge 1\), we have

$$\begin{aligned} \alpha _{n+3}= & {} \frac{1}{(n+1)(n+2)(n+3)(\alpha _{0}-1)}\bigg [a_{1}{\sum _{k=0}^n(n-k+1)\alpha _{k}\alpha _{n-k+1}} \nonumber \\&+a_{2}{\sum _{k=0}^n(k+1)(n-k+1)(n-k+2)}\alpha _{k+1}\alpha _{n-k+2}\nonumber \\&-{\sum _{k=1}^n(n-k+1)(n-k+2)(n-k+3)\alpha _{k}\alpha _{n-k+3}} \nonumber \\&-\alpha _{n}-(n+1)\alpha _{n+1}+(n+1)(n+2)\alpha _{n+2}\bigg ], \end{aligned}$$
(4.6)

where \(n=1,2,3,...\), and for arbitrary chosen constant numbers \(\alpha _{0}\ne 1\), \(\alpha _{1}\) and \(\alpha _{2}\). For example, for \(n=1\), we get

$$\begin{aligned} \alpha _{4}= & {} \frac{1}{24(\alpha _{0}-1)}(2a_{1}\alpha _{0}\alpha _{2}+a_{1}\alpha _{1}^{2}+6a_{2}\alpha _{1}\alpha _{3}+4a_{2}\alpha _{2}^{2}\\&-6\alpha _{1}\alpha _{3}-\alpha _{1}-2\alpha _{2}+6\alpha _{3}); \end{aligned}$$

for \(n=2\), we get

$$\begin{aligned} \alpha _{5}= & {} \frac{1}{60(\alpha _{0}-1)}(3a_{1}\alpha _{0}\alpha _{3}+3a_{1}\alpha _{1}\alpha _{2}+12a_{2}\alpha _{1}\alpha _{4}+18a_{2}\alpha _{2}\alpha _{3} -24\alpha _{1}\alpha _{4}\\&-6\alpha _{2}\alpha _{3}-\alpha _{2}-3\alpha _{3}+12\alpha _{4}), \end{aligned}$$

and so on.

Therefore, the exact analytic solution of Eq. (4.2) can be written as following

$$\begin{aligned} f(\xi )=\alpha _{0}+\alpha _{1}\xi +\alpha _{2}\xi ^{2}+\alpha _{3}\xi ^{3}+{\sum _{n=1}^\infty \alpha _{n+3}\xi ^{n+3}}. \end{aligned}$$
(4.7)

Thus, the exact power series solution of Eq. (1.1) is

$$\begin{aligned} u(x,t)=\, & {} \alpha _{0}t^{-1}+\alpha _{1}t^{-1}(x-\ln t)+\alpha _{2}t^{-1}(x-\ln t)^{2}+\alpha _{3}t^{-1}(x-\ln t)^{3}\nonumber \\&+{\sum _{n=1}^\infty \alpha _{n+3}t^{-1}(x-\ln t)^{n+3}}, \end{aligned}$$
(4.8)

where \(\alpha _{0}\ne 1\), \(\alpha _{1}\) and \(\alpha _{2}\) are arbitrary constants, the other coefficients \(\alpha _{i}\)\((i\ge 3)\) can be determined successively from (4.5) and (4.6).

Remark 4.1

 

  1. (i)

    We note that the recurrence formula (4.6) cannot include (4.5) as its special case, and the condition \(\alpha _{0}-1\ne 0\) is necessary for the two formulas.

  2. (ii)

    In view of \(\alpha _{0}\ne 1\), \(\alpha _{1}\) and \(\alpha _{2}\) are arbitrary constants, so (4.7) is the general solution of Eq. (4.2) in power series form actually.

5 Conclusion and Remarks

In this paper, we get the symmetries and Lie algebra of nonlinear shallow water wave equation by means of Lie group analysis. Then, the nonlinear wave equation is transformed into a ODE and dynamic system by using the traveling wave transform, which derived from the combination of two symmetries. Furthermore, the bifurcations under different parameters are given with the help of Maple symbol calculation procedure, and the exact traveling wave solutions are provided. Finally, The exact analytic solution are obtained by the power series method. The results enrich the types of the solutions of nonlinear wave equations, and the combination of Lie symmetry analysis and dynamic system method is a powerful approach to dealing with exact solutions of nonlinear wave equations. We hope to study the combination approach further in the future.