Open Access

Explicit solutions and conservation laws of the logarithmic-KP equation

Advances in Difference Equations20162016:229

https://doi.org/10.1186/s13662-016-0948-6

Received: 29 March 2016

Accepted: 23 August 2016

Published: 1 September 2016

Abstract

In this paper, we study the logarithmic-KP equation. The analysis depends mainly on the Lie symmetry method. The corresponding vector fields and symmetry reductions are derived. Furthermore, the conservation laws of the equation are constructed.

Keywords

logarithmic-KP equation Lie symmetry method explicit solutions conservation laws

1 Introduction

Nonlinear evolution equations (NLEEs) have been used in many science fields, such as physics, chemistry, engineering, and other fields. The investigation of the explicit solutions of NLEEs gave rise to much research work. A great many of systematic and effective methods are used for investigating NLEEs. Some of the methods are the inverse scattering method [1], the Hirota bilinear method [2, 3], the Bäklund transformation method [46], Darboux transformation [7, 8], the Painlevé analysis [9], the Lie group method [1028], the solitary wave ansatz method [2936], and others.

Conservation laws (CLs) provide a important tool to investigate many problems involving mathematical physics. A systematic method for the determination of conservation laws is the famous Noether theorem [37]. Recently, the direct method was given in [11], a new method to construct the conservation laws was provided in [38].

The generalized KP equation is the so-called logarithmic-KP (log-KP) equation given by [39]
$$\begin{aligned} \bigl(v_{t}+ \bigl(v \ln{ \vert v\vert } \bigr)_{x}+v_{xxx} \bigr)_{x}+v_{yy}=0, \end{aligned}$$
(1)
where \(v(x,t)\) represents the wave profile. In [39], the authors studied the Gaussian solitary waves of the log-KP equation. More explications of the log-KP equation and its applications can be found in [39] and references therein. In [25], the authors studied the log-KdV equation. The KP equation appears in many important fields, such as water waves, ferromagnetic media, and so on. Kadomtsev and Petviashvili first derived the famous KP equation [40]. There are many papers dealing with these types of equations [3136, 39, 40]. We first employ the following transformation [39]:
$$\begin{aligned} v=e^{u}, \end{aligned}$$
(2)
we get
$$\begin{aligned} &u_{xt}+u_{xx}+u_{yy}+u_{xxxx}+3u_{xx}^{2}+u_{y}^{2} \\ &\quad{}+3u_{x}u_{xxx}+u_{x}^{2}+uu_{xx}+3u_{x}^{2}u_{xx}=0. \end{aligned}$$
(3)

In this paper, we use the Lie group method to deal with (3). The outline of the paper is as follows: In Section 2, the vectors fields are derived. In Section 3, symmetry reductions and explicit solutions are constructed. In Section 4, conservation laws are presented using the new conservation law theorem. The conclusions are presented in the final section.

