Some invariant solutions and conservation laws of a type of long-water wave system

Abstract

We propose a generalized long-water wave system that reduces to the standard water wave system. We also obtain the Lax pair and symmetries of the generalized shallow-water wave system and single out some their similarity reductions, group-invariant solutions, and series solutions. We further investigate the corresponding self-adjointness and the conservation laws of the generalized system.

Introduction

The classical dispersiveless long wave equations

$$ \textstyle\begin{cases} u_{t}+uu_{x}+h_{x}=0, \\ h_{t}+(uh)_{x}=0, \end{cases} $$

(1)

have a number of dispersive generalizations [1]. Kupershmidt [2] considered the following extension of (1):

$$ \textstyle\begin{cases} u_{t}=(\frac{1}{2}u^{2}+h-\beta u_{x})_{x}, \\ h_{t}=(uh+\alpha u_{xx}-\beta h_{x})_{x}, \end{cases} $$

(2)

where α, β are arbitrary constants. The invertible change of variables $u=\bar{u}, h=\bar{h}+\gamma \bar{u}x$, turns (2) into

$$ \textstyle\begin{cases} \bar{u}_{t}=(\frac{1}{2}\bar{u}^{2}+\bar{h}+\mu \bar{u}_{x})_{x}, \\ \bar{h}_{t}=(\bar{u}\bar{h}-\mu \bar{h}_{x})_{x},\mu =\gamma +\beta = \pm \sqrt{\alpha +\beta ^{2}}. \end{cases} $$

Broer [1] derived system (2) for $\alpha =\frac{1}{3}, \beta =0$, for which it is the proper Boussinesq equation. In terms of the potential $\varphi :u=\varphi _{x}$, system (2) was derived by Kaup [3]. Later, Matveev and Yavor [4] found algebrogeometrically a large class of almost periodic solutions. Li, Ma, and Zhang [5] used a scaling transformation to transfer a nonlinear long wave equation of Boussinesq class to the Broer–Kaup (BK) system, a type of long water wave equations:

$$ \textstyle\begin{cases} v_{t}=\frac{1}{2}(v^{2}+2w-v_{x})_{x}, \\ w_{t}=(vw+\frac{1}{2}w_{x})_{x}. \end{cases} $$

Furthermore, some exact solutions and Darboux transformations of (1) were obtained by applying the Lax-pair method. In terms of [5], we can study the similarity reduction, exact solutions, and conservation laws of the Boussinesq system through the scalar transformation

$$ v=-u,w=\xi +1+\frac{v_{x}}{2}, $$

that is, we can transform the BK system to the Boussinesq system

$$ \textstyle\begin{cases} \xi _{t}+[(1+\xi )u]_{x}=-\frac{1}{4}u_{xxx}, \\ u_{t}+uu_{x}+\xi _{x}=0, \end{cases} $$

where ξ is the elevation of the water wave, u is the surface velocity of water along the x-direction. Hence, the results of the paper have certain physical sense.

In the paper, we construct a generalized BK system as follows:

$$ \textstyle\begin{cases} v_{t}=\frac{\alpha }{2}(v_{x}-v^{2}-2w)_{x}-\beta v_{x}, \\ w_{t}=-\frac{\alpha }{2}(w_{x}+2wv)_{x}-\beta w_{x}, \end{cases} $$

where α, β are constants, so that some symmetries of (2) are produced by the symmetry group method [6]. It follows that some similarity solutions, group-invariant solutions, and series solutions are produced. In addition, Ibragimov and Avdonina [7] showed how to apply the symmetries of differential equations to study the self-adjointness and conservation laws. Thus we would like to follow the approach to investigate the quasiself-adjointness and conservation laws of the generalized BK system (2).

Integrability of (2)

Set

$$ \varphi _{x}=U\varphi ,\qquad \varphi _{t}=V\varphi , $$

(3)

where

$$\begin{aligned}& U=\begin{pmatrix} -\lambda +\frac{v}{2} & 1 \\ -w & \lambda -\frac{v}{2} \end{pmatrix}, \\& V=\begin{pmatrix} \alpha \lambda ^{2}+\beta \lambda +\frac{\alpha }{4}v_{x}-\frac{\alpha }{4}v^{2}-\frac{\beta }{2}v & -\alpha \lambda -\frac{\alpha }{2}v- \beta \\ \alpha w\lambda +\frac{\alpha }{2}w_{x}+\frac{\alpha }{2}wv+\beta w &- \alpha \lambda ^{2}-\beta \lambda -\frac{\alpha }{4}v_{x}+\frac{\alpha }{4}v^{2}+\frac{\beta }{2}v \end{pmatrix}. \end{aligned}$$

Then the compatibility condition of (3)

$$ V_{t}-U_{x}+UV-VU=0 $$

admits the generalized BK system, which can be directly verified. Hence the generalized BK system (2) is Lax integrable. Using (3), we can get some Darboux transformations for deducing solutions of the system. Here we omit them.

