# Solutions and Conservation Laws of a (2+1)-Dimensional Boussinesq Equation

Authors: Letlhogonolo Daddy Moleleki; Chaudry Masood Khalique
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We study a nonlinear evolution partial differential equation, namely, the (2+1)-dimensional Boussinesq equation. For the first time Lie symmetry method together with simplest equation method is used to find the exact solutions of the (2+1)-dimensional Boussinesq equation. Furthermore, the new conservation theorem due to Ibragimov will be utilized to construct the conservation laws of the (2+1)-dimensional Boussinesq equation.


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## Body

## 1. Introduction

Nonlinear evolution equations (NLEEs) are broadly used as models to represent physical phenomena in numerous fields of sciences, especially in biology, solid state physics, plasma physics, plasma waves, and fluid mechanics. It is therefore of paramount importance that exact solutions of such NLEEs are obtained. However, finding exact solutions of NLEEs is an onerous exercise and only in certain distinctive cases one can explicitly write down their solutions. Nevertheless, in the last few decades important progress has been made and many powerful and effective methods for obtaining exact solutions of NLEEs have been suggested in the literature. Some of the important methods found in the literature include the Darboux transformation method [1], the inverse scattering transform method [2], Hirota’s bilinear method [3], Jacobi elliptic function expansion method [4], the sine-cosine method [5], the auxiliary ordinary differential equation method [6], Lie symmetry analysis [7–11], the F-expansion method [12], and the exp-function expansion method [13].In this paper we consider one such NLEE, namely, the (2+1)-dimensional Boussinesq equation given by(1)utt-uxx-uyy-α(u2)xx-uxxxx=0,
which describes the propagation of gravity waves on the surface of water; in particular it describes the head-on collision of an oblique wave. The (2+1)-dimensional Boussinesq equation (1) combines the two-way propagation of the classical Boussinesq equation with the dependence on a second spatial variable, as that occurs in the two-dimensional Kadomstev-Petviashvili (KP) equation. This equation provides a description of head-on collision of oblique waves and it possesses some interesting properties. The unknown function describes the elevation of the free surface of the fluid. Moreover, (1) involves the two dissipative terms uxx and uyy in addition to the fourth-order spatial derivative uxxxx that represents the dispersion phenomenon. Unlike the standard Boussinesq equation, which is completely integrable, that admits multiple solitons solutions, (1) is not integrable and gives two soliton solutions at most. It is to be noted that if we delete the dissipative term uyy from (1), we obtain the standard Boussinesq equation. The standard Boussinesq equation arises in many physical applications such as nonlinear lattice waves and iron sound waves in plasma and in vibrations in a nonlinear string. It is used in many physical applications such as the percolation of water in porous subsurface of a horizontal layer of material. See also [14].In [15] the authors used a generalized transformation in homogeneous balance method and found some explicit solitary wave solutions of the (2+1)-dimensional Boussinesq equation. Applied homotopy perturbation method was used in [16] to construct numerical solutions of (1). Extended ansätz method was employed in [17] to derive exact periodic solitary wave solutions. Recently, the Hirota bilinear method was used in [18] to obtain two soliton solutions.Lie group method is one of the most effective methods to find solutions of nonlinear partial differential equations (PDEs). Originally, developed by Sophus Lie (1842–1899) in the latter half of the nineteenth century, this method is based upon the study of the invariance under one parameter Lie group of point transformations and it is highly algorithmic [7–9].In the study of PDEs, conservation laws play a vital role in their solution process and the reduction of PDEs. Conservation laws have been broadly used in studying the existence, uniqueness, and stability of solutions of nonlinear PDEs (see, e.g., [19–21]). They have also been used in the development of numerical methods (see, e.g., [22, 23]). Exact solutions (by exploiting a double reduction method) of some classical partial differential equations have been obtained using conserved vectors associated with the Lie point symmetries [24–26].In this paper, for the first time, Lie group analysis in conjunction with the simplest equation method [27, 28] is employed to obtain some exact solutions of (1). In addition to this, conservation laws will be derived for (1) using the new conservation theorem due to Ibragimov [29].


## 2. Solutions of (1)

In this section we obtain exact solutions of (1) using Lie group analysis along with the simplest equation method.


### 2.1. Exact Solutions Using Lie Point Symmetries

In this subsection we first calculate the Lie point symmetries of (1) and later use the translation symmetries to construct the exact solutions.


#### 2.1.1. Lie Point Symmetries

The symmetry group of the (2+1)-dimensional Boussinesq equation (1) will be generated by the vector field of the form
(2)R=ξ1∂∂x+ξ2∂∂y+ξ3∂∂t+η∂∂u,
where ξi, i=1,2,3 and η depend on x, y, t and u. Applying the fourth prolongation pr(4)R to (1) we obtain an overdetermined system of linear partial differential equations. Solving this resultant system one obtains the following five Lie point symmetries:
(3)R1=∂∂x,R2=∂∂t,R3=∂∂y,R4=y∂∂t+t∂∂y,R5=-2αt∂∂t-αx∂∂x-2αy∂∂y+(1+2αu)∂∂u.
We now utilize the symmetry R=R1+R2+cR3, where c is a constant, and reduce the Boussinesq equation (1) to a PDE in two independent variables. Solving the associated Lagrange system for R, we obtain the following three invariants:
(4)f=y-ct,g=t-x,θ=u.
Now treating θ as the new dependent variable and f and g as new independent variables, the Boussinesq equation (1) transforms to
(5)(1-c2)θff+2cθfg+2αθg2+2αθθgg+θgggg=0,
which is a nonlinear PDE in two independent variables. We now use the Lie point symmetries of (5) and transform it to an ordinary differential equation (ODE). The equation (5) has the following three symmetries:
(6)Γ1=∂∂g,Γ2=∂∂f,Γ3=(2αf-2αfc2)∂∂f+(αcf-c2αg+αg)∂∂g+(c2+2c2αθ-2αθ)∂∂θ.
The combination of the first two translational symmetries, Γ=Γ1+νΓ2, where ν is a constant, yields the two invariants
(7)z=f-νg,ψ=θ,
which gives rise to a group-invariant solution ψ=ψ(z), and consequently using these invariants, (5) is transformed into the fourth-order nonlinear ODE
(8)2αν2ψ′2+(1-c2-2cν)ψ′′+2αν2ψψ′′+ν4ψ′′′′=0.
Integrating the above equation four times and taking the constants of integration to be zero (because we are looking for soliton solutions) and reverting back to the original variables, we obtain the following group-invariant solutions of the Boussinesq equation (1):
(9)u(x,y,t)=A1A2sech2[A12(B±z)],
where B is a constant of integration and
(10)A1=c2+2νc-1ν4,A2=2α3ν2,z=νx+y-(c+ν)t.
It is worth noting that the obtained solitary wave solution u(x,y,t) is a regular soliton in the form of a bell-shaped soliton.


