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

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.

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 𝑢_𝑡𝑡−𝑢_𝑥𝑥−𝑢_𝑦𝑦−𝛼𝑢²_𝑥𝑥−𝑢_{𝑥𝑥𝑥𝑥}=0, (1) 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 𝑢_𝑥𝑥 and 𝑢_𝑦𝑦 in addition to the fourth-order spatial derivative 𝑢_{𝑥𝑥𝑥𝑥} 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 𝑢_𝑦𝑦 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 𝑅=𝜉¹𝜕𝜕𝑥+𝜉²𝜕𝜕𝑦+𝜉³𝜕𝜕𝜕𝑡+𝜂,𝜕𝑢 (2) where 𝜉^𝑖, 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: 𝑅₁=𝜕𝜕𝑥,𝑅₂=𝜕𝜕𝑡,𝑅₃=𝜕,𝑅𝜕𝑦₄𝜕=𝑦𝜕𝜕𝑡+𝑡,𝑅𝜕𝑦₅𝜕=−2𝛼𝑡𝜕𝜕𝑡−𝛼𝑥𝜕𝜕𝑥−2𝛼𝑦+()𝜕𝜕𝑦1+2𝛼𝑢.𝜕𝑢 (3) 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: 𝑓=𝑦−𝑐𝑡,𝑔=𝑡−𝑥,𝜃=𝑢. (4) Now treating 𝜃 as the new dependent variable and 𝑓 and 𝑔 as new independent variables, the Boussinesq equation (1) transforms to 1−𝑐²𝜃_𝑓𝑓+2𝑐𝜃_𝑓𝑔+2𝛼𝜃²_𝑔+2𝛼𝜃𝜃_𝑔𝑔+𝜃_{𝑔𝑔𝑔𝑔}=0, (5) 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: Γ₁=𝜕𝜕𝑔,Γ₂=𝜕,Γ𝜕𝑓₃=2𝛼𝑓−2𝛼𝑓𝑐²𝜕+𝜕𝑓𝛼𝑐𝑓−𝑐²𝜕𝛼𝑔+𝛼𝑔+𝑐𝜕𝑔₂+2𝑐²𝜕𝛼𝜃−2𝛼𝜃.𝜕𝜃 (6) The combination of the first two translational symmetries, Γ=Γ₁+𝜈Γ₂, where 𝜈 is a constant, yields the two invariants 𝑧=𝑓−𝜈𝑔,𝜓=𝜃, (7) 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 2𝛼𝜈²𝜓^′2+1−𝑐²−2𝑐𝜈𝜓^′′+2𝛼𝜈²𝜓𝜓^′′+𝜈⁴𝜓^′′′′=0. (8) 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): 𝑢=𝐴𝑥,𝑦,𝑡¹𝐴₂sech²𝐴₁2(𝐵±𝑧), (9) where 𝐵 is a constant of integration and 𝐴₁=𝑐²+2𝜈𝑐−1𝜈⁴,𝐴₂=2𝛼3𝜈²,𝑧=𝜈𝑥+𝑦−(𝑐+𝜈)𝑡. (10) 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 𝐹(𝑧)=^𝑀_𝑖=0𝐴_𝑖(𝐺(𝑧))^𝑖, (11) where 𝐺(𝑧) satisfies the Bernoulli and Riccati equations, 𝑀 is a positive integer that can be determined by balancing procedure, and 𝐴₀,…,𝐴_𝑀 are parameters to be determined.

The Bernoulli equation 𝐺^′(𝑧)=𝑎𝐺(𝑧)+𝑏𝐺²(𝑧), (12) where 𝑎 and 𝑏 are arbitrary constants has the general solution given by 𝐺(𝑧)[𝑎([𝑎(=𝑎cosh𝑧+𝐶)]+sinh𝑧+𝐶)]1−𝑏cosh[𝑎(𝑧+𝐶)]−𝑏sinh[𝑎(𝑧+𝐶)] (13) and represents a solitary wave solution.