2 Lie symmetry analysis

Suppose that (3) is invariant via the one-parameter Lie group
$$ \begin{aligned} &t^{*}=t+\varepsilon\xi_{t}(x,y,t,u)+O\bigl(\varepsilon^{2} \bigr),\\ &x^{*}=x+\varepsilon\xi_{x}(x,y,t,u)+O\bigl(\varepsilon^{2} \bigr),\\ &y^{*}=y+\varepsilon\xi_{y}(x,y,t,u)+O\bigl(\varepsilon^{2} \bigr),\\ &u^{*}=u+\varepsilon\eta(x,y,t,u)+O\bigl(\varepsilon^{2}\bigr), \end{aligned} $$
(4)
where ε is the group parameter, and the vector fields are
$$ V=\xi_{t}(x,y,t,u)\frac{\partial}{\partial{t}}+\xi_{x}(x,y,t,u) \frac {\partial}{\partial{x}}+\xi_{y}(x,y,t,u)\frac{\partial}{\partial{y}}+\eta (x,y,t,u) \frac{\partial}{\partial{u}}. $$
(5)
Here
$$ \begin{aligned} &\xi_{t}(x,y,t,u)=\frac{dt^{*}}{d\varepsilon}\bigg|_{\varepsilon=0},\qquad \xi_{x}(x,y,t,u)=\frac{dx^{*}}{d\varepsilon}\bigg|_{\varepsilon=0},\\ &\xi_{y}(x,y,t,u)=\frac{dy^{*}}{d\varepsilon}\bigg|_{\varepsilon=0}, \qquad\eta(x,y,t,u)= \frac{du^{*}}{d\varepsilon}\bigg|_{\varepsilon=0}. \end{aligned} $$
(6)
Under the assumption of the infinitesimal invariance criterion, one gets
$$\begin{aligned} pr^{(3)}V(\Delta)|_{\Delta=0}=0. \end{aligned}$$
(7)
According to the Lie group theory, one has
$$\begin{aligned} &\eta^{xt}+\eta^{xx}+\eta^{yy}+ \eta^{xxxx}+6\eta^{xx}u_{xx}+2u_{y}\eta ^{y}+3\eta^{x}u_{xxx} \\ &\quad{}+3u_{x}\eta^{xxx}+2u_{x}\eta^{x}+\eta u_{xx}+u\eta^{xx}+6u_{x}u_{xx} \eta ^{x}+3u_{x}^{2}\eta^{xx} =0. \end{aligned}$$
(8)
Putting (4) into (8), and letting all of the powers of derivatives of u be zero, one can obtain overdetermined systems. Solving the systems, one can get
$$ \xi_{t}=c_{1}, \qquad\xi_{x}=c_{3}F(t),\qquad \xi_{y}=c_{2}, \qquad\eta=c_{3}F_{t}, $$
(9)
where \(c_{1}, c_{2}\), and \(c_{3}\) are arbitrary constants, and F is a smooth function of t. Consequently, we have
$$\begin{aligned} V_{1}=\frac{\partial}{\partial{t}},\qquad V_{2}=\frac{\partial}{\partial{y}},\qquad V_{3}=F\frac{\partial}{\partial{x}}+F_{t}\frac{\partial}{\partial{u}}. \end{aligned}$$
(10)
In addition, solving the Lie equation
$$ \begin{aligned} &\frac{d(\bar{x}(\varepsilon))}{d\varepsilon}=\xi_{x}{ \bigl(\bar{x} (\varepsilon),\bar{y}( \varepsilon),\bar{t}(\varepsilon)},\bar {u}(\varepsilon) \bigr),\qquad\bar{x}(0)=x, \\ &\frac{d(\bar{y}(\varepsilon))}{d\varepsilon}=\xi_{y}{ \bigl(\bar{x} (\varepsilon),\bar{y}( \varepsilon),\bar{t}(\varepsilon)},\bar {u}(\varepsilon) \bigr),\qquad\bar{y}(0)=y, \\ &\frac{d(\bar{t}(\varepsilon))}{d\varepsilon}=\xi_{t}{ \bigl(\bar{x} (\varepsilon),\bar{y}( \varepsilon),\bar{t}(\varepsilon)},\bar {u}(\varepsilon) \bigr),\qquad\bar{t}(0)=t, \\ &\frac{d(\bar{u}(\varepsilon))}{d\varepsilon}=\eta{ \bigl(\bar{x} (\varepsilon),\bar{y}(\varepsilon), \bar{t}(\varepsilon)},\bar {u}(\varepsilon) \bigr),\qquad\bar{u}(0)=u, \end{aligned} $$
(11)
where ε is a group parameter, we get the Lie symmetry group,
$$\begin{aligned} g:(x,y,t,u)\rightarrow(\bar{x},\bar{y},\bar{t},\bar{u}). \end{aligned}$$
(12)
The associated one-parameter groups \(g_{i}(\varepsilon)\) generated by \(V_{i}\) for \(i=1, 2, 3 \) are
$$ \begin{aligned} &g_{1}:(x,y,t,u)\mapsto{(x,t+\varepsilon,y,u)},\\ &g_{2}:(x,y,t,u)\mapsto{(x,t,y+\varepsilon,y,u)},\\ &g_{3}:(x,y,t,u)\mapsto{(x+F\varepsilon,y,t, u+F_{t} \varepsilon)}. \end{aligned} $$
(13)
In addition, we get the following associated theorem.

Theorem 1

If \(u=f(x,y,t)\) is a solution of the logarithmic-KP equation, the functions
$$ \begin{aligned} &g_{1}(\varepsilon)\cdot{f(x,y,t)}=f(x,y,t-\varepsilon),\\ &g_{2}(\varepsilon)\cdot{f(x,y,t)}=f(x,y-\varepsilon,t),\\ &g_{3}(\varepsilon)\cdot{f(x,y,t)}=f(x-F\varepsilon,y, t)+F_{t}\varepsilon, \end{aligned} $$
(14)
are also solutions of (3).
Taking the following Gaussian solitary wave solution [30]:
$$\begin{aligned} u(x,t)=\frac{c}{k}+\frac{1}{2}-\frac{3k^{2}+2r^{2}}{12k^{4}}(kx+ry-ct)^{2}, \end{aligned}$$
(15)
we can derive a new explicit solution of (3) using \(g_{3}\),
$$\begin{aligned} u(x,t)=\frac{c}{k}+\frac{1}{2}-\frac{3k^{2}+2r^{2}}{12k^{4}} \bigl(k(x-F \varepsilon)+ry-ct \bigr)^{2}+F_{t}\varepsilon. \end{aligned}$$
(16)
Hence, one can get new solutions of (1),
$$\begin{aligned} v(x,t)=e^{\frac{c}{k}+\frac{1}{2}-\frac{1}{4k^{2}} (k(x-F\varepsilon )+ry-ct )^{2}+F_{t}\varepsilon}. \end{aligned}$$
(17)
In particular, letting \(F(t)=t^{2}\), one can get
$$\begin{aligned} v(x,t)=e^{\frac{c}{k}+\frac{1}{2}-\frac{1}{4k^{2}} (k(x-t^{2}\varepsilon )+ry-ct )^{2}+2t\varepsilon}, \end{aligned}$$
(18)
setting \(F(t)=\sin t\), one has
$$\begin{aligned} v(x,t)=e^{\frac{c}{k}+\frac{1}{2}-\frac{1}{4k^{2}} (k(x-\sin t\varepsilon)+ry-ct )^{2}+\cos t\varepsilon}, \end{aligned}$$
(19)
and setting \(F(t)=\tanh t\), one obtains
$$\begin{aligned} v(x,t)=e^{\frac{c}{k}+\frac{1}{2}-\frac{1}{4k^{2}} (k(x-\tanh t\varepsilon)+ry-ct )^{2}+(1-\tanh t^{2})\varepsilon}, \end{aligned}$$
(20)
setting \(F(t)=e^{t}\), one can arrive at
$$\begin{aligned} v(x,t)=e^{\frac{c}{k}+\frac{1}{2}-\frac{1}{4k^{2}} (k(x-e^{t}\varepsilon )+ry-ct )^{2}+e^{t}\varepsilon}, \end{aligned}$$
(21)
setting \(F(t)=\sin(e^{t})\), one can lead to
$$\begin{aligned} v(x,t)=e^{\frac{c}{k}+\frac{1}{2}-\frac{1}{4k^{2}} (k(x-\sin (e^{t})\varepsilon)+ry-ct )^{2}+\cos(e^{t})e^{t}\varepsilon}, \end{aligned}$$
(22)
setting \(F(t)=\ln(t)\), one can have
$$\begin{aligned} v(x,t)=e^{\frac{c}{k}+\frac{1}{2}-\frac{1}{4k^{2}} (k(x-\ln (t)\varepsilon)+ry-ct )^{2}+\frac{1}{t}\varepsilon}. \end{aligned}$$
(23)