### 2.2. Exact Solutions of (1) Using Simplest Equation Method

In this section we employ the simplest equation method [27, 28] to solve the nonlinear ODE (8). This will then give us the exact solutions for our Boussinesq equation (1). The simplest equations that we will use in our work are the Bernoulli and Riccati equations.Here we first present the simplest equation method and consider the solutions of (8) in the form
(11)F(z)=∑i=0M‍Ai(G(z))i,
where G(z) satisfies the Bernoulli and Riccati equations, M is a positive integer that can be determined by balancing procedure, and A0,…,AM are parameters to be determined.The Bernoulli equation(12)G′(z)=aG(z)+bG2(z),
where a and b are arbitrary constants has the general solution given by
(13)G(z)=a{cosh[a(z+C)]+sinh[a(z+C)]1-bcosh[a(z+C)]-bsinh[a(z+C)]}
and represents a solitary wave solution.The Riccati equation considered in this work is(14)G′(z)=bG2(z)+aG(z)+d,
where a, b, and d are arbitrary constants. Its solutions are
(15)G(z)=-a2b-θ2btanh[12θ(z+C)],G(z)=-a2b-θ2btanh(12θz)+sech(θz/2)Ccosh(θz/2)-(2b/θ)sinh(θz/2),
with θ2=a2-4bd and C is an arbitrary constant of integration.


#### 2.2.1. Solutions of (1) Using the Bernoulli Equation as the Simplest Equation

The balancing procedure givesM=2 so the solutions of (8) are of the form
(16)F(z)=A0+A1G+A2G2.
Inserting (16) into (8) and using the Bernoulli equation (12) and, thereafter, equating the coefficients of powers of Gi to zero, we obtain an algebraic system of six equations in terms of A0, A1, and A2, namely,
(17)-120ν4A2b4-20αν2A22b2=0,-336ν4A2ab3-36αν2A22ab-24ν4A1b4-24αν2A1A2b2=0,-A1a2+2νA1a2c-νA1a4+A1a2c2-2αν2A1A2a2=0,-16αν2A22a2+12νA2b2c-6αν2A12b2-60ν4A1ab3-12αν2A0A1b2-330ν4A2a2b2-42αν2A1abA2+6A2b2c2-6A2b2=0,-15ν4A1a3b+8νA2a2c-3A1ab-4A2a2+4A2a2c2-4αν2A12a2-6αν2A0A1ab-16ν4A2a4+6A1abcν-8αν2A0A2a2+3A1abc2=0,-18αν2A1A2a2-10αν2A12ab-4αν2A0A1b2+10A2abc2+4νA1b2c+20νA2abc-2A1b2+2A1b2c2-20αν2A0A2ab-10A2ab-130ν4A2a3b-50ν4A1a2b2=0.
With the aid of Mathematica, solving the above system of algebraic equations, one possible solution for A0, A1, and A2 is
(18)A0=-(1-c2-2cν+a2ν4)2αν2,A1=-6abν2α,A2=-6b2ν2α.
Thus, reverting back to the original variables, a solution of (1) is
(19)u(t,x,y)=A0+A1a{cosh[a(z+C)]+sinh[a(z+C)]1-bcosh[a(z+C)]-bsinh[a(z+C)]}+A2a2{cosh[a(z+C)]+sinh[a(z+C)]1-bcosh[a(z+C)]-bsinh[a(z+C)]}2,
where z=νx+y-(c+ν)t and C is an arbitrary constant of integration, which represents a solitary wave solution.


#### 2.2.2. Solutions of (1) Using the Riccati Equation as the Simplest Equation

The balancing procedure yieldsM=2 so the solutions of (8) take the form
(20)F(z)=A0+A1G+A2G2.
Inserting (20) into (8) and making use of the Riccati equation (14), we obtain algebraic system of equations in terms of A0, A1, and A2 by equating the coefficients of powers of Gi to zero. The resulting algebraic equations are
(21)-120ν4A2b4-20αν2A22b2=0,-36αν2A22ab-336ν4A2ab3-24ν4A1b4-24αν2A1A2b2=0,-32αν2A22bd-6αν2A12b2-240ν4A2b3d+6A2b2c2+12A2b2cν-6A2b2-12αν2A0A1b2-42αν2A1A2ab-16αν2A22a2-60ν4A1ab3-330A2a2b2=0,-16ν4A2bd3-14ν4A2a2d2+2A1acdν-A1ad+4A2cd2ν+A1ac2d-8ν4A1abd2+2A2c2d2-ν4A1a3d-2αν2A0A1ad-4αν2A0A2d2-2αν2A12d2-2A2d2=0,2A1b2c2-28αν2A22ad-20αν2A0A2ab-36αν2A1A2bd-18αν2A1A2a2-10αν2A12ab-10A2ab+10A2abc2+4A1b2cν-40ν4A1b3d-4αν2A0A1b2-50ν4A1a2b2+20A2abcν-2A1b2-130ν4A2a3b-440ν4A2ab2d=0,2A1a2cν-6A2ad-ν4A1a4+12A2acdν-6αν2A12ad+6A2ac2d+A1a2c2-12αν2A1A2d2-4αν2A0A1bd-120ν4A2abd2+4A1bcdν-2αν2A0A1a2-12αν2A0A2ad-16ν4A1b2d2-A1a2-30ν4A2a3d-2A1bd+2A1bc2d-22ν4A1a2bd=0,-8αν2A0A2a2+3A1abc2-8A2bd+6A1abcν-3A1ab-6αν2A0A1ab-136ν4A2b2d2-4A2a2-12αν2A22d2+8A2a2cν-16αν2A0A2bd-232ν4A2a2bd-8αν2A12bd+16A2bcdν-15ν4A1a3b-16ν4A2a4-60ν4A1ab2d+8A2bc2d+4A2a2c2-30αν2A1A2ad-4αν2A12a2=0.
Solving the above equations, we get
(22)A0=-8bdν4-a2ν4+c2+2cν-12ν2α,A1=-6abν2α,A2=-6b2ν2α,
and consequently, the solutions of (1) are
(23)u(t,x,y)=A0+A1{-a2b-θ2btanh[12θ(z+C)]}+A2{-a2b-θ2btanh[12θ(z+C)]}2,u(t,x,y)=A0+A1{sech(θz2)Ccosh(θz2)-2bθsinh(θz2)-a2b-θ2btanh(12θz)+sech(θz/2)Ccosh(θz/2)-(2b/θ)sinh(θz/2)-a2b-θ2btanh(12θz)sech(θz2)Ccosh(θz2)-(2bθ)sinh(θz2)}+A2{-a2b-θ2btanh(12θz)sech(θz2)Ccosh(θz2)-(2bθ)sinh(θz2)+sech(θz/2)Ccosh(θz/2)-(2b/θ)sinh(θz/2)-a2b-θ2btanh(12θz)sech(θz2)Ccosh(θz2)-(2bθ)sinh(θz2)}2,
where z=νx+y-(c+ν)t and C is an arbitrary constant of integration.The solution (16) is a solitary wave solution in the form of a kink solution.


