Invariant Solutions for a Class of Perturbed Nonlinear Wave Equations

Abstract

Approximate symmetries of a class of perturbed nonlinear wave equations are computed using two newly-developed methods. Invariant solutions associated with the approximate symmetries are constructed for both methods. Symmetries and solutions are compared through discussing the advantages and disadvantages of each method.

1. Introduction

Approximate Lie symmetry is based on the utilization of the perturbation approach in finding symmetries of certain equations. Baikov, Gazizov and Ibragimov [1] proved an approximate Lie theorem enabling one to construct approximate symmetries of differential equations that are stable under small perturbations. Fushchich and Shtelen [2] and later Gazizov [3] introduced approximate symmetries of differential equations with small perturbations and showed that the these symmetries form an approximate Lie algebra. Since then, many authors have used the approximate Lie symmetries to study nonlinear partial differential equations (PDEs) with a small parameter; see, for instance, [4,5,6,7] and the references therein.

Pakdermirli, Yurusoy and Dolapci [8] provided a comparison between several methods that use approximate symmetries. Valenti [9] calculated the solution of a model describing dissipative media using the generator of the first-order approximate symmetries. Bokhari, Kara and Zaman [4] considered some nonlinear evolution equations with a small parameter and their symmetries. On the other hand, a refined invariant subspace method to determine subspaces of solutions to nonlinear wave equations was discussed in [10]. Zhi-Yong, Yu-Fu and Xue-Lin [11] performed classification and gave approximate solutions to a class of perturbed nonlinear wave equation employing the method originated from Fushchich and Shtelen. In [12], the authors introduced a new method to obtain the approximate symmetry of the nonlinear evolution equation from perturbations.

In this paper, we study the approximate symmetries of a class of perturbed nonlinear wave equations given by:

$$u_{tt}+\alpha u_t={\left(g\left(u\right)u_x\right)}_x+{\left(h\left(u\right)u_y\right)}_y+\beta f\left(u\right).$$

(1)

Lie group theory provides a systematic way of finding exact solutions of differential equations. If the problem involves a small parameter, then an approximate solution instead of an exact solution can be sought. We employ two methods in which a combination of Lie symmetries and perturbation theory is used to find approximate Lie symmetries and invariant solutions.

Method I was introduced by Baikov, Gazizov and Ibragimov [1,13]. In this method, an approximate generator is calculated to obtain the solution. The Lie operator is expanded in a perturbation series other than perturbation for dependent variables as in the usual case. In other words, it is assumed that the perturbed differential equation is of the form:

$$F\left(z\right)=F_0\left(z\right)+\epsilon F_1\left(z\right)=0,$$

(2)

where $z=(x,u,u(1),\dots ,u(n\left)\right),$ $F_0\left(z\right)=0$ is the unperturbed equation and $F_1\left(z\right)$ is the perturbed term.

Theorem 1.

[14] Equation (2) is approximately invariant with the generator $X=X^0+\epsilon X^1$ if and only if:

$${\left[XF\right]}_{F\approx 0}=O\left(\epsilon \right)or[X^0{F}^0+\epsilon {(X^1{F}_0+X^0{F}_1]}_{F\approx 0}=O\left(\epsilon \right),$$

in which $X^0$ is a generator of Lie symmetry of $F_0=0$ and $X^1$ is a generator of Lie symmetry of $F_1.$

The exact symmetry of the unperturbed equation $F_0\left(z\right)=0$ denoted by $X^0$ can be obtained using the equation $X^0{F}_0{\left(z\right)|}_{F_0\left(z\right)=0}=0.$ Applying the auxiliary function:

$$H=\frac{1}{\epsilon }{X}^0(F_0\left(z\right)+\epsilon F_1\left(z\right)){|}_{F0+eF1=0},$$

we deduce the vector field $X^1$ from the relation:

$$X^1{F}_0{\left(z\right)|}_{F0=0}+H=0.$$

(3)

After computing the approximate symmetries, the corresponding invariant solutions are constructed via the classical Lie symmetry method [14]. One may refer the reader for some cases of studying unperturbed and perturbed non-linear wave equations to Bokhari, Kara, Karim, Zaman [15] and Zhi-Yong, Yu-Fu and Xue-Lin [12]. Ahmed, Bokhari, Kara and Zaman [16] provided a classification of the symmetries of the unperturbed nonlinear $(2+1)$ dimensional wave equation with its respective commutator table.

Method II is due to Fushchich and Shtelen [2] and later followed by Euler et al. [17] and Euler and Euler [18]. In this method, the dependent variables are expanded in a perturbation series as is done in the usual perturbation analysis (see, e.g., [19,20]). The approximate symmetry of the original equation is defined to be the exact symmetry of the coupled equations.

Consider the general m-th order nonlinear evolution equation:

$$E=E(x,t,u,u_1,u_2,\dots ,u_m,u_t;\epsilon )=0,$$

(4)

where $u_t=\partial u/\partial t,u_k={\partial }^{k}u/\partial x^k,1\le k\le m$, $\epsilon$ is a small parameter and E is a smooth function of the indicated variables. Expanding the dependent variable in the small parameter yields:

$$u=u_0+\epsilon u_1+\dots ,0<\epsilon <1.$$

(5)

Inserting expansion Equation (5) into the original Equation (4) and separating at each order of the perturbed parameter, one has:

$$Order{\epsilon }^0:E_0=0,Order{\epsilon }^1:E_1=0,$$

(6)

and hence, the exact symmetry of system Equation (6) is the approximate symmetry of the original Equation (4).

The outline of this paper is as follows. In Section 2, we construct invariant solutions of a perturbed nonlinear $(1+1)$-dimensional wave equation. In Section 3, we consider Equation (1) with $\beta =0$ and obtain exact and approximate symmetries of the equation using the approximate Lie symmetry Method I. Moreover approximate invariant solutions of the perturbed non-linear wave equation based on the Lie group method are constructed. In Section 4, we discuss Equation (1) with $\alpha =0$ and compute approximate symmetries of the equation with a forcing term using both the approximate Lie symmetry methods. We compare these different methods and discuss the advantages of using one over the other. Moreover, approximate invariant solutions of the nonlinear wave equation with a forcing term based on the Lie group method are constructed.