Similarity solutions and group-invariant solutions

Applying the Lie symmetry analysis, we can get the symmetry of system (2):

$$ X_{1}=\frac{\partial }{\partial t},\qquad X_{2}=\frac{\partial }{ \partial x},\qquad X_{3}= \biggl( \frac{1}{2}x+\beta t \biggr)\frac{ \partial }{\partial x}+t\frac{\partial }{\partial t}- \frac{1}{2}v\frac{ \partial }{\partial v}-w\frac{\partial }{\partial w}. $$

(4)

The vector field $X_{3}$ has the following characteristic equation:

$$ \frac{dt}{t}=\frac{dx}{\frac{1}{2}x+\beta t}= \frac{dv}{-\frac{1}{2}v}= \frac{dw}{-w}, $$

(5)

which gives rise to

$$ \biggl(\beta t+\frac{1}{2}x\biggr)\,dt-t\,dx=0. $$

(6)

One integration factor of (6) is given by

$$ \mu =e^{-\int \frac{3}{2t}\,dt}=t^{-\frac{3}{2}}, $$

which transfers (6) to the complete integration equation

$$ \beta t^{-\frac{1}{2}}\,dt+d \bigl(-t^{-\frac{1}{2}}x \bigr)=0, $$

from which we have the invariant variable $\xi =2\beta t^{\frac{1}{2}}-t ^{-\frac{1}{2}}x$. In terms of Eq. (5), we have the formal invariants

$$ v=t^{-\frac{1}{2}}f(\xi ),\qquad w=t^{-1}g(\xi ), $$

(7)

where $f(\xi )$ and $g(\xi )$ are arbitrary smooth functions of ξ. Substituting (7) into system (2) yields the ordinary differential system

$$ \textstyle\begin{cases} -\frac{1}{2}f(\xi )+\frac{1}{2}xt^{-\frac{1}{2}}f'(\xi )=\frac{\alpha }{2}[f''(\xi )+2f(\xi )f'(\xi )+2g'(\xi )], \\ -g(\xi )+\frac{1}{2}xt^{-\frac{1}{2}}g'(\xi )=-\frac{\alpha }{2}[g''( \xi )-2g'(\xi )f(\xi )-2g(\xi )f'(\xi )]. \end{cases} $$

(8)

Let $\beta =0$, Then system (8) reduces to

$$ \textstyle\begin{cases} f(\tau )+\tau f'(\tau )=-\alpha [f''(\tau )+2f(\tau )f'(\tau )+2g'( \tau )], \\ g(\tau )+\frac{1}{2}\tau g'(\tau )=\frac{\alpha }{2}[g''(\tau )-2g'( \tau )f(\tau )-2g(\tau )f'(\tau )], \end{cases} $$

(9)

where $\tau =-t^{-\frac{1}{2}}x$, which is a reduction of ξ. In facr, system (9) is an ordinary differential system corresponding to the BK system (1).

The group-invariant transformations of the generalized BK system are as follows:

$$ \textstyle\begin{cases} g_{1}:(x,t,v,w)\rightarrow (x,t+\epsilon ,v,w), \\ g_{2}:(x,t,v,w)\rightarrow (x+\epsilon ,t,v,w), \\ g_{3}:(x,t,v,w)\rightarrow (2\beta te^{\epsilon }+(x-2\beta t)e^{ \frac{1}{2}\epsilon },te^{\epsilon },e^{-\frac{1}{2}\epsilon }v,we ^{-\epsilon }). \end{cases} $$

(10)

In what follows, we consider solutions to the BK system. Set $v=V(\rho )$, $w=W(\rho )$, $\rho =x+lt$. Then (2) becomes

$$ \textstyle\begin{cases} lV'=\frac{\alpha }{2}(V''-2VV'-2W')-\beta V', \\ lW'=-\frac{\alpha }{2}(W''+2W'V+2WV')-\beta W', \end{cases} $$

from which we have

$$ \textstyle\begin{cases} lV-\frac{\alpha }{2}(V'-V^{2}-2W)+\beta V=c_{1}, \\ lW+\frac{\alpha }{2}(W'+2WV)-\beta W'=c_{2}. \end{cases} $$

(11)

A special solution to (11) is given by

$$ V=\frac{1}{c+\xi },\qquad W=\frac{1}{(c+\xi )^{2}} $$

(12)

in the case of $l=-\beta$, $c_{1}=c_{2}=0$. Hence we get a set of solutions to the generalized BK system (2):

$$ v=\frac{1}{c+x-\beta t},\qquad w=\frac{1}{(c+x-\beta t)^{2}}. $$

(13)

Applying the group-invariant transformation (10), we can deduce some other new solutions to system (2):

$$ \textstyle\begin{cases} g_{1}:\quad v=\frac{1}{c+x-\beta (t+\epsilon )},\qquad w=\frac{1}{[c+x- \beta (t+\epsilon )]^{2}}, \\ g_{2}:\quad v=\frac{1}{c+x-\beta t+\epsilon },\qquad w=\frac{1}{(c+x-\beta t+ \epsilon )^{2}}, \\ g_{3}:\quad v=\frac{e^{-\frac{1}{2}\epsilon }}{c+\beta te^{\epsilon }+(x-2 \beta t)e^{\frac{1}{2}\epsilon }},\qquad w=\frac{e^{-\epsilon }}{[c+ \beta te^{\epsilon }+(x-2\beta t)e^{\frac{1}{2}\epsilon }]^{2}}. \end{cases} $$