The Riccati equation considered in this work is 𝐺^′(𝑧)=𝑏𝐺²(𝑧)+𝑎𝐺(𝑧)+𝑑, (14) where 𝑎, 𝑏, and 𝑑 are arbitrary constants. Its solutions are 𝐺(𝑧)𝑎=−−𝜃2𝑏12𝑏tanh2𝜃(,𝐺(𝑧)𝑎𝑧+𝐶)=−−𝜃2𝑏12𝑏tanh2+()𝜃𝑧sech𝜃𝑧/2()−()(),𝐶cosh𝜃𝑧/22𝑏/𝜃sinh𝜃𝑧/2 (15) 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 gives 𝑀=2 so the solutions of (8) are of the form 𝐹(𝑧)=𝐴₀+𝐴₁𝐺+𝐴₂𝐺². (16) Inserting (16) into (8) and using the Bernoulli equation (12) and, thereafter, equating the coefficients of powers of 𝐺^𝑖 to zero, we obtain an algebraic system of six equations in terms of 𝐴₀, 𝐴₁, and 𝐴₂, namely, −120𝜈⁴𝐴₂𝑏⁴−20𝛼𝜈²𝐴²₂𝑏²=0,−336𝜈⁴𝐴₂𝑎𝑏³−36𝛼𝜈²𝐴²₂𝑎𝑏−24𝜈⁴𝐴₁𝑏⁴−24𝛼𝜈²𝐴₁𝐴₂𝑏²=0,−𝐴₁𝑎²+2𝜈𝐴₁𝑎²𝑐−𝜈𝐴₁𝑎⁴+𝐴₁𝑎²𝑐²−2𝛼𝜈²𝐴₁𝐴₂𝑎²=0,−16𝛼𝜈²𝐴²₂𝑎²+12𝜈𝐴₂𝑏²𝑐−6𝛼𝜈²𝐴²₁𝑏²−60𝜈⁴𝐴₁𝑎𝑏³−12𝛼𝜈²𝐴₀𝐴₁𝑏²−330𝜈⁴𝐴₂𝑎²𝑏²−42𝛼𝜈²𝐴₁𝑎𝑏𝐴₂+6𝐴₂𝑏²𝑐²−6𝐴₂𝑏²=0,−15𝜈⁴𝐴₁𝑎³𝑏+8𝜈𝐴₂𝑎²𝑐−3𝐴₁𝑎𝑏−4𝐴₂𝑎²+4𝐴₂𝑎²𝑐²−4𝛼𝜈²𝐴²₁𝑎²−6𝛼𝜈²𝐴₀𝐴₁𝑎𝑏−16𝜈⁴𝐴₂𝑎⁴+6𝐴₁𝑎𝑏𝑐𝜈−8𝛼𝜈²𝐴₀𝐴₂𝑎²+3𝐴₁𝑎𝑏𝑐²=0,−18𝛼𝜈²𝐴₁𝐴₂𝑎²−10𝛼𝜈²𝐴²₁𝑎𝑏−4𝛼𝜈²𝐴₀𝐴₁𝑏²+10𝐴₂𝑎𝑏𝑐²+4𝜈𝐴₁𝑏²𝑐+20𝜈𝐴₂𝑎𝑏𝑐−2𝐴₁𝑏²+2𝐴₁𝑏²𝑐²−20𝛼𝜈²𝐴₀𝐴₂𝑎𝑏−10𝐴₂𝑎𝑏−130𝜈⁴𝐴₂𝑎³𝑏−50𝜈⁴𝐴₁𝑎²𝑏²=0. (17) With the aid of Mathematica, solving the above system of algebraic equations, one possible solution for 𝐴₀, 𝐴₁, and 𝐴₂ is 