2. Perturbed Nonlinear (1 + 1)-Dimension Wave Equation

Consider the perturbed nonlinear wave equation (see e.g., [21]):

$$F_0\left(z\right)+\epsilon F_1\left(z\right)=u_{tt}-{\left(u^2{u}_x\right)}_x+\epsilon u_t=0.$$

(7)

The approximate group generator of Equation (7) is of the form:

$$X=X^0+\epsilon X^1\equiv ({\tau }_0+\epsilon {\tau }_1)\frac{\partial }{\partial t}+({\xi }_0+\epsilon {\xi }_1)\frac{\partial }{\partial x}+({\eta }_0+\epsilon {\eta }_1)\frac{\partial }{\partial u},$$

(8)

where ${\tau }_j,{\xi }_j,{\eta }_j(j=0,1)$ are all unknown functions of $t,x,$ and $u.$ The infinitesimal generator for the unperturbed equation is a vector field in the three-dimensional space (two independent variables and one dependent variable):

$$X^0={\tau }_0\frac{\partial }{\partial t}+{\xi }_0\frac{\partial }{\partial x}+{\eta }_0\frac{\partial }{\partial u}.$$

(9)

The prolongation of the infinitesimal symmetry generator is given by:

$$X^{0^{\left(2\right)}}=X^0+{\eta }_0^t\frac{\partial }{\partial u_t}+{\eta }_0^x\frac{\partial }{\partial u_x}+{\eta }_0^{tt}\frac{\partial }{\partial u_{tt}}+{\eta }_0^{xt}\frac{\partial }{\partial u_{xt}}+{\eta }_0^{xx}\frac{\partial }{\partial u_{xx}}.$$

(10)

The symmetry criterion of Equation (10) yields the relation:

$$X^{0^{\left(2\right)}}\left(u_{tt}-{\left(u^2{u}_x\right)}_x\right){|}_{u_{tt}-{\left(u^2{u}_x\right)}_x}=0.$$

(11)

Comparing coefficients of $u_x,u_x^2,\dots ,$ we obtain the following system of determining equations

$$\begin{array}{c}{\xi }_{0_u}=0,{\xi }_{0_t}=0,{\tau }_{0_u}=0,{\tau }_{0_x}=0,{\eta }_{0_{uu}}=0,\hfill \\ 2{\eta }_0+2u{\eta }_{0_u}-4u{\xi }_{0_x}+4u{\tau }_{0_t}=0,2u^2{{\eta }_0}_{xu}-u{{\eta }_0}_{xx}+4u{{\eta }_0}_x=0,\hfill \\ -2{{\eta }_0}_{tu}+{{\tau }_0}_{tt}=0,2u{\eta }_0+2u^2{{\eta }_0}_x+2u^2{{\tau }_0}_t=0,u^2{{\eta }_0}_{xx}-{{\eta }_0}_{tt}=0\hfill \end{array}$$

Solving this system of PDEs, we obtain:

$${\xi }_0={\alpha }_0+{\alpha }_{1}x,{\tau }_0={\alpha }_{3}t+{\alpha }_{1}t+{\alpha }_2,{\eta }_0=-{\alpha }_{3}u,$$

(12)

where ${\alpha }_0,{\alpha }_1,{\alpha }_2,{\alpha }_3$ are arbitrary constants. Thus,

$$X^0=({\alpha }_0+{\alpha }_{1}x)\frac{\partial }{\partial x}+({\alpha }_{3}t+{\alpha }_{1}t+{\alpha }_2)\frac{\partial }{\partial t}-{\alpha }_{3}u\frac{\partial }{\partial u}$$

(13)

To determine the auxiliary function $H,$ we consider:

$$H=\frac{1}{\epsilon }\left[X^{0^{\left(2\right)}}[f_0\left(z\right)+\epsilon F_1\left(z\right)]\right]{|}_{\{F_0\left(z\right)+\epsilon F_1\left(z\right)=0\}},$$

(14)

or:

$$H=\frac{1}{\epsilon }\left[X^{0^{\left(2\right)}}[u_{tt}-2u{u}_x^2-u^2{u}_{xx}+\epsilon u_t]\right]{|}_{\{u_{tt}-2u{u}_x^2-u^2{u}_{xx}+\epsilon u_t=0\}},$$

(15)

where $X^{0^{\left(2\right)}}$ is the second prolongation of $X^0$. This implies that:

$$H=\frac{1}{\epsilon }\left[\eta (-2u_x^2-2u{u}_{xx})+{\eta }^x(-4u{u}_x^2)+{\eta }^t\left(\epsilon \right)+{\eta }^{xx}(-u^2)+{\eta }^{tt}\right]{|}_{\{u_{tt}-2u{u}_x^2-u^2{u}_{xx}+\epsilon u_t=0\}}.$$

(16)

Hence,

$$\begin{array}{c}\eta =-{\alpha }_{3}u,{\eta }^x=-{\alpha }_3{u}_x,{\eta }^t={\alpha }_4{u}_t-{\alpha }_3{u}_t,\hfill \\ {\eta }^{tt}={\alpha }_4{u}_{tt}-2{\alpha }_3{u}_{tt},{\eta }^{xx}=-{\alpha }_4{u}_{xx}-2{\alpha }_3{u}_{xx}.\hfill \end{array}$$

Substituting $\eta ,{\eta }^x,{\eta }^t,{\eta }^{tt},{\eta }^{xx}$ and $u_{tt}=2u{u}_x^2+u^2{u}_{xx}-\epsilon u_t$ into Equation (16) gives:

$$H={\alpha }_3{u}_t.$$

(17)

The determining equation for deformations is written as:

$$X^{1^{\left(2\right)}}(u_{tt}-u^2{u}_{xx}-2u{u}_x^2){|}_{u_{tt}=u^2{u}_{xx}+2u{u}_x^2}+H=0,$$

(18)

where $X^{1^{\left(2\right)}}$ denotes the second prolongation of the operator:

$$X^1={\tau }_1\frac{\partial }{\partial t}+{\xi }_1\frac{\partial }{\partial x}+{\eta }_1\frac{\partial }{\partial u}.$$

(19)

We obtain the following system of the determining equations for Equation (18):

$$\begin{array}{c}{\xi }_{1_u}=0,{\xi }_{1_t}=0,{\tau }_{1_u}=0,{\tau }_{1_x}=0,{\eta }_{1_{uu}}=0,\hfill \\ 2{\eta }_1+2u{\eta }_{1_u}-4u{\xi }_{1_x}+4u{\tau }_{1_t}=0,2u^2{{\eta }_1}_{xu}-u{{\eta }_1}_{xx}+4u{{\eta }_1}_x=0,\hfill \\ -2{{\eta }_1}_{tu}+{{\tau }_1}_{tt}-{\alpha }_3=0,2u{\eta }_1+2u^2{{\eta }_1}_x+2u^2{{\tau }_1}_t=0,u^2{{\eta }_1}_{xx}-{{\eta }_1}_{tt}=0\hfill \end{array}$$

Solving the above system yields:

$${\tau }_1={\beta }_1+{\beta }_{3}t+\frac{1}{6}{\alpha }_3{t}^2,{\xi }_1={\beta }_2+({\beta }_3+{\beta }_4)x,{\eta }_1=({\beta }_4-\frac{1}{3}{\alpha }_{3}u)u.$$

(20)

Substituting Equations (12) and (20) into Equation (8), we obtain the following approximate symmetries for Equation (7):

$$\begin{array}{c}\hfill X_1=\frac{\partial }{\partial t},X_2=\frac{\partial }{\partial x},X_3=t\frac{\partial }{\partial t}+x\frac{\partial }{\partial x}+\frac{\epsilon }{6}(t^2\frac{\partial }{\partial t}-2tu\frac{\partial }{\partial u}),\\ \hfill X_4=x\frac{\partial }{\partial x}+u\frac{\partial }{\partial u},X_5=\epsilon X_1,X_6=\epsilon X_2,X_7=\epsilon X_4,X_8=\epsilon X_3.\end{array}$$

In Table 1, we show that the generators span an eight-dimensional approximate Lie algebra and, hence, generate an eight-parameter approximate transformation group.