Taking $\beta =0$, we can obtain group-invariant solutions to the BK system (1). In particular, we can get the series solutions to the BK system. Indeed, let

$$ f(\tau )=\sum_{n=0}^{\infty }c_{n} \tau ^{n},\qquad g(\tau )=\sum_{m=0} ^{\infty }c_{m}\tau ^{m}, $$

(14)

and substituting into (9), we have that

$$\begin{aligned}& \begin{gathered} c_{0}+\sum_{n=1}^{\infty }c_{n} \tau ^{n}+\tau c_{1}+\sum_{n=1}^{ \infty }(n+1)c_{n+1} \tau ^{n+1}\\ \quad =-\alpha \Biggl[2c_{2}+\sum _{n=1}^{\infty }(n+2) (n+1)c _{n+2}\tau ^{n}\Biggr] +2\Biggl(c_{0}+\sum _{n=1}^{\infty }c_{n}\tau ^{n}\Biggr) \Biggl(c_{1}+ \sum _{n=1}^{\infty }(n+1)c_{n+1}\tau ^{n}\Biggr)\\ \qquad {}+2d_{1} +2\sum_{m=1}^{\infty }(m+1)d_{m+1} \tau ^{m}, \end{gathered} \\& d_{0}+\sum_{m=1}^{\infty }d_{m} \tau ^{m}+\frac{1}{2}\tau d_{1}+ \frac{1}{2}\sum_{m=1}^{\infty }(m+1)d_{m+1} \tau ^{m+1} \\& \quad = \frac{\alpha }{2}[2d_{2}+\sum _{m=1}^{\infty }(m+2) (m+1)d_{m+2}\tau ^{m}-2\Biggl(c_{0} +\sum_{n=1}^{\infty }c_{n} \tau ^{n}\Biggr) \Biggl(d_{1}+\sum _{m=1}^{ \infty }(m+1)d_{m+1}\tau ^{m}\Biggr)\\& \qquad {}-2\Biggl(d_{0}+\sum _{m=1}^{\infty }d_{m}\tau ^{m}\Biggr) \Biggl(c_{1}+\sum _{n=1}^{\infty }(n+1)c_{n+1}\tau ^{n}\Biggr), \end{aligned}$$

from which we infer that

$$\begin{aligned}& c_{2}=-c_{0}c_{1}-\frac{1}{2\alpha }c_{0}-d_{1}, \\& d_{2}=c_{0}d_{1}+d_{0}c_{1}+ \frac{1}{\alpha }d_{0}, \\& c_{3}=-\frac{1}{3\alpha }c_{1}- \frac{2}{3}c_{0}c_{2}-\frac{1}{6}c _{1}^{2}-\frac{2}{3}d_{1}, \\& d_{3}=\frac{1}{2\alpha }d_{1}+\frac{2}{3}c_{0}d_{2}- \frac{1}{3}d_{1}c _{1}+\frac{2}{3}d_{0}c_{2}, \\& \cdots \cdots , \\& c_{n+2}=\frac{1}{\alpha (n+1)(n+2)}\Biggl[-c_{n}-2\alpha c_{0}(n+1)c_{n+1}-2 \alpha c_{1}c_{n}\\& \hphantom{c_{n+2}=}{}- \alpha \sum_{i,j=2}^{n}c_{i}c_{j+1}(j+1) \tau ^{i+j}-2 \alpha (n+1)d_{n+1}\Biggr], \\& d_{n+2}=\frac{1}{(n+1)(n+2)}\Biggl[\frac{2}{\alpha }d_{n}+2(n+1)c_{0}d_{n+1} +2d_{1}c_{n}+2\sum_{i,j=2}^{n}c_{i}d_{j+1} \tau ^{i+j}\\& \hphantom{d_{n+2}=}{}+2(n+1)d_{0}c_{n+1}+2c _{1}d_{n} +2\sum_{i,j=2}^{n}(j+1)d_{i}c_{j+1} \tau ^{i+j}\Biggr], \end{aligned}$$

where $c_{0}$, $d_{0}$, $c_{1}$, $d_{1}$ are arbitrary parameters. Inserting these expressions into (14), we get the series solutions of the BK system. The second equation of system (9) can be reduced to

$$ g''(\tau )-\frac{1}{\alpha } \tau g'(\tau )-\frac{2}{\alpha }g(\tau )=0 $$

(15)

under the condition

$$ (fg)'=0\quad \Rightarrow\quad fg=c. $$

(16)

As long as the solution of (15) is obtained, we can get the solution $f(\tau )$ from (16). If $g_{1}(\tau )$ is the known solution of (15), then we assume that $g(\tau )=u(\tau )g_{1}(\tau )$. If $u(\tau )$ is known, then the solution $g(\tau )$ to Eq. (15) can be presented. It is easy to see that

$$ g''(\tau )=g_{1}( \tau )u''(\tau )+2u'(\tau )g_{1}'(\tau )+u(\tau )g _{1}''( \tau ). $$