𝐴₀=−1−𝑐²−2𝑐𝜈+𝑎²𝜈⁴2𝛼𝜈²,𝐴₁=−6𝑎𝑏𝜈²𝛼,𝐴₂=−6𝑏²𝜈²𝛼. (18) Thus, reverting back to the original variables, a solution of (1) is 𝑢𝑡,𝑥,𝑦=𝐴₀+𝐴₁𝑎cosh[𝑎(𝑧+𝐶)]+sinh[𝑎(𝑧+𝐶)]1−𝑏cosh[𝑎(𝑧+𝐶)]−𝑏sinh[𝑎(𝑧+𝐶)]+𝐴₂𝑎²cosh[𝑎(𝑧+𝐶)]+sinh[𝑎(𝑧+𝐶)]1−𝑏cosh[𝑎(𝑧+𝐶)]−𝑏sinh[𝑎(𝑧+𝐶)]², (19) 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 yields 𝑀=2 so the solutions of (8) take the form 𝐹(𝑧)=𝐴₀+𝐴₁𝐺+𝐴₂𝐺². (20) 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 𝐺^𝑖 to zero. The resulting algebraic equations are −120𝜈⁴𝐴₂𝑏⁴−20𝛼𝜈²𝐴²₂𝑏²=0,−36𝛼𝜈²𝐴²₂𝑎𝑏−336𝜈⁴𝐴₂𝑎𝑏³−24𝜈⁴𝐴₁𝑏⁴−24𝛼𝜈²𝐴₁𝐴₂𝑏²=0,−32𝛼𝜈²𝐴²₂𝑏𝑑−6𝛼𝜈²𝐴²₁𝑏²−240𝜈⁴𝐴₂𝑏³𝑑+6𝐴₂𝑏²𝑐²+12𝐴₂𝑏²𝑐𝜈−6𝐴₂𝑏²−12𝛼𝜈²𝐴₀𝐴₁𝑏²−42𝛼𝜈²𝐴₁𝐴₂𝑎𝑏−16𝛼𝜈²𝐴²₂𝑎²−60𝜈⁴𝐴₁𝑎𝑏³−330𝐴₂𝑎²𝑏²=0,−16𝜈⁴𝐴₂𝑏𝑑³−14𝜈⁴𝐴₂𝑎²𝑑²+2𝐴₁𝑎𝑐𝑑𝜈−𝐴₁𝑎𝑑+4𝐴₂𝑐𝑑²𝜈+𝐴₁𝑎𝑐²𝑑−8𝜈⁴𝐴₁𝑎𝑏𝑑²+2𝐴₂𝑐²𝑑²−𝜈⁴𝐴₁𝑎³𝑑−2𝛼𝜈²𝐴₀𝐴₁𝑎𝑑−4𝛼𝜈²𝐴₀𝐴₂𝑑²−2𝛼𝜈²𝐴²₁𝑑²−2𝐴₂𝑑²=0,2𝐴₁𝑏²𝑐²−28𝛼𝜈²𝐴²₂𝑎𝑑−20𝛼𝜈²𝐴₀𝐴₂𝑎𝑏−36𝛼𝜈²𝐴₁𝐴₂𝑏𝑑−18𝛼𝜈²𝐴₁𝐴₂𝑎²−10𝛼𝜈²𝐴²₁𝑎𝑏−10𝐴₂𝑎𝑏+10𝐴₂𝑎𝑏𝑐²+4𝐴₁𝑏²𝑐𝜈−40𝜈⁴𝐴₁𝑏³𝑑−4𝛼𝜈²𝐴₀𝐴₁𝑏²−50𝜈⁴𝐴₁𝑎²𝑏²+20𝐴₂𝑎𝑏𝑐𝜈−2𝐴₁𝑏²−130𝜈⁴𝐴₂𝑎³𝑏−440𝜈⁴𝐴₂𝑎𝑏²𝑑=0,2𝐴₁𝑎²𝑐𝜈−6𝐴₂𝑎𝑑−𝜈⁴𝐴₁𝑎⁴+12𝐴₂𝑎𝑐𝑑𝜈−6𝛼𝜈²𝐴²₁𝑎𝑑+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−8𝛼𝜈²𝐴²₁𝑏𝑑+16𝐴₂𝑏𝑐𝑑𝜈−15𝜈⁴𝐴₁𝑎³𝑏−16𝜈⁴𝐴₂𝑎⁴−60𝜈⁴𝐴₁𝑎𝑏²𝑑+8𝐴₂𝑏𝑐²𝑑+4𝐴₂𝑎²𝑐²−30𝛼𝜈²𝐴₁𝐴₂𝑎𝑑−4𝛼𝜈²𝐴²₁𝑎²=0. (21) Solving the above