Remark 1

Many new explicit solutions can be derived via the solutions obtained [30].

3 Symmetry reductions and explicit solutions

3.1 Symmetry reductions

In the present subsection, we will present symmetry reductions and explicit solutions of (3).

(1) \(V_{1}\).

For the generator \(V_{1}\), we have
$$\begin{aligned} f_{xx}+f_{yy}+f_{xxxx}+3f_{xx}^{2}+f_{y}^{2}+3f_{x}f_{xxx}+f_{x}^{2}+ff_{xx}+3f_{x}^{2}f_{xx}=0. \end{aligned}$$
(24)
For this equation, we found that it also is a PDE. In order to reduce this equation, once again, we use the Lie group method to deal with this equation. As in the previous step, one can get the corresponding vectors,
$$\begin{aligned} \Upsilon_{1}=\frac{\partial}{\partial{x}},\qquad \Upsilon_{2}= \frac{\partial }{\partial{y}}. \end{aligned}$$
(25)

(1.1) \(\Upsilon_{1}\).

For \(\Upsilon_{1}\), we have
$$\begin{aligned} g_{yy}+g_{y}^{2}=0. \end{aligned}$$
(26)
Solving this equation, one can get
$$\begin{aligned} g=\ln(c_{1}y+c_{2}). \end{aligned}$$
(27)
That is to say,
$$\begin{aligned} u=\ln(c_{1}y+c_{2}). \end{aligned}$$
(28)
Also, one can get
$$\begin{aligned} v=c_{1}y+c_{2}. \end{aligned}$$
(29)

(1.2) \(\Upsilon_{2}\).

For \(\Upsilon_{2}\), we get
$$\begin{aligned} g_{xx}+g_{xxxx}+3g_{xx}^{2}+3g_{x}g_{xxx}+g_{x}^{2}+gg_{xx}+3g_{x}^{2}g_{xx}=0. \end{aligned}$$
(30)

(2) \(V_{2}\).

In the case of \(V_{2}\), we get the group-invariant solution,
$$\begin{aligned}& u=f(x,t), \end{aligned}$$
(31)
$$\begin{aligned}& f_{xt}+f_{xx}+f_{xxxx}+3f_{xx}^{2}+3f_{x}f_{xxx}+f_{x}^{2}+ff_{xx}+3f_{x}^{2}f_{xx}=0. \end{aligned}$$
(32)
As in the previous step, we get the associated vectors,
$$\begin{aligned} \Gamma_{1}=\frac{\partial}{\partial{x}},\qquad \Gamma_{2}= \frac{\partial }{\partial{t}}. \end{aligned}$$
(33)

(2.1) \(\Gamma_{1}\).

For \(\Gamma_{1}\), we get the trivial solution,
$$\begin{aligned} u=c_{1}. \end{aligned}$$
(34)

(2.2) \(\Gamma_{2}\).

For \(\Gamma_{2}\), we arrive at
$$\begin{aligned} g_{xx}+g_{xxxx}+3g_{xx}^{2}+3g_{x}g_{xxx}+g_{x}^{2}+gg_{xx}+3g_{x}^{2}g_{xx}=0. \end{aligned}$$
(35)

(3) \(V_{3}\).