(17)

Substituting (17) into Eq. (15) yields

$$\begin{aligned}& g_{1}(\tau )u''(\tau )+ \biggl(2g_{1}'(\tau )-\frac{1}{\alpha } \tau g_{1}(\tau ) \biggr)u'(\tau )+ \biggl(g_{1}''( \tau )-\frac{1}{ \alpha }\tau g_{1}'(\tau )- \frac{2}{\alpha } g_{1}(\tau ) \biggr)u( \tau )=0. \end{aligned}$$

Since

$$ g_{1}''(\tau )-\frac{1}{\alpha } \tau g_{1}'(\tau )-\frac{2}{\alpha }g _{1}(\tau )=0, $$

we have

$$ g_{1}(\tau )u''(\tau )+ \biggl(2g_{1}'(\tau )-\frac{1}{\alpha }\tau g _{1}(\tau ) \biggr)u'(\tau )=0. $$

(18)

Assume that $u'(\tau )=z(\tau )$. Then Eq. (18) becomes

$$ g_{1}(\tau )z'(\tau )+ \biggl(2g_{1}'( \tau )-\frac{1}{\alpha }\tau g _{1}(\tau ) \biggr)z(\tau )=0, $$

which has the solution

$$ z=\frac{c}{g_{1}^{2}(\tau )}e^{\int \frac{1}{\alpha }\tau\, d\tau }=\frac{c}{g _{1}^{2}(\tau )}e^{\frac{1}{2\alpha }\tau ^{2}}, $$

where c is a constant. Thus we have

$$\begin{aligned}& \begin{gathered} u(\tau )=c \int^{\tau }\frac{1}{g_{1}^{2}(\tau )}e^{\frac{\tau ^{2}}{2 \alpha }}\,d \tau +\bar{c}, \\ g(\tau )=g_{1}(\tau ) \biggl[c \int^{\tau } \frac{1}{g_{1}^{2}(\tau )}e^{\frac{\tau ^{2}}{2\alpha }}\,d \tau +\bar{c} \biggr]. \end{gathered} \end{aligned}$$

(19)

Substituting (19) into Eq. (16), we can get $f(\tau )$. Thus a type of special solutions to system (9) can be obtained.

The self-adjointness of system (2)

Ibragimov [8] introduced a few related notations of the strict self-adjointness, the nonlinear self-adjointness, and the quasiself-adjointness. Let us recall them.

Let H be a Hilbert space with the scalar product $(u,v)$ defined by

$$ (Fu,v)=\bigl(u,F^{*}v\bigr),\quad u,v\in H, $$

(20)

where $F^{*}$ is the adjoint operator to a linear operator F. A special Hilbert space is given by

$$ H=\biggl\{ \int _{R^{n}} \bigl\vert f(x) \bigr\vert ^{2}\,dx \biggr\} $$

along with an inner product

$$ (u,v)= \int _{R^{n}} u(x)v(x)\,dx. $$

Let F be a linear differential operator in H whose action on the function u is expressed by $F[u]$. Then Eq. (20) becomes

$$ \bigl(F[u],v\bigr)=\bigl(u,F^{*}[v]\bigr), $$

which means that

$$ vF[u]-uF^{*}[v]=D_{i}\bigl(\xi ^{i}\bigr), $$

(21)

where $D_{i}=\frac{\partial }{\partial x^{i}}+u_{i}^{\alpha } \partial _{u^{\alpha }}+u_{ij}^{\alpha }\partial _{u_{j}^{\alpha }}+ \cdots$ .

For the differential equations

$$ F_{\alpha }(x,u,u_{x_{i}},u_{x_{i}x_{j}}, \dots )=0,\quad \alpha =1, \dots ,m, $$

(22)

where $u=(u^{1},\ldots ,u^{m})$. The adjoint equations to (22) are as follows:

$$ F_{\alpha }^{*}(x,u,v,u_{x_{i}},v_{x_{i}}, \ldots)=0,\quad \alpha =1,\ldots,m, $$

(23)

with $F_{\alpha }^{*}=\frac{\delta \mathcal{\varphi }}{\delta u^{ \alpha }}$. The Lagrangian φ for (22) is defined by

$$\begin{aligned}& \begin{gathered} \mathcal{\varphi }=v^{\beta }F_{\beta }=:\sum _{\beta =1}^{m}v^{\beta }F_{\beta }, \\ \frac{\delta }{\delta u^{\alpha }}=\frac{\partial }{\partial u^{ \alpha }}+\sum_{j=1}^{\infty }(-1)^{j}D_{i_{1}} \cdots D_{i_{j}}\frac{ \partial }{\partial u_{i_{1}\cdots i_{j}}^{\alpha }}. \end{gathered} \end{aligned}$$

(24)

Definition 1

([7, 8])

The differential Eqs. (22) are said to be strictly self-adjoint if their adjoint Eqs. (23) are equivalent to (23) upon the substitution $v=u$. That is, the equation

$$ F^{*}(x,u,u,u_{x_{i}},u_{x_{i}},\ldots )=\lambda F(x,u,u_{x},\ldots ) $$

holds with a coefficient λ.