## 2.1. Exact Solutions Using Lie Point Symmetries

In this subsection we first calculate the Lie point symmetries of (1) and later use the translation symmetries to construct the exact solutions.


### 2.1.1. Lie Point Symmetries

The symmetry group of the (2+1)-dimensional Boussinesq equation (1) will be generated by the vector field of the form
(2)R=ξ1∂∂x+ξ2∂∂y+ξ3∂∂t+η∂∂u,
where ξi, i=1,2,3 and η depend on x, y, t and u. Applying the fourth prolongation pr(4)R to (1) we obtain an overdetermined system of linear partial differential equations. Solving this resultant system one obtains the following five Lie point symmetries:
(3)R1=∂∂x,R2=∂∂t,R3=∂∂y,R4=y∂∂t+t∂∂y,R5=-2αt∂∂t-αx∂∂x-2αy∂∂y+(1+2αu)∂∂u.
We now utilize the symmetry R=R1+R2+cR3, where c is a constant, and reduce the Boussinesq equation (1) to a PDE in two independent variables. Solving the associated Lagrange system for R, we obtain the following three invariants:
(4)f=y-ct,g=t-x,θ=u.
Now treating θ as the new dependent variable and f and g as new independent variables, the Boussinesq equation (1) transforms to
(5)(1-c2)θff+2cθfg+2αθg2+2αθθgg+θgggg=0,
which is a nonlinear PDE in two independent variables. We now use the Lie point symmetries of (5) and transform it to an ordinary differential equation (ODE). The equation (5) has the following three symmetries:
(6)Γ1=∂∂g,Γ2=∂∂f,Γ3=(2αf-2αfc2)∂∂f+(αcf-c2αg+αg)∂∂g+(c2+2c2αθ-2αθ)∂∂θ.
The combination of the first two translational symmetries, Γ=Γ1+νΓ2, where ν is a constant, yields the two invariants
(7)z=f-νg,ψ=θ,
which gives rise to a group-invariant solution ψ=ψ(z), and consequently using these invariants, (5) is transformed into the fourth-order nonlinear ODE
(8)2αν2ψ′2+(1-c2-2cν)ψ′′+2αν2ψψ′′+ν4ψ′′′′=0.
Integrating the above equation four times and taking the constants of integration to be zero (because we are looking for soliton solutions) and reverting back to the original variables, we obtain the following group-invariant solutions of the Boussinesq equation (1):
(9)u(x,y,t)=A1A2sech2[A12(B±z)],
where B is a constant of integration and
(10)A1=c2+2νc-1ν4,A2=2α3ν2,z=νx+y-(c+ν)t.
It is worth noting that the obtained solitary wave solution u(x,y,t) is a regular soliton in the form of a bell-shaped soliton.


## 2.1.1. Lie Point Symmetries

The symmetry group of the (2+1)-dimensional Boussinesq equation (1) will be generated by the vector field of the form
(2)R=ξ1∂∂x+ξ2∂∂y+ξ3∂∂t+η∂∂u,
where ξi, i=1,2,3 and η depend on x, y, t and u. Applying the fourth prolongation pr(4)R to (1) we obtain an overdetermined system of linear partial differential equations. Solving this resultant system one obtains the following five Lie point symmetries:
(3)R1=∂∂x,R2=∂∂t,R3=∂∂y,R4=y∂∂t+t∂∂y,R5=-2αt∂∂t-αx∂∂x-2αy∂∂y+(1+2αu)∂∂u.
We now utilize the symmetry R=R1+R2+cR3, where c is a constant, and reduce the Boussinesq equation (1) to a PDE in two independent variables. Solving the associated Lagrange system for R, we obtain the following three invariants:
(4)f=y-ct,g=t-x,θ=u.
Now treating θ as the new dependent variable and f and g as new independent variables, the Boussinesq equation (1) transforms to
(5)(1-c2)θff+2cθfg+2αθg2+2αθθgg+θgggg=0,
which is a nonlinear PDE in two independent variables. We now use the Lie point symmetries of (5) and transform it to an ordinary differential equation (ODE). The equation (5) has the following three symmetries:
(6)Γ1=∂∂g,Γ2=∂∂f,Γ3=(2αf-2αfc2)∂∂f+(αcf-c2αg+αg)∂∂g+(c2+2c2αθ-2αθ)∂∂θ.
The combination of the first two translational symmetries, Γ=Γ1+νΓ2, where ν is a constant, yields the two invariants
(7)z=f-νg,ψ=θ,
which gives rise to a group-invariant solution ψ=ψ(z), and consequently using these invariants, (5) is transformed into the fourth-order nonlinear ODE
(8)2αν2ψ′2+(1-c2-2cν)ψ′′+2αν2ψψ′′+ν4ψ′′′′=0.
Integrating the above equation four times and taking the constants of integration to be zero (because we are looking for soliton solutions) and reverting back to the original variables, we obtain the following group-invariant solutions of the Boussinesq equation (1):
(9)u(x,y,t)=A1A2sech2[A12(B±z)],
where B is a constant of integration and
(10)A1=c2+2νc-1ν4,A2=2α3ν2,z=νx+y-(c+ν)t.
It is worth noting that the obtained solitary wave solution u(x,y,t) is a regular soliton in the form of a bell-shaped soliton.