Table 1. Approximate commutators of approximate symmetry of the perturbed non-linear wave equation.

	${\mathit{X}}_1$	${\mathit{X}}_2$	${\mathit{X}}_3$	${\mathit{X}}_4$	${\mathit{X}}_5$	${\mathit{X}}_6$	${\mathit{X}}_7$	${\mathit{X}}_8$
$X_1$	0	0	$X_1+\frac{1}{3}(X_8-X_7)$	0	0	0	0	$X_5$
$X_2$	0	0	$X_2$	$X_2$	0	0	$X_6$	$X_6$
$X_3$	0	0	0	0	$-X_5$	$-X_6$	0	0
$X_4$	0	0	0	0	0	$-X_6$	0	0
$X_5$	0	0	$X_6$	$X_6$	0	0	0	0
$X_6$	0	0	$X_6$	0	0	0	0	0
$X_7$	0	$-X_6$	0	0	0	0	0	0
$X_8$	$-X_5$	$-X_6$	0	0	0	0	0	0

Approximate Invariant Solution

Using the symmetry $X=X_3-X_4,$ we obtain:

$$X=t\frac{\partial }{\partial t}-u\frac{\partial }{\partial u}+\frac{\epsilon }{6}\left(t^2\frac{\partial }{\partial t}-2tu\frac{\partial }{\partial u}\right).$$

(21)

The approximate invariants of Equation (21) can be written as:

$$E(t,x,u,\epsilon )=E^0(t,x,u)+\epsilon E^1(t,x,u)+O\left(\epsilon \right),$$

which lead to the system:

$$\begin{array}{c}t\frac{\partial E^0}{\partial t}-u\frac{\partial E^0}{\partial u}=0,\hfill \\ t\frac{\partial E^1}{\partial t}-u\frac{\partial E^1}{\partial u}=-\frac{1}{6}\left(t^2\frac{\partial E^0}{\partial t}-2tu\frac{\partial E^0}{\partial u}\right),\hfill \end{array}$$

(22)

Solving Equation (22) gives two functionally independent invariants:

$$\begin{array}{c}\hfill E_1=E_1^0(t,x,u)+\epsilon E_1^1(t,x,u),E_2=E_2^0(t,x,u)+\epsilon E_2^1(t,x,u),\end{array}$$

(23)

for generator Equation (21).

The first equation in Equation (22) has two functionally independent solutions,

$$E_1^0=x,E_2^0=tu.$$

Substituting $E_1^0=x$ into the second equation in Equation (22) and taking its simplest solution $E_1^1=0,$ we obtain one invariant in Equation (23),

$$E_1=x.$$

(24)

Now, we substitute the solution $E_2^0=tu$ of the first equation in Equation (22) into the second equation in Equation (22) and get a non-homogeneous linear equation:

$$t\frac{\partial E_1^1}{\partial t}-u\frac{\partial E_2^1}{\partial u}=\frac{1}{6}{t}^{2}u.$$

The corresponding characteristic equation are:

$$\frac{dt}{t}=-\frac{du}{u}=6\frac{d{E}_2^1}{t^{2}u}$$

for which the first integral $tu=\lambda =const.$ We obtain:

$$E_2^1=\frac{1}{6}{t}^{2}u+c.$$

(25)

Assuming $t=0,$ we get the second invariant in Equation (23),

$$E_2=tu+\frac{\epsilon }{6}{t}^{2}u.$$

Note that invariants’ Equations (24) and (25) are functionally independent. Letting $E_2=\varphi \left(E_1\right),$ i.e.,

$$\left(1+\frac{\epsilon t}{6}\right)tu=\varphi \left(x\right)$$

and solving for $tu$ in the first order of precision,

$$tu={\left(1+\frac{\epsilon t}{6}\right)}^{-1}\varphi \left(x\right)=\left(1+\frac{\epsilon t}{6}\right)\varphi \left(x\right)+O\left(\epsilon \right).$$

The approximately invariant solution is given by:

$$u=\left(\frac{1}{t}-\frac{\epsilon }{6}\right)\varphi \left(x\right).$$

(26)

From Equation (7), we obtain:

$$\frac{d\varphi }{dx}=\pm \sqrt{1+c{\varphi }^{-4}}.$$

Setting $c=0,$ we have $\varphi \left(x\right)=\pm x,$ and:

$$u=\pm \left(\frac{x}{t}-\epsilon \frac{x}{6}\right).$$

3. Perturbed Nonlinear (2 + 1)-Dimension Wave Equation

Consider the perturbed nonlinear wave equation:

$$u_{tt}+\epsilon u_t={\left(g\left(u\right)u_x\right)}_x+{\left(h\left(u\right)u_y\right)}_y,$$

(27)

where $\epsilon$ is a small parameter. Putting $g\left(u\right)=h\left(u\right)=u$ gives:

$$u_{tt}+\epsilon u_t={\left(u{u}_x\right)}_x+{\left(u{u}_y\right)}_y.$$

(28)

The first method is used to obtain a complete approximate symmetry classification of Equation (28) with the first order of precision $o\left(\epsilon \right).$ The approximate group generator of Equation (28) is of the form:

$$\begin{array}{cc}\hfill X& =X_0+\epsilon X_1\hfill \\ & =({\tau }_0+\epsilon {\tau }_1)\frac{\partial }{\partial t}+({\xi }_0+\epsilon {\xi }_1)\frac{\partial }{\partial x}+({\theta }_0+\epsilon {\theta }_1)\frac{\partial }{\partial y}+({\eta }_0+\epsilon {\eta }_1)\frac{\partial }{\partial u},\hfill \end{array}$$

(29)

where ${\tau }_i,{\xi }_i,{\theta }_i$ and ${\eta }_i,i=0,1,$ are unknown functions of $t,x,y$ and $u.$

3.1. Exact Symmetries

To find the exact symmetries, we solve the determining equation:

$$X_0^{\left(2\right)}{F}_0\left(z\right){|}_{F_0\left(z\right)=0}=0,$$

(30)

where $F_0\left(z\right)=u_{tt}-{\left(u{u}_x\right)}_x-{\left(u{u}_y\right)}_y$ is the unperturbed part of Equation (28) and $X_0^{\left(2\right)}$ is the second prolongation of the infinitesimal generator $X_0$ given by:

$$\begin{array}{cc}\hfill X_0^{\left(2\right)}=X_0& +{\eta }_0^t\frac{\partial }{\partial u_t}+{\eta }_0^x\frac{\partial }{\partial u_x}+{\eta }_0^y\frac{\partial }{\partial u_y}+{\eta }_0^{tt}\frac{\partial }{\partial u_{tt}}+{\eta }_0^{tx}\frac{\partial }{\partial u_{tx}}\hfill \\ & +{\eta }_0^{ty}\frac{\partial }{\partial u_{ty}}+{\eta }_0^{xx}\frac{\partial }{\partial u_{xx}}+{\eta }_0^{xy}\frac{\partial }{\partial u_{xy}}+{\eta }_0^{yy}\frac{\partial }{\partial u_{yy}}.\hfill \end{array}$$