Definition 2

([8])

Upon a substitution

$$ v=\varphi (u), $$

(25)

if (23) becomes (22), then we call (22) is quasiself-adjoint.

Definition 3

([7, 8])

Upon a substitution

$$ v=\varphi (x,u)\neq 0, $$

(26)

if (26) solves the adjoint Eqs. (23) for all the solutions of (22), then we call system (22) nonlinearly self-adjoint, that is, we have the following equations:

$$ F_{\alpha }^{*}(x,u,\varphi ,\ldots )= \lambda _{\alpha }^{\beta }F_{ \beta }(x,u,\ldots ). $$

(27)

It is easy to find that the strictly self-adjoint and quasiself-adjoint equations both are particular cases of the nonlinear self-adjoint equations.

For the generalized BK system (2), denoted by

$$ \textstyle\begin{cases} F=v_{t}-\frac{\alpha }{2}(v_{x}-v^{2}-2w)_{x}+\beta v_{x}, \\ G=w_{t}+\frac{\alpha }{2}(w_{x}+2wv)_{x}+\beta w_{x}, \end{cases} $$

the formal Lagrangian $\mathcal{L}$ can be written as $\mathcal{L}=pF+qG$, and the adjoint system of (2) is as follows:

$$ \textstyle\begin{cases} \frac{\delta \mathcal{\mathcal{L}}}{\delta v}=2\alpha pv_{x}-p_{t}-\frac{ \alpha }{2}p_{xx}+\alpha (pv)_{x}-\beta p_{x}-\alpha wq_{x}=0, \\ \frac{\delta \mathcal{\mathcal{L}}}{\delta w}=-\alpha p_{x}-q_{t}- \alpha (qv)_{x}+\frac{\alpha }{2}q_{xx}-\beta q_{x}=0. \end{cases} $$

(28)

Setting $p=\varphi (v,w)$ and $q=\psi (v,w)$ and substituting into (27), along with (28), we have

$$ \frac{\delta \mathcal{\mathcal{L}}}{\delta v}\biggm|_{p=\varphi ,q=\psi }= \lambda _{1}F+\mu _{1}G,\qquad \frac{\delta \mathcal{\mathcal{L}}}{\delta w}\biggm|_{ p=\varphi , q=\psi }= \lambda _{2}F+\mu _{2}G, $$

(29)

where $\lambda _{1}$, $\lambda _{2}$, $\mu _{1}$, $\mu _{2}$ are undetermined functions. It is easy to get

$$ \textstyle\begin{cases} p_{t}=\varphi _{v}v_{t}+\varphi _{w}w_{t},\qquad p_{x}=\varphi _{v}v_{x}+ \varphi _{w}w_{x}, \\ p_{xx}=\varphi _{vv}v_{x}^{2}+2\varphi _{vw}v_{x}w_{x}+\varphi _{ww}w _{x}^{2}+\varphi _{v}v_{xx}+\varphi _{w}w_{xx}, \\ q_{t}=\psi _{v}v_{t}+\psi _{w}w_{t},\qquad q_{x}=\psi _{v}v_{x}+\psi _{w}w _{x}, \\ q_{xx}=\psi _{vv}v_{x}^{2}+2\psi _{vw}v_{x}w_{x}+\psi _{ww}w_{x}^{2}+\psi _{v}v_{xx}+\psi _{w}w_{xx}. \end{cases} $$

Inserting all these results into (29) yields that

$$ \lambda _{1}=\mu _{1}=\lambda _{2}=\mu _{2}=0. $$

Therefore, for all solutions of system (2), (28) holds. Thus system (2) is nonlinearly self-adjoint.

Another expression of system (2) and some properties

Set

$$ v(x,t)=V \biggl(x,\frac{\alpha }{2}t \biggr)-\frac{\beta }{\alpha }, \qquad w(x,t)=W \biggl(x,\frac{\alpha }{2}t \biggr). $$

Then system (2) becomes

$$ \textstyle\begin{cases} V_{t}=V_{xx}-2VV_{x}-2W_{x}, \\ W_{t}=-W_{xx}-2W_{x}V-2WV_{x}, \end{cases} $$

(30)

which has the infinitesimal symmetries

$$ X=(2c_{1}t+c_{2})\partial _{t}+(c_{1}x+c_{3}t+c_{4}) \partial _{x}+\biggl(c _{1}v-\frac{1}{2}c_{3} \biggr)\partial _{V}+2c_{1}\partial _{W}, $$

where $c_{1}$, $c_{2}$, $c_{3}$, $c_{4}$ are constants. Obviously, when $c_{1}=c_{2}=c_{3}=0$ and $c_{4}=1$, we get $X_{1}=\partial _{x}$. When $c_{1}=c_{3}=c_{4}=0$ and $c_{2}=1$, we have $X_{2}=\partial _{t}$. When $c_{2}=c_{3}=c_{4}=0$ and $c_{1}=1$, we find $X_{3}=2t\partial _{t}+x \partial _{x}+\partial _{V}+2\partial _{W}$; $X_{i}$ $(i=1,2,3)$ all are particular cases of X.