## 2.2. Exact Solutions of (1) Using Simplest Equation Method

In this section we employ the simplest equation method [27, 28] to solve the nonlinear ODE (8). This will then give us the exact solutions for our Boussinesq equation (1). The simplest equations that we will use in our work are the Bernoulli and Riccati equations.Here we first present the simplest equation method and consider the solutions of (8) in the form
(11)F(z)=∑i=0M‍Ai(G(z))i,
where G(z) satisfies the Bernoulli and Riccati equations, M is a positive integer that can be determined by balancing procedure, and A0,…,AM are parameters to be determined.The Bernoulli equation(12)G′(z)=aG(z)+bG2(z),
where a and b are arbitrary constants has the general solution given by
(13)G(z)=a{cosh[a(z+C)]+sinh[a(z+C)]1-bcosh[a(z+C)]-bsinh[a(z+C)]}
and represents a solitary wave solution.The Riccati equation considered in this work is(14)G′(z)=bG2(z)+aG(z)+d,
where a, b, and d are arbitrary constants. Its solutions are
(15)G(z)=-a2b-θ2btanh[12θ(z+C)],G(z)=-a2b-θ2btanh(12θz)+sech(θz/2)Ccosh(θz/2)-(2b/θ)sinh(θz/2),
with θ2=a2-4bd and C is an arbitrary constant of integration.


### 2.2.1. Solutions of (1) Using the Bernoulli Equation as the Simplest Equation

The balancing procedure givesM=2 so the solutions of (8) are of the form
(16)F(z)=A0+A1G+A2G2.
Inserting (16) into (8) and using the Bernoulli equation (12) and, thereafter, equating the coefficients of powers of Gi to zero, we obtain an algebraic system of six equations in terms of A0, A1, and A2, namely,
(17)-120ν4A2b4-20αν2A22b2=0,-336ν4A2ab3-36αν2A22ab-24ν4A1b4-24αν2A1A2b2=0,-A1a2+2νA1a2c-νA1a4+A1a2c2-2αν2A1A2a2=0,-16αν2A22a2+12νA2b2c-6αν2A12b2-60ν4A1ab3-12αν2A0A1b2-330ν4A2a2b2-42αν2A1abA2+6A2b2c2-6A2b2=0,-15ν4A1a3b+8νA2a2c-3A1ab-4A2a2+4A2a2c2-4αν2A12a2-6αν2A0A1ab-16ν4A2a4+6A1abcν-8αν2A0A2a2+3A1abc2=0,-18αν2A1A2a2-10αν2A12ab-4αν2A0A1b2+10A2abc2+4νA1b2c+20νA2abc-2A1b2+2A1b2c2-20αν2A0A2ab-10A2ab-130ν4A2a3b-50ν4A1a2b2=0.
With the aid of Mathematica, solving the above system of algebraic equations, one possible solution for A0, A1, and A2 is
(18)A0=-(1-c2-2cν+a2ν4)2αν2,A1=-6abν2α,A2=-6b2ν2α.
Thus, reverting back to the original variables, a solution of (1) is
(19)u(t,x,y)=A0+A1a{cosh[a(z+C)]+sinh[a(z+C)]1-bcosh[a(z+C)]-bsinh[a(z+C)]}+A2a2{cosh[a(z+C)]+sinh[a(z+C)]1-bcosh[a(z+C)]-bsinh[a(z+C)]}2,
where z=νx+y-(c+ν)t and C is an arbitrary constant of integration, which represents a solitary wave solution.


### 2.2.2. Solutions of (1) Using the Riccati Equation as the Simplest Equation

The balancing procedure yieldsM=2 so the solutions of (8) take the form
(20)F(z)=A0+A1G+A2G2.
Inserting (20) into (8) and making use of the Riccati equation (14), we obtain algebraic system of equations in terms of A0, A1, and A2 by equating the coefficients of powers of Gi to zero. The resulting algebraic equations are
(21)-120ν4A2b4-20αν2A22b2=0,-36αν2A22ab-336ν4A2ab3-24ν4A1b4-24αν2A1A2b2=0,-32αν2A22bd-6αν2A12b2-240ν4A2b3d+6A2b2c2+12A2b2cν-6A2b2-12αν2A0A1b2-42αν2A1A2ab-16αν2A22a2-60ν4A1ab3-330A2a2b2=0,-16ν4A2bd3-14ν4A2a2d2+2A1acdν-A1ad+4A2cd2ν+A1ac2d-8ν4A1abd2+2A2c2d2-ν4A1a3d-2αν2A0A1ad-4αν2A0A2d2-2αν2A12d2-2A2d2=0,2A1b2c2-28αν2A22ad-20αν2A0A2ab-36αν2A1A2bd-18αν2A1A2a2-10αν2A12ab-10A2ab+10A2abc2+4A1b2cν-40ν4A1b3d-4αν2A0A1b2-50ν4A1a2b2+20A2abcν-2A1b2-130ν4A2a3b-440ν4A2ab2d=0,2A1a2cν-6A2ad-ν4A1a4+12A2acdν-6αν2A12ad+6A2ac2d+A1a2c2-12αν2A1A2d2-4αν2A0A1bd-120ν4A2abd2+4A1bcdν-2αν2A0A1a2-12αν2A0A2ad-16ν4A1b2d2-A1a2-30ν4A2a3d-2A1bd+2A1bc2d-22ν4A1a2bd=0,-8αν2A0A2a2+3A1abc2-8A2bd+6A1abcν-3A1ab-6αν2A0A1ab-136ν4A2b2d2-4A2a2-12αν2A22d2+8A2a2cν-16αν2A0A2bd-232ν4A2a2bd-8αν2A12bd+16A2bcdν-15ν4A1a3b-16ν4A2a4-60ν4A1ab2d+8A2bc2d+4A2a2c2-30αν2A1A2ad-4αν2A12a2=0.
Solving the above equations, we get
(22)A0=-8bdν4-a2ν4+c2+2cν-12ν2α,A1=-6abν2α,A2=-6b2ν2α,
and consequently, the solutions of (1) are
(23)u(t,x,y)=A0+A1{-a2b-θ2btanh[12θ(z+C)]}+A2{-a2b-θ2btanh[12θ(z+C)]}2,u(t,x,y)=A0+A1{sech(θz2)Ccosh(θz2)-2bθsinh(θz2)-a2b-θ2btanh(12θz)+sech(θz/2)Ccosh(θz/2)-(2b/θ)sinh(θz/2)-a2b-θ2btanh(12θz)sech(θz2)Ccosh(θz2)-(2bθ)sinh(θz2)}+A2{-a2b-θ2btanh(12θz)sech(θz2)Ccosh(θz2)-(2bθ)sinh(θz2)+sech(θz/2)Ccosh(θz/2)-(2b/θ)sinh(θz/2)-a2b-θ2btanh(12θz)sech(θz2)Ccosh(θz2)-(2bθ)sinh(θz2)}2,
where z=νx+y-(c+ν)t and C is an arbitrary constant of integration.The solution (16) is a solitary wave solution in the form of a kink solution.