For this case, we get
$$\begin{aligned} u=\frac{g(y,t)}{F}+\frac{F_{t}x}{F}. \end{aligned}$$
(36)
Plugging (36) into (3), one arrives at
$$\begin{aligned} \frac{1}{F^{2}} \bigl(FF_{tt}+Fg_{yy}+g_{y}^{2} \bigr) =0. \end{aligned}$$
(37)
By solving this equation, one obtains
$$\begin{aligned} g=-\frac{1}{2}\ln \biggl( \frac{F_{tt}F}{ (F_{1} \sin (\frac{\sqrt{F_{tt}}y}{\sqrt{F}} )-F_{2} \cos (\frac {\sqrt{F_{tt}}y}{\sqrt{F}} ) )^{2}} \biggr) F. \end{aligned}$$
(38)
In this way, we get
$$\begin{aligned} u=-\frac{1}{2}\ln \biggl( \frac{F_{tt}F}{ (F_{1} \sin (\frac{\sqrt{F_{tt}}y}{\sqrt{F}} )-F_{2} \cos (\frac {\sqrt{F_{tt}}y}{\sqrt{F}} ) )^{2}} \biggr) +\frac{F_{t}x}{F}. \end{aligned}$$
(39)
Thus, one can get the explicit solution of (1),
$$\begin{aligned} v =&e^{u}=e^{-\frac{1}{2}\ln ( \frac{F_{tt}F}{ (F_{1} \sin (\frac{\sqrt{F_{tt}}y}{\sqrt{F}} )-F_{2} \cos (\frac {\sqrt{F_{tt}}y}{\sqrt{F}} ) )^{2}} ) +\frac{F_{t}x}{F}} \\ =& \biggl( \frac{F_{tt}F}{ (F_{1} \sin (\frac{\sqrt{F_{tt}}y}{\sqrt{F}} )-F_{2} \cos (\frac {\sqrt{F_{tt}}y}{\sqrt{F}} ) )^{2}} \biggr)^{-\frac {1}{2}}e^{\frac{F_{t}x}{F}}, \end{aligned}$$
(40)
where \(F_{tt}\neq{0}\), \(F_{1}\) and \(F_{2}\) are functions of t.
In particular, if we set \(F=t\), we have
$$\begin{aligned} u=\frac{g(y,t)}{t}+\frac{x}{t}. \end{aligned}$$
(41)
Substituting (41) into (3), we obtain
$$\begin{aligned} \frac{1}{t^{2}} \bigl(tg_{yy}+g_{y}^{2} \bigr) =0. \end{aligned}$$
(42)
Solving this equation, one can get
$$\begin{aligned} g=t\ln \biggl( \frac{F_{1}(t)y+F_{2}(t)}{t} \biggr). \end{aligned}$$
(43)
Therefore, one gets
$$\begin{aligned} u=\ln \biggl( \frac{F_{1}(t)y+F_{2}(t)}{t} \biggr)+\frac{x}{t}. \end{aligned}$$
(44)
Therefore, one can derive the explicit solution of (1),
$$\begin{aligned} v=e^{u}= \biggl( \frac{F_{1}(t)y+F_{2}(t)}{t} \biggr)e^{\frac{x}{t}}. \end{aligned}$$
(45)

(3) \(V_{1}+V_{2}\).

For this case, we get
$$\begin{aligned} u=f(\xi,\tau),\qquad \xi=x,\qquad\tau=y-t. \end{aligned}$$
(46)
Plugging (46) into (3), one arrives at
$$\begin{aligned} -f_{\xi\tau}+f_{\xi\xi}+f_{\tau\tau}+f_{\xi\xi\xi\xi}+3f_{\xi\xi }^{2}+f_{\tau}^{2}+3f_{\xi}f_{\xi\xi\xi}+f_{\xi}^{2}+ff_{\xi\xi}+3f_{\xi}^{2}f_{\xi\xi}=0. \end{aligned}$$
(47)
As in the previous step, we obtain the associated vectors,
$$\begin{aligned} \Gamma_{1}=\frac{\partial}{\partial{\xi}},\qquad \Gamma_{2}= \frac{\partial }{\partial{\tau}}. \end{aligned}$$
(48)

(3.1) \(\Gamma_{1}\).

For \(\Gamma_{1}\), we get
$$\begin{aligned} f=g(\tau) \end{aligned}$$
(49)
and
$$\begin{aligned} g_{\tau}^{2}+g_{\tau\tau}=0. \end{aligned}$$
(50)

(3.2) \(\Gamma_{2}\).

For \(\Gamma_{2}\), we have
$$\begin{aligned} g_{\xi\xi}+g_{\xi\xi\xi\xi}+3g_{\xi\xi}^{2}+3g_{\xi}g_{\xi\xi\xi}+g_{\xi}^{2}+gg_{\xi\xi}+3g_{\xi}^{2}g_{\xi\xi}=0. \end{aligned}$$
(51)

(3.3) \(\Gamma_{2}+\lambda\Gamma_{1}\) (traveling wave transformation).

For this case, we get
$$\begin{aligned} f=g(\pi),\qquad \pi=\xi-\tau=x-\lambda(y-t) \end{aligned}$$
(52)
and
$$\begin{aligned} &\lambda g_{\pi\pi}+g_{\pi\pi}+\lambda^{2}g_{\pi\pi}+g_{\pi\pi\pi\pi }+3g_{\pi\pi}^{2}+ \lambda^{2} g_{\pi}^{2}+3g_{\pi}g_{\pi\pi\pi} \\ &\quad{}+g_{\pi}^{2}+gg_{\pi\pi}+3g_{\pi}^{2}g_{\pi\pi}=0. \end{aligned}$$
(53)

4 Conservation laws

In the present section, we derive the conservation laws of the logarithmic-KP equation.