Next, we consider the characteristic equation of X so that we can obtain the similarity reductions of system (30). The characteristic equation of X reads as

$$ \frac{dt}{2c_{1}t+c_{2}}=\frac{dx}{c_{1}x+c_{3}t+c_{4}}= \frac{dV}{-c _{1}v+\frac{1}{2}c_{3}}=\frac{dW}{-2c_{1}W}. $$

(31)

Case 1: $c_{1}=1$.

$$ \xi =\frac{x-c_{3}t+c_{4}-c_{2}c_{3}}{\sqrt{2t+c_{2}}},\qquad V= \frac{1}{2}c_{3}+ \frac{f(\xi )}{\sqrt{2t+c_{2}}},\qquad W=\frac{g( \xi )}{2(2t+c_{2})}. $$

(32)

System (30) reduces to

$$ \textstyle\begin{cases} -f(\xi )-\xi f'(\xi )+2f(\xi )f'(\xi )-f''(\xi )+g'(\xi )=0, \\ -2g(\xi )-\xi g'(\xi )+g''(\xi )+2g(\xi )f'(\xi )+2g'(\xi )f(\xi )=0. \end{cases} $$

(33)

Case 2: $c_{1}=c_{2}=0$. Equation (31) becomes

$$ \frac{dt}{0}=\frac{dx}{c_{3}t+c_{4}}=\frac{dV}{\frac{1}{2}c_{3}}= \frac{dW}{0}. $$

We take

$$ \xi =t,\qquad W=W(t),\qquad V=\frac{c_{3}x}{2(c_{3}t+c_{4})}- \frac{1}{2}c_{3}f(t). $$

Then system (30) reduces to

$$ \textstyle\begin{cases} c_{3}\xi f'(\xi )+c_{3}f(\xi )+c_{4}f'(\xi )=0, \\ c_{3}\xi W'(\xi )+c_{4}W'(\xi )+c_{3}W(\xi )=0. \end{cases} $$

(34)

The two equations are in fact the same.

Case 3: $c_{1}=0$, $c_{2}\neq0$. Equation (31) reduces to

$$ \frac{dt}{c_{2}}=\frac{dx}{c_{3}t+c_{4}}=\frac{dV}{\frac{1}{2}c_{3}}= \frac{dW}{0}. $$

We choose

$$ \xi =c_{2}x-\frac{1}{2}c_{3}t^{2}-c_{4}t, \qquad V= \frac{c_{3}t}{2c_{3}}+\frac{f(\xi )}{c_{2}},\qquad W=g(\xi ). $$

Thus system (30) turns to

$$ \textstyle\begin{cases} -2c_{4}f'(\xi )+c_{3}-2c_{2}^{2}f''(\xi )+4f(\xi )f'(\xi )+4c_{2}^{2}g'( \xi )=0, \\ -c_{4}g'(\xi )+c_{2}^{2}g''(\xi )+2g'(\xi )f(\xi )+2g(\xi )f'(\xi )=0. \end{cases} $$

(35)

System (35) has the particular solutions

$$ f(\xi )=\frac{1}{2}\xi ^{2}-\xi ^{-1},\qquad g(\xi )=c\xi , $$

where ξ satisfies the constraint

$$ \xi ^{3}-\frac{3}{2}\xi ^{2}+c-2=0. $$

Thus from (32) we get a set of new solutions of system (2):

$$ \textstyle\begin{cases} v(x,t)=\frac{1}{2}c_{3}+\frac{1}{\sqrt{\alpha t+c_{2}}}(\frac{1}{2}\frac{(x-c _{3}t+c_{4}-c_{2}c_{3})^{2}}{2t+c_{2}}-\frac{\sqrt{2t+c_{3}}}{x-c _{3}t+c_{4}-c_{2}c_{3}})-\frac{\beta }{\alpha }, \\ w(x,t)=\frac{c}{2}\frac{x-c_{3}t+c_{4}-c_{2}c_{3}}{(\alpha t+c_{2})^{ \frac{3}{2}}}. \end{cases} $$

In what follows, we consider the series solutions of (33).

Setting

$$ f(\xi )=\sum_{i=0}^{\infty }a_{i} \xi ^{i}, \qquad g(\xi )=\sum_{i=0}^{\infty }b_{i} \xi ^{i} $$

and substituting into system (33), we infer that

$$\begin{aligned}& \textstyle\begin{cases} a_{0}+2a_{0}a_{1}-2a_{2}+b_{1}=0, \\ 4a_{0}a_{2}+2a_{1}^{2}-6a_{3}+2b_{2}=0, \\ -a_{2}+2(3a_{0}a_{3}+3a_{1}a_{2})-12a_{4}+3b_{3}=0, \end{cases}\displaystyle \\& a_{n}-na_{n}+2\sum_{i,j=1}^{n}a_{i}(j+1)a_{j+1}-(n+2)!a_{n+2}+(n+1)b _{n+1}=0, \end{aligned}$$