(31)

Equation (30) takes the form:

$$\begin{array}{cc}\hfill ({\eta }_0(-u_{xx}-u_{yy})& +{\eta }_0^x(-2u_x)+{\eta }_0^y(-2u_y)+{\eta }_0^{xx}(-u)\hfill \\ & +{\eta }_0^{yy}(-u)+{\eta }_0^{tt}){|}_{u_{tt}={\left(u{u}_x\right)}_x+{\left(u{u}_y\right)}_y}=0,\hfill \end{array}$$

(32)

where:

$$\begin{array}{cc}\hfill {\eta }_0^x& =D_x{\eta }_0-(u_x{D}_x{\xi }_0+u_y{D}_x{\theta }_0+u_t{D}_x{\tau }_0),\hfill \\ \hfill {\eta }_0^y& =D_y{\eta }_0-(u_x{D}_y{\xi }_0+u_y{D}_y{\theta }_0+u_t{D}_y{\tau }_0),\hfill \\ \hfill {\eta }_0^t& =D_t{\eta }_0-(u_x{D}_t{\xi }_0+u_y{D}_t{\theta }_0+u_t{D}_t{\tau }_0),\hfill \\ \hfill {\eta }_0^{xx}& =D_x{\eta }_0^x-u_{xx}{D}_x{\xi }_0-u_{xy}{D}_x{\theta }_0-u_{xt}{D}_x{\tau }_0,\hfill \\ \hfill {\eta }_0^{yy}& =D_y{\eta }_0^y-u_{xy}{D}_y{\xi }_0-u_{yy}{D}_y{\theta }_0-u_{yt}{D}_y{\tau }_0),\hfill \\ \hfill {\eta }_0^{tt}& =D_t{\eta }_0^t-u_{xt}{D}_t{\xi }_0-u_{yt}{D}_t{\theta }_0+u_{tt}{D}_t{\tau }_0).\hfill \end{array}$$

(33)

Here, $D_x,D_y$ and $D_t$ denote the total derivative operators with respect to $x,y$ and $t,$ respectively,

$$\begin{array}{cc}\hfill D_x& =\frac{\partial }{\partial x}+u_x\frac{\partial }{\partial u}+u_{xx}\frac{\partial }{\partial u_x}+u_{xy}\frac{\partial }{\partial u_y}+u_{xt}\frac{\partial }{\partial u_t}+\cdots +u_{xtt}\frac{\partial }{\partial u_{tt}},\hfill \\ \hfill D_y& =\frac{\partial }{\partial y}+u_y\frac{\partial }{\partial u}+u_{xy}\frac{\partial }{\partial u_x}+u_{yy}\frac{\partial }{\partial u_y}+u_{ty}\frac{\partial }{\partial u_t}+\cdots +u_{ytt}\frac{\partial }{\partial u_{tt}},\hfill \\ \hfill D_t& =\frac{\partial }{\partial t}+u_t\frac{\partial }{\partial u}+u_{xt}\frac{\partial }{\partial u_x}+u_{yt}\frac{\partial }{\partial u_y}+u_{tt}\frac{\partial }{\partial u_t}+\cdots +u_{ttt}\frac{\partial }{\partial u_{tt}}.\hfill \end{array}$$

(34)

Equation (32) gives the following system of equations:

$$\begin{array}{c}{{\xi }_0}_u=0,{{\xi }_0}_t=0,{{\tau }_0}_u=0,{{\tau }_0}_y=0,{{\tau }_0}_x=0,{{\theta }_0}_u=0,{{\theta }_0}_t=0,\hfill \\ {{\eta }_0}_{uu}=0,-2{{\eta }_0}_x-2u{{\eta }_0}_{xu}+u{{\xi }_0}_{xx}+u{{\xi }_0}_{yy}=0,-{{\eta }_0}_u+2{{\xi }_0}_x-2{{\tau }_0}_t=0,\hfill \\ -2{{\eta }_0}_y+u{{\theta }_0}_{xx}+u{{\theta }_0}_{yy}=0,-2{{\eta }_0}_y+2{{\theta }_0}_y+{{\eta }_0}_u-2{{\tau }_0}_t=0,-u{{\eta }_0}_{xx}+{{\eta }_0}_{tt}-u{{\eta }_0}_{yy}=0,\hfill \\ 2{{\eta }_0}_{ut}-{{\tau }_0}_{tt}=0,2u{{\xi }_0}_x-2u{{\tau }_0}_t-{\eta }_0=0,{{\theta }_0}_x+{{\xi }_0}_y=0,-2u{{\tau }_0}_t+2u{{\theta }_0}_y-{\eta }_0=0\hfill \end{array}$$

Solving this system of PDEs, one has:

$$\begin{array}{cccc}{\xi }_0=a_{3}x+a_{1}y+a_2,\hfill & \hfill {\theta }_0=a_{3}y-a_{1}x+a_6,& {\tau }_0=a_{4}t+a_5,\hfill & \hfill {\eta }_0=2u(a_3-a_4),\end{array}$$

(35)

where $a_1,a_2,a_3,a_4,a_5$ and $a_6$ are arbitrary constants. Hence, the infinitesimal generator for Equation (28) is:

$$\begin{array}{cc}\hfill X_0=& (a_{4}t+a_5)\frac{\partial }{\partial t}+(a_{3}x+a_{1}y+a_2)\frac{\partial }{\partial x}\hfill \\ & +(a_{3}y-a_{1}x+a_6)\frac{\partial }{\partial y}+\left(2u(a_3-a_4)\right)\frac{\partial }{\partial u}.\hfill \end{array}$$

(36)

3.2. Approximate Symmetries

The auxiliary function H is given by:

$$H=\frac{1}{\epsilon }\left[X_0^{\left(k\right)}\left(F_0\left(z\right)+\epsilon F_1\left(z\right)\right){|}_{F_0\left(z\right)+\epsilon F_1\left(z\right)=0}\right].$$

(37)

Substituting the generator $X_0$ into Equation (36) and:

$$F_0\left(z\right)+\epsilon F_1\left(z\right)=u_{tt}+\epsilon u_t-{\left(u{u}_x\right)}_x-{\left(u{u}_y\right)}_y$$

into Equation (37), we obtain:

$$H=a_4{u}_t.$$

(38)

Now, we calculate operator $X_1$ by solving the inhomogeneous determining equation:

$$X_1^{\left(2\right)}{F}_0\left(z\right){|}_{F_0\left(z\right)}+H=0,$$

which can be written as:

$$\left[X_1^{\left(2\right)}\left(u_{tt}-{\left(u{u}_x\right)}_x-{\left(u{u}_y\right)}_y\right){|}_{u_{tt}={\left(u{u}_x\right)}_x+{\left(u{u}_y\right)}_y}\right]+a_4{u}_t=0.$$

(39)