4.1 Necessary preliminaries

For a conserved vector the following conservation equation holds:
$$\begin{aligned} D_{t}\bigl(C^{t}\bigr)+D_{x}\bigl(C^{x} \bigr)+D_{y}\bigl(C^{y}\bigr)=0, \end{aligned}$$
(54)
where \(C^{t} = C^{t}(t, x, y, u, \ldots), C^{x} = C^{x}(t, x, y, u, \ldots), C^{y} = C^{y}(t, x, y, u, \ldots)\).
A formal Lagrangian for (3) is
$$\begin{aligned} {L} =&p(x,y,t) \bigl[u_{xt}+u_{xx}+u_{yy}+u_{xxxx}+3u_{xx}^{2} \\ &{}+u_{y}^{2}+3u_{x}u_{xxx}+u_{x}^{2}+uu_{xx}+3u_{x}^{2}u_{xx} \bigr]. \end{aligned}$$
(55)
Here \(p(x,y,t)\) is a new dependent variable.

Theorem 2

[29]

Every Lie point, Lie-Bäcklund, and nonlocal symmetry of equation (3) provides a conservation law for this equation and the adjoint equation. Then the elements of conservation vector are given by the following formula:
$$\begin{aligned} C^{i} =&\xi^{i}L+W^{\alpha}\biggl[ \frac{\partial{L}}{\partial{u_{i}^{\alpha}}}-D_{j}\biggl(\frac{\partial{L}}{\partial{u_{ij}^{\alpha}}}\biggr) +D_{j}{D_{k}}\biggl(\frac{\partial{L}}{\partial{u_{ijk}^{\alpha}}}\biggr)-\cdots \biggr] +D_{j}\bigl(W^{\alpha}\bigr) \biggl[\biggl(\frac{\partial{L}}{\partial{u_{ij}^{\alpha}}} \biggr) \\ &{}-D_{k}\biggl(\biggl(\frac{\partial{L}}{\partial{u_{ijk}^{\alpha}}}\biggr)\biggr)+\cdots \biggr]+D_{j}D_{k}\bigl(W^{\alpha}\bigr) \biggl[ \frac{\partial{L}}{ \partial{u_{ijk}^{\alpha}}}-\cdots \biggr], \end{aligned}$$
(56)
where \(W^{\alpha}=\eta^{\alpha}-\xi^{j}u_{j}^{\alpha}\).

4.2 Conservation laws

The adjoint equation of (3) has the form
$$\begin{aligned} F =&-2 u_{ y} p_{ y}-2 pu_{yy}+p_{ yy}+p_{ xt}+6 u_{ x} p_{ x} u_{xx}+p_{xx}+u p_{ xx} \\ &{}+3 u_{x}^{2} p_{xx} -3 u_{ xx} p_{xx}-3 u_{ x} p_{xxx}+p_{ xxxx}=0. \end{aligned}$$
(57)
It is easily found that on substituting u instead of p in equation (57), equation (3) is not recovered. Thus, equation (3) is not self-adjoint. The Lagrangian is
$$\begin{aligned} {L} =&p \bigl[u_{xt}+u_{xx}+u_{yy}+u_{xxxx}+3u_{xx}^{2}+u_{y}^{2} \\ &{}+3u_{x}u_{xxx}+u_{x}^{2}+uu_{xx}+3u_{x}^{2}u_{xx} \bigr]. \end{aligned}$$
(58)
From (56), one gets
$$\begin{aligned}& C^{t}=\xi_{t} L+W \biggl(-D_{x}\frac{\partial{L}}{\partial{u_{tx}}} \biggr)+D_{x}(W)\frac{\partial{L}}{\partial{u_{tx}}}, \end{aligned}$$
(59)
$$\begin{aligned}& \begin{aligned}[b] C^{x}={}&\xi{L}+ W \biggl( \frac{\partial{L}}{\partial{u_{x}}}- D_{x} \frac {\partial{L}}{\partial{u_{xx}}}- D_{t} \frac{\partial{L}}{\partial {u_{xt}}}+ D_{x}^{2} \frac{\partial{L}}{\partial{u_{xxx}}}-D_{x}^{3} (W) \frac {\partial{L}}{\partial{u_{xxxx}}} \biggr)\\ &{}+D_{x}(W) \biggl( \frac{\partial{L}}{\partial{u_{xx}}}- D_{x} \frac {\partial{L}}{\partial{u_{xxx}}}+D_{x}^{2} \frac{\partial{L}}{\partial {u_{xxxx}}} \biggr)\\ &{}+D_{x}^{2} (W) \biggl( \frac{\partial{L}}{\partial{u_{xxx}}} -D_{x} (W) \frac{\partial{L}}{\partial{u_{xxxx}}} \biggr), \end{aligned} \end{aligned}$$
(60)
$$\begin{aligned}& C^{y}=\xi_{y} L+W \biggl(\frac{\partial{L}}{\partial{u_{y}}}-D_{y} \frac{\partial {L}}{\partial{u_{yy}}} \biggr)+D_{y}(W) \frac{\partial{L}}{\partial{u_{yy}}}, \end{aligned}$$
(61)
which leads to
$$\begin{aligned}& C^{t}=-v_{x}W+vW_{x}, \end{aligned}$$
(62)
$$\begin{aligned}& \begin{aligned}[b] C^{x}={}&W \bigl(u_{x}v-v_{x}-v_{x}u-3u_{x}^{2}v_{x}-v_{t}-v_{xxx}+3u_{x}v_{xx} \bigr)\\ &{}+(W_{x}) \bigl(v+3vu_{xx}+uv+3u_{x}^{2}v-3u_{x}v_{x}+v_{xx} \bigr)\\ &{}+W_{xx}(3u_{x}v-v_{x}), \end{aligned} \end{aligned}$$
(63)
$$\begin{aligned}& C^{y}=2vWu_{y}-Wv_{y}+vW_{y}. \end{aligned}$$
(64)