equations, we get 𝐴₀=−8𝑏𝑑𝜈⁴−𝑎²𝜈⁴+𝑐²+2𝑐𝜈−12𝜈²𝛼,𝐴₁=−6𝑎𝑏𝜈²𝛼,𝐴₂=−6𝑏²𝜈²𝛼, (22) and consequently, the solutions of (1) are 𝑢𝑡,𝑥,𝑦=𝐴₀+𝐴₁−𝑎−𝜃2𝑏12𝑏tanh2𝜃(𝑧+𝐶)+𝐴₂−𝑎−𝜃2𝑏12𝑏tanh2𝜃(𝑧+𝐶)²,𝑢𝑡,𝑥,𝑦=𝐴₀+𝐴₁−𝑎−𝜃2𝑏12𝑏tanh2+()𝜃𝑧sech𝜃𝑧/2𝐶cosh(𝜃𝑧/2)−(2𝑏/𝜃)sinh(𝜃𝑧/2)+𝐴₂−𝑎−𝜃2𝑏12𝑏tanh2+()𝜃𝑧sech𝜃𝑧/2𝐶cosh(𝜃𝑧/2)−(2𝑏/𝜃)sinh(𝜃𝑧/2)², (23) 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 a 𝑘th-order system of PDEs of 𝑛 independent variables 𝑥=(𝑥¹,𝑥²,…,𝑥^𝑛) and 𝑚 dependent variables 𝑢=(𝑢¹,𝑢²,…,𝑢^𝑚), namely, 𝐸_𝛼𝑥,𝑢,𝑢₍₁₎,…,𝑢_(𝑘)=0,𝛼=1,…,𝑚. (24) Here 𝑢₍₁₎,𝑢₍₂₎,…,𝑢_(𝑘) denote the collections of all first, second, …, 𝑘th-order partial derivatives; that is, 𝑢^𝛼_𝑖=𝐷_𝑖(𝑢^𝛼), 𝑢^𝛼_𝑖𝑗=𝐷_𝑗𝐷_𝑖(𝑢^𝛼),…, respectively, with the total derivative operator with respect to 𝑥^𝑖 defined by 𝐷_𝑖=𝜕𝜕𝑥^𝑖+𝑢^𝛼_𝑖𝜕𝜕𝑢^𝛼+𝑢^𝛼_𝑖𝑗𝜕𝜕𝑢^𝛼_𝑗+⋯,𝑖=1,…,𝑛. (25) The Euler-Lagrange operator, for each 𝛼, is defined by 𝛿𝛿𝑢^𝛼=𝜕𝜕𝑢^𝛼+_𝑠≥1(−1)^𝑠𝐷_𝑖1⋯𝐷_𝑖𝑠𝜕𝜕𝑢^𝛼_{𝑖1𝑖2⋯𝑖𝑠},𝛼=1,…,𝑚, (26) and the Lie-Bäcklund operator is given by 𝑋=𝜉^𝑖𝜕𝜕𝑥^𝑖+𝜂^𝛼𝜕𝜕𝑢^𝛼,𝜉^𝑖,𝜂^𝛼∈𝒜, (27) where 𝒜 is the space of differential functions. The operator (27) can be written as 𝑋=𝜉^𝑖𝜕𝜕𝑥^𝑖+𝜂^𝛼𝜕𝜕𝑢^𝛼+_𝑠≥1𝜁^𝛼_{𝑖1𝑖2⋯𝑖𝑠}𝜕𝜕𝑢^𝛼_{𝑖1𝑖2⋯𝑖𝑠}, (28) where 