## 2.2.1. Solutions of (1) Using the Bernoulli Equation as the Simplest Equation

The balancing procedure givesM=2 so the solutions of (8) are of the form
(16)F(z)=A0+A1G+A2G2.
Inserting (16) into (8) and using the Bernoulli equation (12) and, thereafter, equating the coefficients of powers of Gi to zero, we obtain an algebraic system of six equations in terms of A0, A1, and A2, namely,
(17)-120ν4A2b4-20αν2A22b2=0,-336ν4A2ab3-36αν2A22ab-24ν4A1b4-24αν2A1A2b2=0,-A1a2+2νA1a2c-νA1a4+A1a2c2-2αν2A1A2a2=0,-16αν2A22a2+12νA2b2c-6αν2A12b2-60ν4A1ab3-12αν2A0A1b2-330ν4A2a2b2-42αν2A1abA2+6A2b2c2-6A2b2=0,-15ν4A1a3b+8νA2a2c-3A1ab-4A2a2+4A2a2c2-4αν2A12a2-6αν2A0A1ab-16ν4A2a4+6A1abcν-8αν2A0A2a2+3A1abc2=0,-18αν2A1A2a2-10αν2A12ab-4αν2A0A1b2+10A2abc2+4νA1b2c+20νA2abc-2A1b2+2A1b2c2-20αν2A0A2ab-10A2ab-130ν4A2a3b-50ν4A1a2b2=0.
With the aid of Mathematica, solving the above system of algebraic equations, one possible solution for A0, A1, and A2 is
(18)A0=-(1-c2-2cν+a2ν4)2αν2,A1=-6abν2α,A2=-6b2ν2α.
Thus, reverting back to the original variables, a solution of (1) is
(19)u(t,x,y)=A0+A1a{cosh[a(z+C)]+sinh[a(z+C)]1-bcosh[a(z+C)]-bsinh[a(z+C)]}+A2a2{cosh[a(z+C)]+sinh[a(z+C)]1-bcosh[a(z+C)]-bsinh[a(z+C)]}2,
where z=νx+y-(c+ν)t and C is an arbitrary constant of integration, which represents a solitary wave solution.


## 2.2.2. Solutions of (1) Using the Riccati Equation as the Simplest Equation

The balancing procedure yieldsM=2 so the solutions of (8) take the form
(20)F(z)=A0+A1G+A2G2.
Inserting (20) into (8) and making use of the Riccati equation (14), we obtain algebraic system of equations in terms of A0, A1, and A2 by equating the coefficients of powers of Gi to zero. The resulting algebraic equations are
(21)-120ν4A2b4-20αν2A22b2=0,-36αν2A22ab-336ν4A2ab3-24ν4A1b4-24αν2A1A2b2=0,-32αν2A22bd-6αν2A12b2-240ν4A2b3d+6A2b2c2+12A2b2cν-6A2b2-12αν2A0A1b2-42αν2A1A2ab-16αν2A22a2-60ν4A1ab3-330A2a2b2=0,-16ν4A2bd3-14ν4A2a2d2+2A1acdν-A1ad+4A2cd2ν+A1ac2d-8ν4A1abd2+2A2c2d2-ν4A1a3d-2αν2A0A1ad-4αν2A0A2d2-2αν2A12d2-2A2d2=0,2A1b2c2-28αν2A22ad-20αν2A0A2ab-36αν2A1A2bd-18αν2A1A2a2-10αν2A12ab-10A2ab+10A2abc2+4A1b2cν-40ν4A1b3d-4αν2A0A1b2-50ν4A1a2b2+20A2abcν-2A1b2-130ν4A2a3b-440ν4A2ab2d=0,2A1a2cν-6A2ad-ν4A1a4+12A2acdν-6αν2A12ad+6A2ac2d+A1a2c2-12αν2A1A2d2-4αν2A0A1bd-120ν4A2abd2+4A1bcdν-2αν2A0A1a2-12αν2A0A2ad-16ν4A1b2d2-A1a2-30ν4A2a3d-2A1bd+2A1bc2d-22ν4A1a2bd=0,-8αν2A0A2a2+3A1abc2-8A2bd+6A1abcν-3A1ab-6αν2A0A1ab-136ν4A2b2d2-4A2a2-12αν2A22d2+8A2a2cν-16αν2A0A2bd-232ν4A2a2bd-8αν2A12bd+16A2bcdν-15ν4A1a3b-16ν4A2a4-60ν4A1ab2d+8A2bc2d+4A2a2c2-30αν2A1A2ad-4αν2A12a2=0.
Solving the above equations, we get
(22)A0=-8bdν4-a2ν4+c2+2cν-12ν2α,A1=-6abν2α,A2=-6b2ν2α,
and consequently, the solutions of (1) are
(23)u(t,x,y)=A0+A1{-a2b-θ2btanh[12θ(z+C)]}+A2{-a2b-θ2btanh[12θ(z+C)]}2,u(t,x,y)=A0+A1{sech(θz2)Ccosh(θz2)-2bθsinh(θz2)-a2b-θ2btanh(12θz)+sech(θz/2)Ccosh(θz/2)-(2b/θ)sinh(θz/2)-a2b-θ2btanh(12θz)sech(θz2)Ccosh(θz2)-(2bθ)sinh(θz2)}+A2{-a2b-θ2btanh(12θz)sech(θz2)Ccosh(θz2)-(2bθ)sinh(θz2)+sech(θz/2)Ccosh(θz/2)-(2b/θ)sinh(θz/2)-a2b-θ2btanh(12θz)sech(θz2)Ccosh(θz2)-(2bθ)sinh(θz2)}2,
where z=νx+y-(c+ν)t and C is an arbitrary constant of integration.The solution (16) is a solitary wave solution in the form of a kink solution.


## 3. Conservation Laws for (1)

In this section we obtain conservation laws for the (2+1)-dimensional Boussinesq equation (1) using Ibragimov theorem [29], but first we give some definitions and notations which we will utilize later.