Equation (39) generates the following system of equations:

$$\begin{array}{c}{{\xi }_1}_u=0,{{\xi }_1}_t=0,{{\tau }_1}_u=0,{{\tau }_1}_y=0,{{\tau }_1}_x=0,{{\theta }_1}_u=0,\hfill \\ {{\theta }_1}_t=0,{{\eta }_1}_{uu}=0,-2{\eta }_{1}x-2u{{\eta }_1}_{xu}+u{{\xi }_1}_{xx}+u{{\xi }_1}_{yy}=0,-{{\eta }_1}_u+2{{\xi }_1}_x-2{{\tau }_1}_t=0,\hfill \\ -2{{\eta }_1}_y+u{{\theta }_1}_{xx}+u{{\theta }_1}_{yy}=0,-2{{\eta }_1}_y+2{{\theta }_1}_y+{{\eta }_1}_u-2{{\tau }_1}_t=0,-u{{\eta }_1}_{xx}+{{\eta }_1}_{tt}-u{{\eta }_1}_{yy}=0,\hfill \\ 2{{\eta }_1}_{ut}-{{\tau }_1}_{tt}+a_4=0,2u{{\eta }_1}_x-2u{{\tau }_1}_t-{\eta }_1=0,{{\theta }_1}_x+{{\xi }_1}_y=0,-2u{{\tau }_1}_t+2u{{\theta }_1}_y-{\eta }_1=0.\hfill \end{array}$$

Solving this system of PDEs, we obtain:

$$\begin{array}{cc}{\xi }_1=\hfill & b_{3}x+b_{1}y+b_2,{\theta }_1=b_{3}y-b_{1}x+b_6,\hfill \\ {\tau }_1=\hfill & \frac{a_4{t}^2}{10}+b_{4}t+b_5,{\eta }_1=2u\left(b_3-\frac{1}{5}{a}_{4}t-b_4\right),\hfill \end{array}$$

(40)

where $b_1,b_2,b_3,b_4,b_5$ and $b_6$ are arbitrary constants. Thus, the approximate symmetries of Equation (28) are:

$$\begin{array}{c}{X}_1=y\frac{\partial }{\partial x}-x\frac{\partial }{\partial y},X_2=\frac{\partial }{\partial x},X_3=x\frac{\partial }{\partial x}+y\frac{\partial }{\partial x}+y\frac{\partial }{\partial y}+2u\frac{\partial }{\partial u},\hfill \\ X_4=t\frac{\partial }{\partial t}-2u\frac{\partial }{\partial u}+\epsilon \left(\frac{t^2}{10}\frac{\partial }{\partial t}-\frac{2}{5}tu\frac{\partial }{\partial u}\right),\hfill \\ X_5=\frac{\partial }{\partial t},X_6=\frac{\partial }{\partial y},X_7=\epsilon X_1,X_8=\epsilon X_2,\hfill \\ X_9=\epsilon X_3,X_{10}=\epsilon X_5,X_{11}=\epsilon X_6,X_{12}=\epsilon X_4.\hfill \end{array}$$

Remark 1.

$$X_{12}=\epsilon \left(t\frac{\partial }{\partial t}-2ut\frac{\partial }{\partial u}\right)$$

In Table 2, we show that the previous generators span a twelve-dimensional approximate Lie algebra and, hence, generate a twelve-parameter approximate transformations group.

Table 2. Approximate commutator table of approximate symmetries of the perturbed non-linear wave equation.

	${\mathit{X}}_1$	${\mathit{X}}_2$	${\mathit{X}}_3$	${\mathit{X}}_4$	${\mathit{X}}_5$	${\mathit{X}}_6$	${\mathit{X}}_7$	${\mathit{X}}_8$	${\mathit{X}}_9$	${\mathit{X}}_{10}$	${\mathit{X}}_{11}$	${\mathit{X}}_{12}$
$X_1$	0	$X_6$	0	0	0	$-X_2$	0	$X_{11}$	0	0	0	0
$X_2$	$-X_6$	0	$X_2$	0	0	0	$-X_{11}$	0	$X_8$	0	0	0
$X_3$	0	$-X_2$	0	0	0	$-X_6$	0	$-X_8$	0	0	$X_{11}$	0
$X_4$	0	0	0	0	$-X_5-\frac{2}{5}{X}_{12}$	0	0	0	0	$-X_{10}$	0	0
$X_5$	0	0	0	$X_5+\frac{2}{5}{X}_{12}$	0	0	0	0	0	0	0	$X_{10}$
$X_6$	$X_2$	0	$X_6$	0	0	0	$X_8$	0	$X_{11}$	0	0	0
$X_7$	0	$X_{11}$	0	0	0	$-X_8$	0	0	0	0	0	0
$X_8$	$-X_{11}$	0	$X_8$	0	0	0	0	0	0	0	0	0
$X_9$	0	$-X_8$	0	0	0	$-X_{11}$	0	0	0	0	0	0
$X_{10}$	0	0	0	$X_{10}$	0	0	0	0	0	0	0	0
$X_{11}$	0	0	$X_{11}$	0	0	0	0	0	0	0	0	0
$X_{12}$	0	0	0	0	$-X_{10}$	0	0	0	0	0	0	0

3.3. Approximate Invariant Solutions

Reconsider Equation (28):

$$u_{tt}-\epsilon u_t={\left(u{u}_x\right)}_x+{\left(u{u}_y\right)}_y.$$

(41)

and the symmetry:

$$X_4=t\frac{\partial }{\partial t}-2u\frac{\partial }{\partial u}+\frac{\epsilon }{10}\left(t^2\frac{\partial }{\partial t}-4ut\frac{\partial }{\partial u}\right).$$

(42)

The approximate invariant for Equation (42) is of the form:

$$E(t,x,y,u,\epsilon )=E^0(t,x,y,u)+\epsilon E^1(t,x,y,u)+o\left(\epsilon \right),$$

determined by the equation $X\left(E\right)=o\left(\epsilon \right).$ Using the notation:

$$X=X^0+\epsilon X^1,$$

where:

$$X^0=t\frac{\partial }{\partial t}-2u\frac{\partial }{\partial u},X^1=\frac{1}{10}\left(t^2\frac{\partial }{\partial t}-4tu\frac{\partial }{\partial u}\right);$$

for operator Equation (42), we write the determining equation $X\left(E\right)=o\left(\epsilon \right)$ for the approximate invariants in the form:

$$\begin{array}{c}{X}^0\left(E^0\right)+\epsilon \left[X^0\left(E^1\right)+X^1\left(E^0\right)\right]=0,X^0\left(E^0\right)=0,\hfill \\ X^0\left(E^1\right)+X^1\left(E^0\right)=0,\hfill \end{array}$$

or:

$$\begin{array}{cc}\hfill t\frac{\partial E^0}{\partial t}-2u\frac{\partial E^0}{\partial u}& =0\hfill \\ \hfill t\frac{\partial E^1}{\partial t}-2u\frac{\partial E^1}{\partial u}& =-\frac{1}{10}\left(t^2\frac{\partial E^0}{\partial t}-4tu\frac{\partial E^0}{\partial u}\right).\hfill \end{array}$$

(43)

Solving Equation (43) gives two functionally independent invariants:

$$\begin{array}{c}\hfill E_1=E_1^0(t,x,y,u)+\epsilon E_1^1(t,x,y,u),E_2=E_2^0(t,x,y,u)+\epsilon E_2^1(t,x,y,u),\end{array}$$

(44)

for generator Equation (42).