In particular:

1. For the case \(V_{1}=\partial_{y}\), we get \(W=-u_{y}\), and
$$\begin{aligned}& C^{t}=v_{x}u_{y}-vu_{yx}, \end{aligned}$$
(65)
$$\begin{aligned}& \begin{aligned}[b] C^{x}={}&-u_{y} \bigl(u_{x}v-v_{x}-v_{x}u-3u_{x}^{2}v_{x}-v_{t}-v_{xxx}+3u_{x}v_{xx} \bigr)\\ &{}-u_{xy}\bigl(v+3vu_{xx}+uv+3u_{x}^{2}v-3u_{x}v_{x}+v_{xx} \bigr)\\ &{}-u_{yxx}(3u_{x}v-v_{x}), \end{aligned} \end{aligned}$$
(66)
$$\begin{aligned}& C^{y}=-2vu_{y}^{2}+u_{y}v_{y}-vu_{yy}. \end{aligned}$$
(67)
2. For \(V_{2}=\partial_{t}\), one has \(W=-u_{t}\), and
$$\begin{aligned}& C^{t}=u_{t}v_{x}-vu_{tx}, \end{aligned}$$
(68)
$$\begin{aligned}& \begin{aligned}[b] C^{x}={}&-u_{t} \bigl(u_{x}v-v_{x}-v_{x}u-3u_{x}^{2}v_{x}-v_{t}-v_{xxx}+3u_{x}v_{xx} \bigr)\\ &{}-(u_{tx}) \bigl(v+3vu_{xx}+uv+3u_{x}^{2}v-3u_{x}v_{x}+v_{xx} \bigr)\\ &{}-u_{txx}(3u_{x}v-v_{x}), \end{aligned} \end{aligned}$$
(69)
$$\begin{aligned}& C^{y}=-2vu_{t}u_{y}+u_{t}v_{y}-vu_{ty}. \end{aligned}$$
(70)
3. For the case \(V_{3}=F\partial_{x}+F_{t}\partial_{u}\), we have \(W=F_{t}-Fu_{x}\), and we arrive at
$$\begin{aligned}& \begin{aligned}[b] C^{t}&=-v_{x}(F_{t}-Fu_{x})+v(F_{t}-Fu_{x})_{x}\\ &=Fv_{x}u_{x}-F_{t}v_{x}+F_{tx}v-F_{x}vu_{x}-Fvu_{xx}, \end{aligned} \end{aligned}$$
(71)
$$\begin{aligned}& \begin{aligned}[b] C^{x}={}&(F_{t}-Fu_{x}) \bigl(u_{x}v-v_{x}-v_{x}u-3u_{x}^{2}v_{x}-v_{t}-v_{xxx}+3u_{x}v_{xx} \bigr)\\ &{}+(F_{tx}-F_{x}u_{x}-Fu_{xt}) \bigl(v+3vu_{xx}+uv+3u_{x}^{2}v-3u_{x}v_{x}+v_{xx} \bigr)\\ &{}+(F_{txx}-F_{xx}u_{x}-F_{x}u_{xx}-F_{x}u_{xt}-Fu_{xxt}) (3u_{x}v-v_{x}), \end{aligned} \end{aligned}$$
(72)
$$\begin{aligned}& \begin{aligned}[b] C^{y}&=2vWu_{y}-Wv_{y}+vW_{y},\\ &=2vu_{y}F_{t}-2vu_{y}Fu_{x}-F_{t}v_{y}+Fu_{x}v_{y}+vF_{ty}-vF_{y}u_{x}-Fvu_{xy}. \end{aligned} \end{aligned}$$
(73)

5 Concluding remarks

In this paper, we studied the logarithmic-KP equation. The Lie group method was applied to conduct the analysis for this work. Symmetry reductions and explicit solutions were obtained. These solutions maybe explain some complex physical phenomena. It is to be noted that conservation laws were also constructed. We hope that the results obtained may be useful in further numerical analysis. Comparing with [30], it can be seen that our results are new. In the future work, we will try to employ more methods, such as nonclassical Lie groups, the nonlocal symmetry method, and other methods, to derive more novel exact solutions of the logarithmic types of equations.