### 3.1. Fundamental Operators and Their Relationship

Let us consider akth-order system of PDEs of n independent variables x=(x1,x2,…,xn) and m dependent variables u=(u1,u2,…,um), namely,
(24)Eα(x,u,u(1),…,u(k))=0,α=1,…,m.
Here u(1),u(2),…,u(k) denote the collections of all first, second, …, kth-order partial derivatives; that is, uiα=Di(uα), uijα=DjDi(uα),…, respectively, with the total derivative operator with respect to xi defined by
(25)Di=∂∂xi+uiα∂∂uα+uijα∂∂ujα+⋯,i=1,…,n.
The Euler-Lagrange operator, for each α, is defined by
(26)δδuα=∂∂uα+∑s≥1‍(-1)sDi1⋯Dis∂∂ui1i2⋯isα,α=1,…,m,
and the Lie-Bäcklund operator is given by
(27)X=ξi∂∂xi+ηα∂∂uα,ξi,ηα∈𝒜,
where 𝒜 is the space of differential functions. The operator (27) can be written as
(28)X=ξi∂∂xi+ηα∂∂uα+∑s≥1‍ζi1i2⋯isα∂∂ui1i2⋯isα,
where
(29)ζiα=Di(Wα)+ξjuijα,ζi1⋯isα=Di1⋯Dis(Wα)+ξjuji1⋯isα,s>1.
Here Wα is the Lie characteristic function defined by
(30)Wα=ηα-ξiujα.
We can write the Lie-Bäcklund operator (28) in characteristic form as
(31)X=ξiDi+Wα∂∂uα+∑s≥1‍Di1⋯Dis(Wα)∂∂ui1i2⋯isα.
The Noether operators associated with a Lie-Bäcklund symmetry operator X are defined as
(32)Ni=ξi+Wαδδuiα+∑s≥1‍Di1⋯Dis(Wα)δδuii1i2⋯isα,i=1,…,n,
where the Euler-Lagrange operators with respect to derivatives of uα are obtained from (26) by replacing uα by the corresponding derivatives. For example,
(33)δδuiα=∂∂uiα+∑s≥1‍(-1)sDj1⋯Djs∂∂uij1j2⋯jsα,i=1,…,n,α=1,…,m,
and the Euler-Lagrange, Lie-Bäcklund, and Noether operators are connected by the operator identity
(34)X+Di(ξi)=Wαδδuα+DiNi.
The n-tuple vector T=(T1,T2,…,Tn), Tj∈𝒜, j=1,…,n, is a conserved vector of (24) if Ti satisfies
(35)DiTi|(11)=0,
which defines a local conservation law of system (24).The system of adjoint equations to (24) is defined by
(36)Eα*(x,u,v,…,u(k),v(k))=0,α=1,…,m,
where
(37)Eα*(x,u,v,…,u(k),v(k))=δ(vβEβ)δuα,α=1,…,m,v=v(x)
and v=(v1,v2,…,vm) are new dependent variables.The system of (24) is known as self-adjoint if the substitution of v=u into the system of adjoint equations (36) yields the same system (24).Let us now assume the system of (24) admits the symmetry generator
(38)X=ξi∂∂xi+ηα∂∂uα.Then the system of adjoint equations (36) admits the operator
(39)Y=ξi∂∂xi+ηα∂∂uα+η*α∂∂vα,η*α=-[λβαvβ+vαDi(ξi)],
where the operator (39) is an extension of (38) to the variable vα and the λβα are obtainable from
(40)X(Eα)=λαβEβ.
We now state the following theorem.Theorem 1 (see [29]).
Every Lie point, Lie-Bäcklund, and nonlocal symmetry (38) admitted by the system of (24) gives rise to a conservation law for the system consisting of (24) and the adjoint equation (36), where the components Ti of the conserved vector T=(T1,…,Tn) are determined by
(41)Ti=ξiL+WαδLδuiα+∑s≥1‍Di1⋯Dis(Wα)δLδuii1i2⋯isα,i=1,…,n,
with Lagrangian given by
(42)L=vαEα(x,u,…,u(k)).