The first equation in Equation (43) has two functionally independent solutions:

$$E_1^0=xy,E_2^0=t^{2}u.$$

Substituting $E_1^0=xy$ into the second equation in Equation (43) and taking its simplest solution $E_1^1=0,$ we obtain one invariant in Equation (44),

$$E_1=xy$$

(45)

Note that the dependent variable u does not appear in Equation (45). Now, we substitute the solution $E_2^0=t^{2}u$ of the first equation in Equation (43) into the second equation in Equation (43) and obtain non-homogeneous linear equation:

$$t\frac{\partial {E_1}^1}{\partial t}-2u\frac{\partial {E_2}^1}{\partial u}=\frac{1}{5}\left(t^{3}u\right).$$

The corresponding characteristic equations are:

$$\frac{dt}{t}=-\frac{du}{2u}=5\frac{d{E}_2^1}{t^{2}u},$$

with the first integral $t^{2}u=\lambda =const.$ Therefore, the second equation:

$$\frac{dt}{t}=5\frac{d{E}_2^1}{t^{3}u}$$

gives:

$${E_2}^1=\frac{1}{5}{t}^{3}u+c.$$

(46)

Assuming that $c=0,$ we obtain the second invariant in Equation (44),

$$E_2=t^{2}u+\frac{\epsilon }{5}{t}^{3}u.$$

(47)

Note that $E_1$ and $E_2$ are functionally independent. Letting $E_2=\psi \left(E1\right),$ i.e.,

$$\left(t^{2}u+\frac{\epsilon }{5}{t}^{3}u\right)=\psi \left(xy\right)$$

and solving for $t^{2}u$ in the first order of precision,

$$t^{2}u={\left(1+\frac{\epsilon }{5}t\right)}^{-1}\varphi \left(xy\right)=\left(1-\frac{\epsilon }{5}t\right)\varphi \left(xy\right)+o\left(\epsilon \right),$$

yield the approximate invariant solution:

$$u(t,x,y)=\left(\frac{1}{t^2}-\frac{\epsilon }{5t}\right)\varphi \left(xy\right).$$

(48)

From Equation (28), we obtain:

$${\left(\frac{\partial \varphi }{\partial x}\right)}^2+\frac{{\partial }^2\varphi }{\partial x^2}·\varphi +{\left(\frac{\partial \varphi }{\partial y}\right)}^2+\frac{{\partial }^2\varphi }{\partial y^2}·\varphi -6\varphi =0.$$

(49)

Case I: Let $\psi \left(xy\right)$ be of the form $\varphi \left(xy\right)={\left(xy\right)}^{\alpha }.$ From Equation (49), one obtains:

$$[2{\alpha }^2-\alpha ]\left(x^{\alpha -2}{y}^{\alpha }+x^{\alpha }{y}^{\alpha -2}\right)-6=0$$

For $\alpha =2,$ we have:

$$u(t,x,y)=\left(\frac{1}{t^2}-\frac{\epsilon }{5t}\right)\left(x^2{y}^2\right)s.t{x}^2+y^2=1.$$

An approximate solution for this case is depicted in Figure 1.

Figure 1. CaseI: approximate invariant solution of Equation (28) for $t=\pi ,-1\le x\le 1,-1\le y\le 1,$ $\epsilon =0.1.$

Case II: Let $\varphi \left(xy\right)=A\left(x\right)B\left(x\right).$ Equation (49) gives the following equation:

$$B\left(\frac{{A^{\prime }}^2+A{A}^{″}}{A}\right)+A\left(\frac{{B^{\prime }}^2+B{B}^{″}}{B}\right)-6=0$$

(50)

where:

$$A^{\prime }=\frac{\partial A}{\partial x},A^{″}=\frac{{\partial }^{2}A}{\partial x^2},B^{\prime }=\frac{\partial B}{\partial y},B^{″}=\frac{{\partial }^{2}B}{\partial y^2}$$

Equation (50) leads to the following ordinary differential equations:

$$A{A}^{″}+{A^{\prime }}^2-d_1{A}^2-c_{1}A=0,$$

(51)

and:

$$B{B}^{″}+{B^{\prime }}^2-d_2{B}^2-c_{2}B=0,$$

(52)

where $c_1,c_2,d_1,d_2$ are constants.

Let $\gamma \left(x\right)=A^2\left(x\right).$ From Equations (51) and (52), we obtain:

$$(2ln(d_{1}A\left(x\right)+c_1)-1)\left(\frac{A^2\left(x\right)}{2d_1}-\frac{c_1^2}{2d_1^2}\right)+\frac{c_1}{d_1^2}{A}^2\left(x\right)=\frac{x^2}{2}+c_{3}x+c_4,$$

(53)

and:

$$(2ln(d_{2}B\left(x\right)+c_2)-1)\left(\frac{B^2\left(x\right)}{2d_2}-\frac{c_2^2}{2d_2^2}\right)+\frac{c_2}{d_2^2}{B}^2\left(x\right)=\frac{x^2}{2}+c_{5}x+c_6,$$

(54)

where $c_3,c_4,c_5,c_6$ are arbitrary constants. Therefore, a solution in this case is of the form:

$$u(t,x,y)=\left(\frac{1}{t^2}-\frac{\epsilon }{5t}\right)A\left(x\right)B\left(x\right).$$

(55)

We plot an approximate solution for this case in Figure 2.

Figure 2. Case II:approximate invariant solution of Equation (28) for $t=100,-1\le x\le 1,-1\le y\le 1,$ $\epsilon =0.1.$

4. Nonlinear Wave Equation with a Forcing Term

In this section, we discuss the nonlinear (2 + 1)-dimensional wave equation with a forcing term:

$$u_{tt}-{\left(u{u}_x\right)}_x-{\left(u{u}_y\right)}_y=\epsilon f\left(u\right).$$

(56)

4.1. Approximate Symmetries by Method I

Exact symmetries of the unperturbed part ($\epsilon =0$) of Equation (56) are given by:

$$\begin{array}{c}\hfill {\xi }_0=a_{3}x+a_{1}y+a_2,{\theta }_0=a_{3}y-a_{1}x+a_6,{\tau }_0=a_{4}t+a_5,{\eta }_0=2u(a_3-a_4),\end{array}$$

(57)

where $a_1,a_2,a_3,a_4,a_5$ and $a_6$ are arbitrary constants.