(36)

$$\begin{aligned}& \textstyle\begin{cases} -2b_{0}+2b_{2}+2b_{0}a_{1}+2b_{1}a_{0}=0, \\ -3b_{1}+6b_{3}+2(2b_{0}a_{2}+b_{1}a_{1})+2(b_{1}a_{1}+2b_{2}a_{0})=0, \\ -4b_{2}+12b_{4}+2(3b_{0}a_{3}+2b_{1}a_{2}+a_{1}b_{2})+2(b_{1}a_{2}+2b _{2}a_{1}+3b_{3}a_{0})=0, \\ \cdots \end{cases}\displaystyle \\& -2b_{n}-(n+1)b_{n+1}+(n+2)!b_{n+2}+2\sum _{i,j=1}^{n}b_{i}(j+1)a_{j} +2 \sum_{i,j=1}^{n}a_{i}(j+1)b_{j+1}=0, \end{aligned}$$

(37)

from which we get

$$ \textstyle\begin{cases} a_{2}=\frac{1}{2}a_{0}+a_{0}a_{1}+\frac{1}{2}b_{1}, \\ b_{2}=b_{0}-a_{1}b_{0}-a_{0}b_{1}, \\ a_{3}=\frac{1}{3}(2a_{0}a_{2}+a_{1}^{2}+b_{2}), \\ b_{3}=\frac{1}{2}b_{1}-\frac{1}{3}(a_{1}b_{1}+2a_{2}b_{0})- \frac{1}{3}(b_{1}a_{1}+2b_{2}a_{0}), \\ \cdots \end{cases} $$

where $a_{0}$, $b_{0}$, $a_{1}$, $b_{1}$ are arbitrary parameters. Thus we obtain the following formal series solutions of system (33):

$$\begin{aligned}& f(\xi )=a_{0}+a_{1}\xi + \biggl(\frac{1}{2}a_{0}+a_{0}a_{1}+ \frac{1}{2}b_{1} \biggr)\xi ^{2} + \frac{1}{3}\bigl(2a_{0}a_{2}+a_{1}^{2}+b_{2} \bigr)\xi ^{3}+\sum_{i=4}^{\infty }a _{i}\xi ^{i}, \end{aligned}$$

(38)

$$\begin{aligned}& g(\xi )=b_{0}+b_{1}\xi +(b_{0}-a_{1}b_{0}-a_{0}b_{1}) \xi ^{2} \\& \hphantom{g(\xi )=}{}+\biggl[ \frac{1}{2}b_{1}- \frac{1}{3}(2a_{1}b_{1}+2a_{2}b_{0}+2b_{2}a_{0}) \biggr]\xi ^{3}+\sum_{i=4}^{\infty }b_{i} \xi ^{i}, \end{aligned}$$

(39)

where $a_{i},b_{i}$ ($i=4,5,\ldots$) satisfy (36) and (37). Substituting (38)and (39) into (32), we can get the series solutions of the generalized BK system.

Next, we consider the solutions to system (34). It is easy to see that

$$ g(\xi )=f(\xi )=-\xi -\frac{c_{4}}{c_{3}}\quad \text{or}\quad g( \xi )=f(\xi )=\frac{\hat{c}}{\xi +\frac{c_{4}}{c_{3}}}, $$

(40)

where ĉ is an integration constant.

System (35) is solvable similarly to system (33), and we omit the computations.

Conservation laws

In this section, we consider the conservation laws of the generalized BK system by using the method in [7, 8]. From the identity

$$ X+D_{i}\bigl(\xi ^{i}\bigr)=W^{\alpha } \frac{\delta }{\delta u^{\alpha }}+D_{i}N ^{i} $$

we find that

$$ X(\mathcal{L})+D_{i}\bigl(\xi ^{i} \bigr)\mathcal{L}=W^{\alpha }\frac{\delta \mathcal{L}}{\delta u^{\alpha }}+D_{i} \bigl[N^{i}(\mathcal{L})\bigr], $$

(41)

where

$$ \textstyle\begin{cases} X=\xi ^{i}\partial _{x_{i}}+\eta ^{\alpha }\frac{\partial }{\partial u ^{\alpha }}+\xi _{i}\frac{\partial }{\partial u_{i}^{\alpha }}+\cdots , \\ N^{i}=\xi ^{i}+W^{\alpha }\frac{\delta }{\delta u_{i}^{\alpha }}+\sum_{s=1}^{\infty }D_{i_{1}}\cdots D_{i_{s}}(w_{\alpha })\frac{\delta }{ \delta u_{ii_{1}\cdots i_{s}}},\quad i=1,2,\ldots ,n, \\ W^{\alpha }=\eta ^{\alpha }-\xi ^{j}u_{j}^{\alpha },\quad \alpha =1,\ldots ,m, \end{cases} $$

and $\mathcal{L}$ is the Euler–Lagrange function, which satisfies

$$ \frac{\delta \mathcal{L}}{\delta u^{\alpha }}=0,\quad \alpha =1,\ldots ,m. $$

Since system (28) holds, we can investigate the conservation laws by using (41), where the components of the conservation laws are the following:

$$ C^{i}=N^{i}(\mathcal{L}),\quad i=1,\ldots ,n, $$

(42)

which satisfy the conservation equations

$$ D_{i}\bigl(C^{i}\bigr)_{(22)}=0. $$

(43)