𝜁^𝛼_𝑖=𝐷_𝑖𝑊^𝛼+𝜉^𝑗𝑢^𝛼_𝑖𝑗,𝜁^𝛼_{𝑖1⋯𝑖𝑠}=𝐷_𝑖1⋯𝐷_𝑖𝑠𝑊^𝛼+𝜉^𝑗𝑢^𝛼_{𝑗𝑖1⋯𝑖𝑠},𝑠>1. (29) Here 𝑊^𝛼 is the Lie characteristic function defined by 𝑊^𝛼=𝜂^𝛼−𝜉^𝑖𝑢^𝛼_𝑗. (30) We can write the Lie-Bäcklund operator (28) in characteristic form as 𝑋=𝜉^𝑖𝐷_𝑖+𝑊^𝛼𝜕𝜕𝑢^𝛼+_𝑠≥1𝐷_𝑖1⋯𝐷_𝑖𝑠𝑊^𝛼𝜕𝜕𝑢^𝛼_{𝑖1𝑖2⋯𝑖𝑠}. (31) The Noether operators associated with a Lie-Bäcklund symmetry operator 𝑋 are defined as 𝑁^𝑖=𝜉^𝑖+𝑊^𝛼𝛿𝛿𝑢^𝛼_𝑖+_𝑠≥1𝐷_𝑖1⋯𝐷_𝑖𝑠𝑊^𝛼𝛿𝛿𝑢^𝛼_{𝑖𝑖1𝑖2⋯𝑖𝑠},𝑖=1,…,𝑛, (32) where the Euler-Lagrange operators with respect to derivatives of 𝑢^𝛼 are obtained from (26) by replacing 𝑢^𝛼 by the corresponding derivatives. For example, 𝛿𝛿𝑢^𝛼_𝑖=𝜕𝜕𝑢^𝛼_𝑖+_𝑠≥1(−1)^𝑠𝐷_𝑗1⋯𝐷_𝑗𝑠𝜕𝜕𝑢^𝛼_{𝑖𝑗1𝑗2⋯𝑗𝑠},𝑖=1,…,𝑛,𝛼=1,…,𝑚, (33) and the Euler-Lagrange, Lie-Bäcklund, and Noether operators are connected by the operator identity 𝑋+𝐷_𝑖𝜉^𝑖=𝑊^𝛼𝛿𝛿𝑢^𝛼+𝐷_𝑖𝑁^𝑖. (34) The 𝑛-tuple vector 𝑇=(𝑇¹,𝑇²,…,𝑇^𝑛), 𝑇^𝑗∈𝒜, 𝑗=1,…,𝑛, is a conserved vector of (24) if 𝑇^𝑖 satisfies 𝐷_𝑖𝑇^𝑖|₍₁₁₎=0, (35) which defines a local conservation law of system (24).

The system of adjoint equations to (24) is defined by 𝐸^∗_𝛼𝑥,𝑢,𝑣,…,𝑢_(𝑘),𝑣_(𝑘)=0,𝛼=1,…,𝑚, (36) where 𝐸^∗_𝛼𝑥,𝑢,𝑣,…,𝑢_(𝑘),𝑣_(𝑘)=𝛿𝑣^𝛽𝐸_𝛽𝛿𝑢^𝛼,𝛼=1,…,𝑚,𝑣=𝑣(𝑥) (37) and 𝑣=(𝑣¹,𝑣²,…,𝑣^𝑚) 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 𝑋=𝜉^𝑖𝜕𝜕𝑥^𝑖+𝜂^𝛼𝜕𝜕𝑢^𝛼. (38)