Consider the auxiliary function:

$$H=\frac{1}{\epsilon }\left[X^0(F_0\left(z\right)+\epsilon F_1\left(z\right)\right]{|}_{F_0\left(z\right)+\epsilon F_1\left(z\right)=0},$$

(58)

where:

$$X^0={\tau }_0\frac{\partial }{\partial t}+{\xi }_0\frac{\partial }{\partial x}+{\theta }_0\frac{\partial }{\partial y}+{\eta }_0\frac{\partial }{\partial u}.$$

Using Equations (56) and (57), one obtains:

$$H=-2a_3\left(u{f}^{\prime }\left(u\right)+f\left(u\right)\right)+2a_4\left(u{f}^{\prime }\left(u\right)-2f\left(u\right)\right).$$

(59)

Now we calculate the operator $X_1$ with the condition that

$$X^1\left(F_0\left(z\right)\right){|}_{F_0\left(z\right)=0}+H=0.$$

(60)

Condition Equation (60) can be written as:

$$\begin{array}{c}\hfill \left[X_1^{\left(2\right)}(u_{tt}-{\left(u{u}_x\right)}_x-{\left(u{u}_y\right)}_y){|}_{u_{tt}={\left(u{u}_x\right)}_x+{\left(u{u}_y\right)}_y}\right]-2c_3\left(u{f}^{\prime }\left(u\right)+f\left(u\right)\right)\\ \hfill +2c_4\left(u{f}^{\prime }\left(u\right)-2f\left(u\right)\right)=0,\end{array}$$

(61)

where $X_1^{\left(2\right)}$ is the second prolongation of $X_1.$

Equation (61) yields the following system of equations:

$$\begin{array}{c}{{\xi }_1}_u=0,{{\xi }_1}_t=0,{{\tau }_1}_u=0,{{\tau }_1}_y=0,{{\tau }_1}_x=0,\hfill \\ {{\theta }_1}_u=0,{{\theta }_1}_t=0,{{\eta }_1}_{uu}=0,-2{{\eta }_1}_x-2u{{\eta }_1}_{xu}+u{{\xi }_1}_{xx}+u{{\xi }_1}_{yy}=0,\hfill \\ -{{\eta }_1}_u+2{{\xi }_1}_x-2{{\tau }_1}_t=0-2{{\eta }_1}_y+u{{\theta }_1}_{xx}+u{{\theta }_1}_{yy}=0,-2{{\eta }_1}_y+2{{\theta }_1}_y+{{\eta }_1}_u-2{{\tau }_1}_t=0,\hfill \\ -u{{\eta }_1}_{xx}+{{\eta }_1}_{tt}-u{{\eta }_1}_{yy}-2a_3(u{f}^{\prime }\left(u\right)+f\left(u\right))=0,2{{\eta }_1}_{ut}-{{\tau }_1}_{tt}=0,\hfill \\ 2u{{\xi }_1}_x-2u{{\tau }_1}_t-{\eta }_1=0,{{\theta }_1}_x+{{\xi }_1}_y=0,-2u{{\tau }_1}_t+2u{{\theta }_1}_y-{\eta }_1=0,\hfill \end{array}$$

Solving this system of PDEs, we obtain:

$$\begin{array}{cc}{\xi }_1=\hfill & b_{3}x+b_{1}y+b_2,{\theta }_1=b_{3}y-b_{1}x+b_6,\hfill \\ {\tau }_1=\hfill & b_{4}t+b_5,{\eta }_1=2u(b_3-b_4),\hfill \end{array}$$

(62)

where $b_1,b_2,b_3,b_4,b_5$ and $b_6$ are arbitrary constants.

Case I: $a_3=0.$ The scaling operator:

$$X_3=x\frac{\partial }{\partial x}+y\frac{\partial }{\partial y}+2u\frac{\partial }{\partial u}$$

is not stable, and hence, Equation (56) does not inherit symmetries of its unperturbed part.

Case II: Solving the first order linear differential equation $u{f}^{\prime }\left(u\right)+f\left(u\right)=0,$ we obtain $f\left(u\right)=k_1/u$, where $k_1$ is a constant. The approximate symmetry generator of Equation (56) is given by:

$$\begin{array}{cc}\hfill X=& X_0+\epsilon X_1\hfill \\ \hfill =& [(a_3+\epsilon b_3)x+(a_1+\epsilon b_1)y+(a2+\epsilon b_2)]\frac{\partial }{\partial x}\hfill \\ \hfill & +[(a_3+\epsilon b_3)y-(a_1+\epsilon b_1)x+(a_6+\epsilon b_6)]\frac{\partial }{\partial y}\hfill \\ & +[(a_4+\epsilon b_4)t+(a_5+\epsilon b_5)]\frac{\partial }{\partial t}+\left[2u((a_3+\epsilon b_3)-(a_4+\epsilon b_4))\right]\frac{\partial }{\partial u}\hfill \end{array}$$

(63)

These additional symmetries are actually the same as those obtained from the unperturbed equation that are considered as trivial symmetries. To summarize: in this case, Method I only gives trivial symmetries.

4.2. Approximate Symmetries by Method II

We expand the dependent variable to the first order of $\epsilon$ as follows:

$$u=v+\epsilon w+o\left(\epsilon \right),\epsilon \to 0.$$

Taylor expansion of f in the first order of precision is given by:

$$f\left(u\right)=\frac{k_1}{v}\left(1-\frac{\epsilon w}{v}+o\left(\epsilon \right)\right)=\frac{k_1}{v}-\frac{\epsilon k_{1}w}{v^2}+o\left(\epsilon \right),\epsilon \to 0.$$

Substituting the above expansion into Equation (56) and separating at each order of perturbation parameter, one may obtain:

$$\begin{array}{cc}\hfill v_{tt}-{v_x}^2-v{v}_{xx}-{v_y}^2-v{v}_{yy}& =0,\hfill \\ \hfill w_{tt}-2v_x{w}_x-v{w}_{xx}-w{v}_{xx}-2v_y{w}_y-v{w}_{yy}-w{v}_{yy}& =\frac{k_1}{v}.\hfill \end{array}$$

(64)

Now, the infinitesimal generator for the problem is:

$$\begin{array}{c}\hfill X=\tau (t,x,y,,v,w)\frac{\partial }{\partial t}+\xi (t,x,y,v,w)\frac{\partial }{\partial x}+\theta (t,x,y,v,w)\frac{\partial }{\partial y}\\ \hfill +\varphi (t,x,y,v,w)\frac{\partial }{\partial v}+\eta (t,x,y,v,w)\frac{\partial }{\partial w}.\end{array}$$

(65)

Using standard Lie group analysis, we obtain the infinitesimals as follows:

$$\begin{array}{c}\tau =c_{4}t+c_5,\xi =c_{1}x-c_{3}y+c_6,\theta =c_{3}x+c_{1}y+c_2,\hfill \\ \varphi =2v(-c_4+c_1),\eta =-2w(c_1-2c_4),\hfill \end{array}$$

(66)

where $c_1,c_2,c_3,c_4,c_5$ and $c_6$ are arbitrary constants. Hence, we have the following symmetries:

$$\begin{array}{c}{X}_1=x\frac{\partial }{\partial x}+y\frac{\partial }{\partial y}+2v\frac{\partial }{\partial v}-2w\frac{\partial }{\partial v},X_2=\frac{\partial }{\partial y},X_3=-y\frac{\partial }{\partial x}+x\frac{\partial }{\partial y},\hfill \\ X_4=t\frac{\partial }{\partial t}-2v\frac{\partial }{\partial v}+4w\frac{\partial }{\partial w},X_5=\frac{\partial }{\partial t},X_6=\frac{\partial }{\partial x}.\hfill \end{array}$$

(67)

Table 3 shows that Equation (67) spans a sixth-dimensional Lie algebra.