Declarations

Acknowledgements

The authors would like to thank to the editor and reviewers for their constructive comments and suggestions that improved this paper. This paper is funded by International Graduate Exchange Program of Beijing Institute of Technology.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
School of Mechatronic Engineering, Beijing Institute of Technology

References

  1. Ablowitz, MJ, Segur, H: Solitons and Inverse Scattering Transform. SIAM, Philadelphia (1981) View ArticleMATHGoogle Scholar
  2. Hirota, R: The Direct Method in Soliton Theory. Cambridge University Press, Cambridge (2004) View ArticleMATHGoogle Scholar
  3. Lv, X, Peng, MS: Nonautonomous motion study on accelerated and decelerated solitons for the variable-coefficient Lenells-Fokas model. Chaos 23, 013122 (2013) MathSciNetView ArticleMATHGoogle Scholar
  4. Ma, WX, et al.: Bäcklund transformation and its superposition principle of a Blaszak-Marciniak four-field lattice. J. Math. Phys. 40, 6071-6086 (1999) MathSciNetView ArticleMATHGoogle Scholar
  5. Tian, B, Gao, YT, Zhu, HW: Variable-coefficient higher-order nonlinear Schrodinger model in optical fibers: variable-coefficient bilinear form, Backlund transformation, brightons and symbolic computation. Phys. Lett. A 366, 223-229 (2007) View ArticleMATHGoogle Scholar
  6. Gao, YT, Tian, B: On the non-planar dust-ion-acoustic waves in cosmic dusty plasmas with transverse perturbations. Europhys. Lett. 77, 15001 (2007) View ArticleGoogle Scholar
  7. Li, YS, Ma, WX, Zhang, JE: Darboux transformations of classical Boussinesq system and its new solutions. Phys. Lett. A 275, 60-66 (2000) MathSciNetView ArticleMATHGoogle Scholar
  8. Lv, X: Soliton behavior for a generalized mixed nonlinear Schrodinger model with N-fold Darboux transformation. Chaos 23, 033137 (2013) MathSciNetView ArticleMATHGoogle Scholar
  9. Lou, SY, Wu, QX: Painlevé integrability of two sets of nonlinear evolution equations with nonlinear dispersions. Phys. Lett. A 262, 344-349 (1999) MathSciNetView ArticleMATHGoogle Scholar
  10. Ibragimov, NH (ed.): CRC Handbook of Lie Group Analysis of Differential Equations, Vols. 1-3. CRC Press, Boca Raton (1994) Google Scholar
  11. Bluman, GW, Cheviakov, A, Anco, S: Applications of Symmetry Methods to Partial Differential Equations. Springer, New York (2010) View ArticleMATHGoogle Scholar
  12. Olver, PJ: Application of Lie Group to Differential Equation. Springer, New York (1986) View ArticleGoogle Scholar
  13. Ovsiannikov, LV: Group Analysis of Differential Equations. Academic Press, New York (1982) MATHGoogle Scholar
  14. Wang, GW, Xu, TZ, Ebadi, G, Johnson, S, Strong, AJ, Biswas, A: Singular solitons, shock waves, and other solutions to potential KdV equation. Nonlinear Dyn. 76, 1059-1068 (2014) MathSciNetView ArticleMATHGoogle Scholar
  15. Wang, GW, Xu, TZ, Johnson, S, Biswas, A: Solitons and Lie group analysis to an extended quantum Zakharov-Kuznetsov equation. Astrophys. Space Sci. 349, 317-327 (2014) View ArticleGoogle Scholar
  16. Wang, GW, Liu, XQ, Zhang, YY: Symmetry reduction, exact solutions and conservation laws of a new fifth-order nonlinear integrable equation. Commun. Nonlinear Sci. Numer. Simul. 18, 2313-2320 (2013) MathSciNetView ArticleMATHGoogle Scholar
  17. Vaneeva, OO, Kuriksha, O, Sophocleous, C: Enhanced group classification of Gardner equations with time-dependent coefficients. Commun. Nonlinear Sci. Numer. Simul. 22, 1243-1251 (2015) MathSciNetView ArticleMATHGoogle Scholar
  18. Wang, GW, Xu, TZ, Abazari, R, Jovanoski, Z, Biswas, A: Shock waves and other solutions to the Benjamin-Bona-Mahoney-Burgers equation with dual power-law nonlinearity. Acta Phys. Pol. A 126, 1221-1225 (2014) View ArticleGoogle Scholar
  19. Wang, GW, Xu, TZ, Liu, XQ: New explicit solutions of the fifth-order KdV equation with variable coefficients. Bull. Malays. Math. Soc. 37, 769-778 (2014) MathSciNetMATHGoogle Scholar
  20. Wang, GW, Liu, XQ, Zhang, YY: Lie symmetry analysis to the time fractional generalized fifth-order KdV equation. Commun. Nonlinear Sci. Numer. Simul. 18, 2321-2326 (2013) MathSciNetView ArticleMATHGoogle Scholar