Then the system of adjoint equations (36) admits the operator 𝑌=𝜉^𝑖𝜕𝜕𝑥^𝑖+𝜂^𝛼𝜕𝜕𝑢^𝛼+𝜂^𝛼_∗𝜕𝜕𝑣^𝛼,𝜂^𝛼_∗=−𝜆^𝛼_𝛽𝑣^𝛽+𝑣^𝛼𝐷_𝑖𝜉^𝑖, (39) where the operator (39) is an extension of (38) to the variable 𝑣^𝛼 and the 𝜆^𝛼_𝛽 are obtainable from 𝑋𝐸_𝛼=𝜆^𝛽_𝛼𝐸_𝛽. (40) 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 𝑇^𝑖 of the conserved vector 𝑇=(𝑇¹,…,𝑇^𝑛) are determined by 𝑇^𝑖=𝜉^𝑖𝐿+𝑊^𝛼𝛿𝐿𝛿𝑢^𝛼_𝑖+_𝑠≥1𝐷_𝑖1⋯𝐷_𝑖𝑠𝑊^𝛼𝛿𝐿𝛿𝑢^𝛼_{𝑖𝑖1𝑖2⋯𝑖𝑠},𝑖=1,…,𝑛, (41) with Lagrangian given by 𝐿=𝑣^𝛼𝐸_𝛼𝑥,𝑢,…,𝑢_(𝑘). (42)

3.2. Construction of Conservation Laws for (1)

In this subsection, we obtain conservation laws of (2+1)-dimensional Boussinesq equation 𝑢_𝑡𝑡−𝑢_𝑥𝑥−𝑢_𝑦𝑦−2𝛼𝑢²_𝑥−2𝛼𝑢𝑢_𝑥𝑥−𝑢_{𝑥𝑥𝑥𝑥}=0. (43) Recall that (43) admits the following five Lie point symmetry generators: 𝑅₁=𝜕𝜕𝑥,𝑅₂=𝜕𝜕𝑡,𝑅₃=𝜕,𝑅𝜕𝑦₄𝜕=𝑦𝜕𝜕𝑡+𝑡,𝑅𝜕𝑦₅𝜕=−2𝛼𝑡𝜕𝜕𝑡−𝑥𝛼𝜕𝜕𝑥−2𝛼𝑦+()𝜕𝜕𝑦1+2𝛼𝑢.𝜕𝑢 (44) We now find five conserved vectors corresponding to each of these five Lie point symmetries.