### 3.2. Construction of Conservation Laws for (1)

In this subsection, we obtain conservation laws of (2+1)-dimensional Boussinesq equation(43)utt-uxx-uyy-2αux2-2αuuxx-uxxxx=0.
Recall that (43) admits the following five Lie point symmetry generators:
(44)R1=∂∂x,R2=∂∂t,R3=∂∂y,R4=y∂∂t+t∂∂y,R5=-2αt∂∂t-xα∂∂x-2αy∂∂y+(1+2αu)∂∂u.
We now find five conserved vectors corresponding to each of these five Lie point symmetries.The adjoint equation of (43), by invoking (37), is
(45)E*(t,x,u,v,…,uxxxx,vxxxx)=δδu[v(utt-uxx-uyy-2αux2,v(utt-uxx-uyy-2αux2,-2αuuxx-uxxxx)v(utt-uxx-uyy-2αux2,]=0,
where v=v(t,x,y) is a new dependent variable and (45) gives
(46)vtt-vxx-vyy-2αuvxx-vxxxx=0.
It is obvious from the adjoint equation (46) that (43) is not self-adjoint. By recalling (42), we get the following Lagrangian for the system of (43) and (46):
(47)L=v(utt-uxx-uyy-2αux2-2αuuxx-uxxxx).
(i) We first consider the Lie point symmetry generator R1=∂/∂x. It can be verified from (39) that the operator Y1 is the same as R1 and the Lie characteristic function is W=-ux. Thus, by using (41), the components Ti, i=1,2,3, of the conserved vector T=(T1,T2,T3) are given by
(48)T1=uxvt-vutx,T2=vutt-vuyy-uxvx-2αuuxvx-uxvxxx+vxxuxx-vxuxxx,T3=-uxvy+vuxy.Remark 2.
The conserved vectorT contains the arbitrary solution v of the adjoint equation (46) and hence gives an infinite number of conservation laws.The same remark applies to all the following four cases.(ii) Now for the second symmetry generatorR2=∂/∂t, we have W=-ut. Hence, by invoking (41), the symmetry generator R2 gives rise to the following components of the conserved vector:
(49)T1=-vuxx-vuyy-2αvux2-2αuvuxx-vuxxxx+utvt,T2=-utvx+2αvutux-2αuutvx-utvxxx+vutx+2αuvutx+vxxutx-vxutxx+vutxxx,T3=-vyut+vuty.
(iii) The third symmetry generator, R3=∂/∂y, gives W=-uy and the corresponding components of the conserved vector are
(50)T1=vtuy-vuty,T2=-uyvx+2αvuyux-2αuuyvx-uyvxxx+vuxy+2αuvuxy+vxxuxy-vxuxxy+vuxxxy,T3=vutt-vuxx-2αvux2-2αuvuxx-vuxxxx-uyvy.
(iv) For the symmetry generator R4=y∂/∂t+t∂/∂y the components of the conserved vector, as before, are given by
(51)T1=-yvuxx-yvuyy-2αyvux2-2αyuvuxx-yvuxxxx+yutvt+tuyvt-vuy-tvuty,T2=-yutvx+2αyvutux-2αyuutvx-yutvxxx-tvxuy+2αtvuyux-2αtuvxuy-tuyvxxx+yvutx+2αyuvutx+yutxvxx+tvuxy+2αtuvuxy+tuxyvxx-yvxutxx-tvxuxxy+yvutxxx+tvuxxxy,T3=tvutt-tvuxx-2αtvux2-2αtuvuxx-tvuxxxx-yvyut-tvyuy+vut+yvuty.
(v) Finally, for the symmetry generator
(52)R5=-2αt∂∂t-xα∂∂x-2αy∂∂y+(1+2αu)∂∂u
the value of Y5 is different from R5 and is given by
(53)Y5=-2αt∂∂t-xα∂∂x-2αy∂∂y+(1+2αu)∂∂u-vα∂∂v.
In this case the Lie characteristic function is W=1+2αu+2αtut+xαux+2αyuy. So using (41), one can obtain the conserved vector T whose components are given by
(54)T1=2αtvuxx+2αtvuyy+4α2tvux2+4α2tvuuxx+2αtvuxxxx-vt-2αuvt-2αtutvt-xαuxvt-2αyuyvt+4αvut+xαvutx+2αyvuty,T2=-xαvutt+xαvuyy+2αxuvuxx+vx+4αuvx+vxxx+4α2u2vx+2αuvxxx-8αtvuxut+2αtutvx+4α2tvutux+4α2tuutvx+2αtutvxxx+xαuxvx+2α2xuuxvx+xαuxvxxx-8α2yvuxuy+2αyuyvx+4α2yvuxuy+4α2yuuyvx+2αyuyvxxx-2αtvutx-5αvux-2αyvuxy-4α2tuvutx-10α2uvux-2α2xuvuxx-4α2yuvuxy-3αuxvxx-2αtutxvxx-xαuxxvxx-2αyuxyvxx+2αtvxutxx+4αvxuxx+xαvxuxxx+2αyvxuxxy-2αtvutxxx-5αvuxxx-2αyvuxxxy,T3=-2αyvutt+2αyvuxx+4α2yvux2+4α2yvuuxx+2αyvuxxxx+vy+2αuvy+2αtutvy+xαuxvy+2αyuyvy-4αvuy-xαvuxy-2αtvuty.


## 3.1. Fundamental Operators and Their Relationship

Let us consider akth-order system of PDEs of n independent variables x=(x1,x2,…,xn) and m dependent variables u=(u1,u2,…,um), namely,
(24)Eα(x,u,u(1),…,u(k))=0,α=1,…,m.
Here u(1),u(2),…,u(k) denote the collections of all first, second, …, kth-order partial derivatives; that is, uiα=Di(uα), uijα=DjDi(uα),…, respectively, with the total derivative operator with respect to xi defined by
(25)Di=∂∂xi+uiα∂∂uα+uijα∂∂ujα+⋯,i=1,…,n.
The Euler-Lagrange operator, for each α, is defined by
(26)δδuα=∂∂uα+∑s≥1‍(-1)sDi1⋯Dis∂∂ui1i2⋯isα,α=1,…,m,
and the Lie-Bäcklund operator is given by
(27)X=ξi∂∂xi+ηα∂∂uα,ξi,ηα∈𝒜,
where 𝒜 is the space of differential functions. The operator (27) can be written as
(28)X=ξi∂∂xi+ηα∂∂uα+∑s≥1‍ζi1i2⋯isα∂∂ui1i2⋯isα,
where
(29)ζiα=Di(Wα)+ξjuijα,ζi1⋯isα=Di1⋯Dis(Wα)+ξjuji1⋯isα,s>1.
Here Wα is the Lie characteristic function defined by
(30)Wα=ηα-ξiujα.
We can write the Lie-Bäcklund operator (28) in characteristic form as
(31)X=ξiDi+Wα∂∂uα+∑s≥1‍Di1⋯Dis(Wα)∂∂ui1i2⋯isα.
The Noether operators associated with a Lie-Bäcklund symmetry operator X are defined as
(32)Ni=ξi+Wαδδuiα+∑s≥1‍Di1⋯Dis(Wα)δδuii1i2⋯isα,i=1,…,n,
where the Euler-Lagrange operators with respect to derivatives of uα are obtained from (26) by replacing uα by the corresponding derivatives. For example,
(33)δδuiα=∂∂uiα+∑s≥1‍(-1)sDj1⋯Djs∂∂uij1j2⋯jsα,i=1,…,n,α=1,…,m,
and the Euler-Lagrange, Lie-Bäcklund, and Noether operators are connected by the operator identity
(34)X+Di(ξi)=Wαδδuα+DiNi.
The n-tuple vector T=(T1,T2,…,Tn), Tj∈𝒜, j=1,…,n, is a conserved vector of (24) if Ti satisfies
(35)DiTi|(11)=0,
which defines a local conservation law of system (24).The system of adjoint equations to (24) is defined by
(36)Eα*(x,u,v,…,u(k),v(k))=0,α=1,…,m,
where
(37)Eα*(x,u,v,…,u(k),v(k))=δ(vβEβ)δuα,α=1,…,m,v=v(x)
and v=(v1,v2,…,vm) are new dependent variables.The system of (24) is known as self-adjoint if the substitution of v=u into the system of adjoint equations (36) yields the same system (24).Let us now assume the system of (24) admits the symmetry generator
(38)X=ξi∂∂xi+ηα∂∂uα.Then the system of adjoint equations (36) admits the operator
(39)Y=ξi∂∂xi+ηα∂∂uα+η*α∂∂vα,η*α=-[λβαvβ+vαDi(ξi)],
where the operator (39) is an extension of (38) to the variable vα and the λβα are obtainable from
(40)X(Eα)=λαβEβ.
We now state the following theorem.Theorem 1 (see [29]).
Every Lie point, Lie-Bäcklund, and nonlocal symmetry (38) admitted by the system of (24) gives rise to a conservation law for the system consisting of (24) and the adjoint equation (36), where the components Ti of the conserved vector T=(T1,…,Tn) are determined by
(41)Ti=ξiL+WαδLδuiα+∑s≥1‍Di1⋯Dis(Wα)δLδuii1i2⋯isα,i=1,…,n,
with Lagrangian given by
(42)L=vαEα(x,u,…,u(k)).