Table 3. Commutators span six-dimensional Lie algebra.

	${\mathit{X}}_1$	${\mathit{X}}_2$	${\mathit{X}}_3$	${\mathit{X}}_4$	${\mathit{X}}_5$	${\mathit{X}}_6$
$X_1$	0	$-X_2$	0	0	0	$-X_6$
$X_2$	$X_2$	0	$-X_6$	0	0	0
$X_3$	0	$X_6$	0	0	0	$-X_2$
$X_4$	0	0	0	0	$-X_5$	0
$X_5$	0	0	0	$X_5$	0	0
$X_6$	$X_6$	0	$X_2$	0	0	0

4.3. Approximate Invariant Solution

Using $X_3$ from Equation (67), we retrieve the following characteristic equations:

$$\frac{dx}{-y}=\frac{dy}{x}=\frac{dv}{0}=\frac{dw}{0}.$$

(68)

The equations in Equation (68) yield $\alpha =x^2+y^2$ and suggest that $w=w\left(\alpha \right),v=v\left(\alpha \right).$ Derivatives of dependent variables v and w with respect to x and y are:

$$\begin{array}{c}{v}_t=v_{\alpha }\frac{\partial \alpha }{\partial t}=0,v_{tt}=0\hfill \\ v_x=v-\alpha \frac{\partial \alpha }{\partial x}=2x{v}_{\alpha },v_{xx}=2v_{\alpha }+2x{v}_{\alpha \alpha }\frac{\partial \alpha }{\partial x}=2v_{\alpha }+4x^2{v}_{\alpha \alpha }\hfill \\ v_y=v-\alpha \frac{\partial \alpha }{\partial y}=2y{v}_{\alpha },v_{yy}=2v_{\alpha }+2y{v}_{\alpha \alpha }\frac{\partial \alpha }{\partial y}=2v_{\alpha }+4y^2{v}_{\alpha \alpha }\hfill \\ w_t=w_{\alpha }\frac{\partial \alpha }{\partial t}=0,w_{tt}=0\hfill \\ w_x=w-\alpha \frac{\partial \alpha }{\partial x}=2x{w}_{\alpha },w_{xx}=2w_{\alpha }+2x{w}_{\alpha \alpha }\frac{\partial \alpha }{\partial x}=2w_{\alpha }+4x^2{w}_{\alpha \alpha }\hfill \\ w_y=w-\alpha \frac{\partial \alpha }{\partial y}=2y{w}_{\alpha },w_{yy}=2w_{\alpha }+2y{w}_{\alpha \alpha }\frac{\partial \alpha }{\partial y}=2w_{\alpha }+4y^2{w}_{\alpha \alpha }.\hfill \end{array}$$

These equations lead to the following second order ordinary differential equations:

$$\begin{array}{cc}\hfill \alpha v_{\alpha }^2+\alpha v{v}_{\alpha \alpha }+v{v}_{\alpha }& =0,\hfill \\ \hfill 2\alpha v_{\alpha }{w}_{\alpha }+v{w}_{\alpha }+\alpha v{w}_{\alpha \alpha }+w{v}_{\alpha }+\alpha w{v}_{\alpha \alpha }& =\frac{k_1}{4v}.\hfill \end{array}$$

(69)

Substituting:

$$v\left(\alpha \right)=\frac{H{\left(\alpha \right)}^2}{2}$$

into the first equation of Equation (69), we obtain:

$$\frac{\partial H\left(\alpha \right)}{\partial \alpha }+\alpha \frac{{\partial }^{2}H\left(\alpha \right)}{\partial {\alpha }^2}=0.$$

We have $H\left(\alpha \right)=c_{1}ln\alpha +c_2,$ where $c_1$ and $c_2$ are arbitrary constants of the integration. Thus, $v\left(\alpha \right)=\sqrt{c_2+c_{1}ln\alpha }.$ Put $c_2=0,c_1=1.$ The second equation of Equation (69) is reduced to the following second-order ordinary differential equation:

$$(1+ln\alpha )w_{\alpha }+(\alpha ln\alpha )w_{\alpha \alpha }-\frac{1}{4ln\alpha }w=\frac{k_1}{4}.$$

(70)

Observe that it is not straight forward to obtain a solution for Equation (70). However, we may obtain an asymptotic estimate of the solution of Equation (70) using the asymptotic expansions [22].

Definition 1.

The function $f\left(x\right)=O\left(g\right(x\left)\right)$ as $x\to x_0$ if there exists a constant C such that ${lim}_{x\to x_0}f/g=C.$

In Equation (70), we have $(1+ln\alpha )=O\left(\alpha \right),$ $(\alpha ln{\alpha }^2)=O\left({\alpha }^2\right)$ and $\frac{1}{4ln\alpha }=O\left(1\right)$ as $\alpha \to \infty .$ For large values of $\alpha ,$ Equation (70) is asymptotically equivalent to the following equation:

$${\alpha }^2{w}_{\alpha \alpha }+\alpha w_{\alpha }+w=\frac{k_1}{4}.$$

(71)

The solution of the above non-homogeneous Cauchy–Euler equation is:

$$w\left(\alpha \right)=k_{2}sin(ln\alpha )+k_{3}cos(ln\alpha )+\frac{k_1}{4}$$

where $k_2,k_3$ are constants. Lastly, we re-cast the solution in original coordinates as:

$$\begin{array}{cc}\hfill u(t,x,y)& =v(t,x,y)+\epsilon w(t,x,y)\hfill \\ & =\sqrt{ln(x^2+y^2)}+\epsilon \left(k_{2}sin(ln(x^2+y^2))+k_{3}cos(ln(x^2+y^2))+\frac{k_1}{4}\right).\hfill \end{array}$$

(72)

This is an approximate solution invariant under rotation in $x-y,$ dilation in space and u coordinates. We depict an invariant solution for the unperturbed equation in Figure 3 and an approximate solution of the perturbed one in Figure 4.

Figure 3. Invariant solution of the unperturbed equation of Equation (56) for $-10\le x\le 10$, $-10\le y\le 10,$ $x^2+y^2\ge 1$.

Figure 4. Approximate invariant solution of Equation (56) for $-10\le x\le 10,$ $-10\le y\le 10,$ $x^2+y^2\ge 1,$ $k_1=4,k_2=1,k3=1,$ $\epsilon =0.1$.

5. Concluding Remarks

In this work, we have studied a class of perturbed nonlinear wave equations via Lie symmetry analysis. Two methods have been employed to obtain approximate symmetries used to construct invariant solutions of the equations. There was a case where Method I gives only trivial solutions. We applied Method II to this case and obtained the invariant solutions of the equation. Many problems arising from physical or engineering situations may be dealt with by approximate Lie symmetry analysis. We plan to investigate modified and perturbed forms of Korteweg-de Vries (KdV) equations using this approach.