The adjoint equation of (43), by invoking (37), is 𝐸^∗𝑡,𝑥,𝑢,𝑣,…,𝑢_{𝑥𝑥𝑥𝑥},𝑣_{𝑥𝑥𝑥𝑥}=𝛿𝑣𝑢𝛿𝑢^𝑡𝑡−𝑢_𝑥𝑥−𝑢_𝑦𝑦−2𝛼𝑢²_𝑥,−2𝛼𝑢𝑢_𝑥𝑥−𝑢_{𝑥𝑥𝑥𝑥}=0, (45) where 𝑣=𝑣(𝑡,𝑥,𝑦) is a new dependent variable and (45) gives 𝑣_𝑡𝑡−𝑣_𝑥𝑥−𝑣_𝑦𝑦−2𝛼𝑢𝑣_𝑥𝑥−𝑣_{𝑥𝑥𝑥𝑥}=0. (46) 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): 𝐿=𝑣𝑢_𝑡𝑡−𝑢_𝑥𝑥−𝑢_𝑦𝑦−2𝛼𝑢²_𝑥−2𝛼𝑢𝑢_𝑥𝑥−𝑢_{𝑥𝑥𝑥𝑥}. (47) (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=−𝑢_𝑥. Thus, by using (41), the components 𝑇^𝑖, 𝑖=1,2,3, of the conserved vector 𝑇=(𝑇¹,𝑇²,𝑇³) are given by 𝑇¹=𝑢_𝑥𝑣_𝑡−𝑣𝑢_𝑡𝑥,𝑇²=𝑣𝑢_𝑡𝑡−𝑣𝑢_𝑦𝑦−𝑢_𝑥𝑣_𝑥−2𝛼𝑢𝑢_𝑥𝑣_𝑥−𝑢_𝑥𝑣_𝑥𝑥𝑥+𝑣_𝑥𝑥𝑢_𝑥𝑥−𝑣_𝑥𝑢_𝑥𝑥𝑥,𝑇³=−𝑢_𝑥𝑣_𝑦+𝑣𝑢_𝑥𝑦. (48)

Remark 2. The conserved vector 𝑇 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 generator 𝑅₂=𝜕/𝜕𝑡, we have 𝑊=−𝑢_𝑡. Hence, by invoking (41), the symmetry generator 𝑅₂ gives rise to the following components of the conserved vector: 𝑇¹=−𝑣𝑢_𝑥𝑥−𝑣𝑢_𝑦𝑦−2𝛼𝑣𝑢²_𝑥−2𝛼𝑢𝑣𝑢_𝑥𝑥−𝑣𝑢_{𝑥𝑥𝑥𝑥}+𝑢_𝑡𝑣_𝑡,𝑇²=−𝑢_𝑡𝑣_𝑥+2𝛼𝑣𝑢_𝑡𝑢_𝑥−2𝛼𝑢𝑢_𝑡𝑣_𝑥−𝑢_𝑡𝑣_𝑥𝑥𝑥+𝑣𝑢_𝑡𝑥+2𝛼𝑢𝑣𝑢_𝑡𝑥+𝑣_𝑥𝑥𝑢_𝑡𝑥−𝑣_𝑥𝑢_𝑡𝑥𝑥+𝑣𝑢_{𝑡𝑥𝑥𝑥},𝑇³=−𝑣_𝑦𝑢_𝑡+𝑣𝑢_𝑡𝑦. (49) (iii) The third symmetry generator, 𝑅₃=𝜕/𝜕𝑦, gives 𝑊=−𝑢_𝑦 and the corresponding components of the conserved vector are 𝑇¹=𝑣_𝑡𝑢_𝑦−𝑣𝑢_𝑡𝑦,𝑇²=−𝑢_𝑦𝑣_𝑥+2𝛼𝑣𝑢_𝑦𝑢_𝑥−2𝛼𝑢𝑢_𝑦𝑣_𝑥−𝑢_𝑦𝑣_𝑥𝑥𝑥+𝑣𝑢_𝑥𝑦+2𝛼𝑢𝑣𝑢_𝑥𝑦+𝑣_𝑥𝑥𝑢_𝑥𝑦−𝑣_𝑥𝑢_𝑥𝑥𝑦+𝑣𝑢_{𝑥𝑥𝑥𝑦},𝑇³=𝑣𝑢_𝑡𝑡−𝑣𝑢_𝑥𝑥−2𝛼𝑣𝑢²_𝑥−2𝛼𝑢𝑣𝑢_𝑥𝑥−𝑣𝑢_{𝑥𝑥𝑥𝑥}−𝑢_𝑦𝑣_𝑦. (50) (iv) For