## 3.2. Construction of Conservation Laws for (1)

In this subsection, we obtain conservation laws of (2+1)-dimensional Boussinesq equation(43)utt-uxx-uyy-2αux2-2αuuxx-uxxxx=0.
Recall that (43) admits the following five Lie point symmetry generators:
(44)R1=∂∂x,R2=∂∂t,R3=∂∂y,R4=y∂∂t+t∂∂y,R5=-2αt∂∂t-xα∂∂x-2αy∂∂y+(1+2αu)∂∂u.
We now find five conserved vectors corresponding to each of these five Lie point symmetries.The adjoint equation of (43), by invoking (37), is
(45)E*(t,x,u,v,…,uxxxx,vxxxx)=δδu[v(utt-uxx-uyy-2αux2,v(utt-uxx-uyy-2αux2,-2αuuxx-uxxxx)v(utt-uxx-uyy-2αux2,]=0,
where v=v(t,x,y) is a new dependent variable and (45) gives
(46)vtt-vxx-vyy-2αuvxx-vxxxx=0.
It is obvious from the adjoint equation (46) that (43) is not self-adjoint. By recalling (42), we get the following Lagrangian for the system of (43) and (46):
(47)L=v(utt-uxx-uyy-2αux2-2αuuxx-uxxxx).
(i) We first consider the Lie point symmetry generator R1=∂/∂x. It can be verified from (39) that the operator Y1 is the same as R1 and the Lie characteristic function is W=-ux. Thus, by using (41), the components Ti, i=1,2,3, of the conserved vector T=(T1,T2,T3) are given by
(48)T1=uxvt-vutx,T2=vutt-vuyy-uxvx-2αuuxvx-uxvxxx+vxxuxx-vxuxxx,T3=-uxvy+vuxy.Remark 2.
The conserved vectorT contains the arbitrary solution v of the adjoint equation (46) and hence gives an infinite number of conservation laws.The same remark applies to all the following four cases.(ii) Now for the second symmetry generatorR2=∂/∂t, we have W=-ut. Hence, by invoking (41), the symmetry generator R2 gives rise to the following components of the conserved vector:
(49)T1=-vuxx-vuyy-2αvux2-2αuvuxx-vuxxxx+utvt,T2=-utvx+2αvutux-2αuutvx-utvxxx+vutx+2αuvutx+vxxutx-vxutxx+vutxxx,T3=-vyut+vuty.
(iii) The third symmetry generator, R3=∂/∂y, gives W=-uy and the corresponding components of the conserved vector are
(50)T1=vtuy-vuty,T2=-uyvx+2αvuyux-2αuuyvx-uyvxxx+vuxy+2αuvuxy+vxxuxy-vxuxxy+vuxxxy,T3=vutt-vuxx-2αvux2-2αuvuxx-vuxxxx-uyvy.
(iv) For the symmetry generator R4=y∂/∂t+t∂/∂y the components of the conserved vector, as before, are given by
(51)T1=-yvuxx-yvuyy-2αyvux2-2αyuvuxx-yvuxxxx+yutvt+tuyvt-vuy-tvuty,T2=-yutvx+2αyvutux-2αyuutvx-yutvxxx-tvxuy+2αtvuyux-2αtuvxuy-tuyvxxx+yvutx+2αyuvutx+yutxvxx+tvuxy+2αtuvuxy+tuxyvxx-yvxutxx-tvxuxxy+yvutxxx+tvuxxxy,T3=tvutt-tvuxx-2αtvux2-2αtuvuxx-tvuxxxx-yvyut-tvyuy+vut+yvuty.
(v) Finally, for the symmetry generator
(52)R5=-2αt∂∂t-xα∂∂x-2αy∂∂y+(1+2αu)∂∂u
the value of Y5 is different from R5 and is given by
(53)Y5=-2αt∂∂t-xα∂∂x-2αy∂∂y+(1+2αu)∂∂u-vα∂∂v.
In this case the Lie characteristic function is W=1+2αu+2αtut+xαux+2αyuy. So using (41), one can obtain the conserved vector T whose components are given by
(54)T1=2αtvuxx+2αtvuyy+4α2tvux2+4α2tvuuxx+2αtvuxxxx-vt-2αuvt-2αtutvt-xαuxvt-2αyuyvt+4αvut+xαvutx+2αyvuty,T2=-xαvutt+xαvuyy+2αxuvuxx+vx+4αuvx+vxxx+4α2u2vx+2αuvxxx-8αtvuxut+2αtutvx+4α2tvutux+4α2tuutvx+2αtutvxxx+xαuxvx+2α2xuuxvx+xαuxvxxx-8α2yvuxuy+2αyuyvx+4α2yvuxuy+4α2yuuyvx+2αyuyvxxx-2αtvutx-5αvux-2αyvuxy-4α2tuvutx-10α2uvux-2α2xuvuxx-4α2yuvuxy-3αuxvxx-2αtutxvxx-xαuxxvxx-2αyuxyvxx+2αtvxutxx+4αvxuxx+xαvxuxxx+2αyvxuxxy-2αtvutxxx-5αvuxxx-2αyvuxxxy,T3=-2αyvutt+2αyvuxx+4α2yvux2+4α2yvuuxx+2αyvuxxxx+vy+2αuvy+2αtutvy+xαuxvy+2αyuyvy-4αvuy-xαvuxy-2αtvuty.


## 4. Conclusions

In this paper, for the first time, Lie symmetries as well as the simplest equation method were used to obtain exact solutions of the (2+1)-dimensional Boussinesq equation (1). The solutions obtained were solitary waves and nontopological soliton. It is obvious from the analysis we conducted that the (2+1)-dimensional equation gives rise to a variety of solitary wave solutions that ranges from kink to soliton solutions. The obtained kink and soliton solutions are regular solitons given in the form of hyperbolic tan or the sech2 form. Moreover, the conservation laws for the (2+1)-dimensional Boussinesq equation were also derived by using the new conservation theorem due to Ibragimov [29].



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*Source: 548975-2013-09-19.xml*