For $X_{1}=\frac{\partial }{\partial x}$, we find that

$$ W^{1,1}=-v_{x},\qquad W^{1,2}=-w_{x}. $$

(44)

Substituting (44) into (42) yields

$$ \textstyle\begin{cases} C_{v}^{1}=-\alpha vv_{x}(p+q)-(\alpha +\beta )v_{x}p-\alpha qwv_{x}- \beta qv_{x}+\frac{\alpha }{2}v_{x}(q_{x}-p_{x}) \\ \hphantom{C_{v}^{1}=}{}+\frac{\alpha }{2}v_{xx}-\frac{\alpha }{2}qv_{xx}, \\ C_{w}^{1}=-\alpha pvw_{x}-\beta pw_{x}-\alpha qww_{x}-\alpha pw_{x}- \alpha qvw_{x} \\ \hphantom{C_{w}^{1}=}{}-\beta qw_{x}-\frac{\alpha }{2}p_{x}w_{x}+\frac{\alpha }{2}w_{x}q_{x}+\frac{ \alpha }{2}pw_{xx}-\frac{\alpha }{2}qw_{xx}. \end{cases} $$

For $X_{2}=\frac{\partial }{\partial t}$, we get

$$ \textstyle\begin{cases} C_{v}^{2}=-v_{t}(p+q)=-(p+q)[-\beta v_{x}+\frac{\alpha }{2}(v_{x}-v ^{2}-2w)_{x}], \\ C_{w}^{2}=(p+q)[\beta w_{x}+\frac{\alpha }{2}(w_{x}+2wv)_{x}]. \end{cases} $$

For $X_{3}=(\frac{1}{2}x+\beta t)\partial _{x}+t\partial _{t}- \frac{1}{2}v\partial _{v}-w\partial _{w}$, we infer

$$ \textstyle\begin{cases} W^{3,1}=-\frac{1}{2}v-tv_{t}-(\frac{1}{2}x+\beta t)v_{x}, \\ W^{3,2}=-w-tw_{t}-(\frac{1}{2}x+\beta t)w_{x}, \\ C_{v}^{3}=[-\frac{1}{2}v-tv_{t}-(\frac{1}{2}x+\beta t)v_{x}][\alpha pv+ \beta p+2wq+\alpha p+\alpha qv+\beta \\ \hphantom{C_{v}^{3}=}{}+\frac{\alpha }{2}(p+q)[-\frac{1}{2}v_{x}-v_{xt}-\frac{1}{2}v_{x}-( \frac{1}{2}x+\beta t)v_{xt}], \\ C_{w}^{3}=[-w-tw_{t}-(\frac{1}{2}x+\beta t)w_{x}](\alpha pv+\beta p+2wq+ \alpha p+\alpha qv+\beta ) \\ \hphantom{C_{w}^{3}=}{}+\frac{\alpha }{2}(p+q)[-w_{x}-tw_{xt}-\frac{1}{2}w_{x}-(\frac{1}{2}x+ \beta t)w_{xx}], \end{cases} $$

where $v_{t}$, $w_{t}$ are given by system (2).

Remark

Anco and Bluman [9] proposed a method for constructing conservation laws of differential equations, which uses a formula directly generating the conservation laws and independent of the system having a Lagrangian formulation, in contrast to Noether’s theorem, which requires a Lagrangian. They adopted the linear equations and the adjoint equations of the original differential equations to study conservation laws. Essentially, the algorithm presented by Ibragimov et al. is the same as that of Anco and Bluman. Besides, Anco [10] also gave some comments on the work of Ibragimov.

Conclusions

In the paper, we have investigated various similarity reductions and exact solutions of the generalized BK system and various its conservation laws by the Lie group analysis. We have pointed out that the standard BK system is only a paticular case of the generalized BK system (2) when $\alpha =-1$ and $\beta =0$. In addition, Lou [11, 12] applied the symmetry group method to study some coherent solutions of nonlocal KdV systems and primary branch solutions of a first-order autonomous system. We hope to extend the methods to the systems presented in the paper in the forthcoming days. In addition, Ma [13] obtained some new conservation laws of some discrete evolution equation by symmetries and adjoint symmetries. Zhang, et al. [14, 15] considered symmetry properties of some fractional equations. Therefore there is an open problem how we can look for the fractional systems that correspond to the systems presented in the paper and how we can to solve them. Besides, Liu, Zhang, and Zhou [16] constructed the fractional Volterra hierarchy, gave a definition of the hierarchy in terms of Lax pair and Hamiltonian formalisms, and constructed its tau functions and multisoliton solutions. Bridgman, Hereman, Quispel, and Kamp [17] and El-Nabulsi [18] studied the peakon and Toda lattice. The approaches adopted in [16,17,18] can lead us to investigate some related properties of the generalized BK system presented in the paper. These questions will be discussed in the future.