  21. Wang, GW, Kara, AH: Conservation laws, multipliers, adjoint equations and Lagrangians for Jaulent-Miodek and some families of systems of KdV type equations. Nonlinear Dyn. 81, 753-763 (2015) MathSciNetView ArticleGoogle Scholar
  22. Wang, GW, Kara, AH: Nonlocal symmetry analysis, explicit solutions and conservation laws for the fourth-order Burgers’ equation. Chaos Solitons Fractals 81, 290-298 (2015) MathSciNetView ArticleGoogle Scholar
  23. Wang, GW, Kara, AH, Fakhar, K: Symmetry analysis and conservation laws for the class of time fractional nonlinear dispersive equation. Nonlinear Dyn. 82, 281-287 (2015) MathSciNetView ArticleGoogle Scholar
  24. Wang, GW, Fakhar, K: Lie symmetry analysis, nonlinear self-adjointness and conservation laws to an extended \((2+1)\)-dimensional Zakharov-Kuznetsov-Burgers equation. Comput. Fluids 119, 143-148 (2015) MathSciNetView ArticleGoogle Scholar
  25. Wang, GW, Xu, TZ: Group analysis, explicit solutions and conservation laws of the logarithmic-KdV equation. J. Korean Phys. Soc. 66, 1475-1481 (2015) View ArticleGoogle Scholar
  26. Wang, GW, Xu, TZ, Biswas, A: Topological solitons and conservation laws of the coupled Burgers equation. Rom. Rep. Phys. 66, 274-285 (2014) Google Scholar
  27. Wang, GW, Kara, AH, Buhe, E, Fakhar, K: Group analysis and conservation laws of coupled system for the carbon nanotubes conveying fluid. Rom. J. Phys. 60, 952-960 (2015) Google Scholar
  28. Wang, GW, Kara, AH, Fakhar, K, Vega-Guzman, J, Biswas, A: Group analysis, exact solutions and conservation laws of a generalized fifth order KdV equation. Chaos Solitons Fractals 86, 8-15 (2016) MathSciNetView ArticleGoogle Scholar
  29. Fabian, AL, Kohl, R, Biswas, A: Perturbation of topological solitons due to sine-Gordon equation and its type. Commun. Nonlinear Sci. Numer. Simul. 14, 1227-1244 (2009) MathSciNetView ArticleMATHGoogle Scholar
  30. Collins, T, Kara, AH, Bhrawy, AH, Triki, H, Biswas, A: Dynamics of shallow water waves with logarithmic nonlinearity. Rom. Rep. Phys. 68(3) (2016, in press) Google Scholar
  31. Biswas, A, Ranasinghe, A: 1-Soliton solution of Kadomtsev-Petviashvili equation with power law nonlinearity. Appl. Math. Comput. 214, 645-647 (2009) MathSciNetView ArticleMATHGoogle Scholar
  32. Jawad, AJM, Petkovic, M, Biswas, A: Soliton solutions for nonlinear Calaogero-Degasperis and potential Kadomtsev-Petviashvili equations. Comput. Math. Appl. 62, 2621-2628 (2011) MathSciNetView ArticleMATHGoogle Scholar
  33. Triki, H, Sturdevant, BJM, Hayat, T, Aldossary, OM, Biswas, A: Shock wave solutions of the variants of the Kadomtsev-Petviashvili equation. Can. J. Phys. 89, 979-984 (2011) View ArticleGoogle Scholar
  34. Bhrawy, AH, Abdelkawy, MA, Kumar, S, Biswas, A: Solitons and other solutions to Kadomtsev-Petviashvili equation of B-type. Rom. J. Phys. 58, 729-748 (2013) MathSciNetGoogle Scholar
  35. Ebadi, G, Fard, NY, Bhrawy, AH, Kumar, S, Triki, H, Yildirim, A, Biswas, A: Solitons and other solutions to the \((3+1)\)-dimensional extended Kadomtsev-Petviashvili equation with power law nonlinearity. Rom. Rep. Phys. 65, 27-62 (2013) Google Scholar
  36. Fard, NY, Foroutan, MR, Eslami, M, Mirzazadeh, M, Biswas, A: Solitary waves and other solutions to Kadomtsev-Petviashvili equation with spatio-temporal dispersion. Rom. J. Phys. 60, 1337-1360 (2015) Google Scholar
  37. Noether, E, Variationsprobleme, I: Nachr. Akad. Wiss. Gött. Math.-Phys. Kl. 2, 235-257 (1918); English translation in Transp. Theory Stat. Phys. 1, 186-207 (1971) Google Scholar
  38. Ibragimov, NH: A new conservation theorem. J. Math. Anal. Appl. 333, 311-328 (2007) MathSciNetView ArticleMATHGoogle Scholar
  39. Wazwaz, AM: Gaussian solitary waves for the logarithmic-KdV and the logarithmic-KP equations. Phys. Scr. 89, 095206 (2014) View ArticleGoogle Scholar
  40. Kadomtsev, BB, Petviashvili, VI: On the stability of solitary waves in weakly dispersive media. Sov. Phys. Dokl. 15, 539-541 (1970) MATHGoogle Scholar

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