the symmetry generator 𝑅₄=𝑦𝜕/𝜕𝑡+𝑡𝜕/𝜕𝑦 the components of the conserved vector, as before, are given by 𝑇¹=−𝑦𝑣𝑢_𝑥𝑥−𝑦𝑣𝑢_𝑦𝑦−2𝛼𝑦𝑣𝑢²_𝑥−2𝛼𝑦𝑢𝑣𝑢_𝑥𝑥−𝑦𝑣𝑢_{𝑥𝑥𝑥𝑥}+𝑦𝑢_𝑡𝑣_𝑡+𝑡𝑢_𝑦𝑣_𝑡−𝑣𝑢_𝑦−𝑡𝑣𝑢_𝑡𝑦,𝑇²=−𝑦𝑢_𝑡𝑣_𝑥+2𝛼𝑦𝑣𝑢_𝑡𝑢_𝑥−2𝛼𝑦𝑢𝑢_𝑡𝑣_𝑥−𝑦𝑢_𝑡𝑣_𝑥𝑥𝑥−𝑡𝑣_𝑥𝑢_𝑦+2𝛼𝑡𝑣𝑢_𝑦𝑢_𝑥−2𝛼𝑡𝑢𝑣_𝑥𝑢_𝑦−𝑡𝑢_𝑦𝑣_𝑥𝑥𝑥+𝑦𝑣𝑢_𝑡𝑥+2𝛼𝑦𝑢𝑣𝑢_𝑡𝑥+𝑦𝑢_𝑡𝑥𝑣_𝑥𝑥+𝑡𝑣𝑢_𝑥𝑦+2𝛼𝑡𝑢𝑣𝑢_𝑥𝑦+𝑡𝑢_𝑥𝑦𝑣_𝑥𝑥−𝑦𝑣_𝑥𝑢_𝑡𝑥𝑥−𝑡𝑣_𝑥𝑢_𝑥𝑥𝑦+𝑦𝑣𝑢_{𝑡𝑥𝑥𝑥}+𝑡𝑣𝑢_{𝑥𝑥𝑥𝑦},𝑇³=𝑡𝑣𝑢_𝑡𝑡−𝑡𝑣𝑢_𝑥𝑥−2𝛼𝑡𝑣𝑢²_𝑥−2𝛼𝑡𝑢𝑣𝑢_𝑥𝑥−𝑡𝑣𝑢_{𝑥𝑥𝑥𝑥}−𝑦𝑣_𝑦𝑢_𝑡−𝑡𝑣_𝑦𝑢_𝑦+𝑣𝑢_𝑡+𝑦𝑣𝑢_𝑡𝑦. (51) (v) Finally, for the symmetry generator 𝑅₅𝜕=−2𝛼𝑡𝜕𝜕𝑡−𝑥𝛼𝜕𝜕𝑥−2𝛼𝑦+()𝜕𝜕𝑦1+2𝛼𝑢𝜕𝑢 (52) the value of 𝑌₅ is different from 𝑅₅ and is given by 𝑌₅𝜕=−2𝛼𝑡𝜕𝜕𝑡−𝑥𝛼𝜕𝜕𝑥−2𝛼𝑦+()𝜕𝜕𝑦1+2𝛼𝑢𝜕𝜕𝑢−𝑣𝛼.𝜕𝑣 (53) In this case the Lie characteristic function is 𝑊=1+2𝛼𝑢+2𝛼𝑡𝑢_𝑡+𝑥𝛼𝑢_𝑥+2𝛼𝑦𝑢_𝑦. So using (41), one can obtain the conserved vector 𝑇 whose components are given by 𝑇¹=2𝛼𝑡𝑣𝑢_𝑥𝑥+2𝛼𝑡𝑣𝑢_𝑦𝑦+4𝛼²𝑡𝑣𝑢²_𝑥+4𝛼²𝑡𝑣𝑢𝑢_𝑥𝑥+2𝛼𝑡𝑣𝑢_{𝑥𝑥𝑥𝑥}−𝑣_𝑡−2𝛼𝑢𝑣_𝑡−2𝛼𝑡𝑢_𝑡𝑣_𝑡−𝑥𝛼𝑢_𝑥𝑣_𝑡−2𝛼𝑦𝑢_𝑦𝑣_𝑡+4𝛼𝑣𝑢_𝑡+𝑥𝛼𝑣𝑢_𝑡𝑥+2𝛼𝑦𝑣𝑢_𝑡𝑦,𝑇²=−𝑥𝛼𝑣𝑢_𝑡𝑡+𝑥𝛼𝑣𝑢_𝑦𝑦+2𝛼𝑥𝑢𝑣𝑢_𝑥𝑥+𝑣_𝑥+4𝛼𝑢𝑣_𝑥+𝑣_𝑥𝑥𝑥+4𝛼²𝑢²𝑣_𝑥+2𝛼𝑢𝑣_𝑥𝑥𝑥−8𝛼𝑡𝑣𝑢_𝑥𝑢_𝑡+2𝛼𝑡𝑢_𝑡𝑣_𝑥+4𝛼²𝑡𝑣𝑢_𝑡𝑢_𝑥+4𝛼²𝑡𝑢𝑢_𝑡𝑣_𝑥+2𝛼𝑡𝑢_𝑡𝑣_𝑥𝑥𝑥+𝑥𝛼𝑢_𝑥𝑣_𝑥+2𝛼²𝑥𝑢𝑢_𝑥𝑣_𝑥+𝑥𝛼𝑢_𝑥𝑣_𝑥𝑥𝑥−8𝛼²𝑦𝑣𝑢_𝑥𝑢_𝑦+2𝛼𝑦𝑢_𝑦𝑣_𝑥+4𝛼²𝑦𝑣𝑢_𝑥𝑢_𝑦+4𝛼²𝑦𝑢𝑢_𝑦𝑣_𝑥+2𝛼𝑦𝑢_𝑦𝑣_𝑥𝑥𝑥−2𝛼𝑡𝑣𝑢_𝑡𝑥−5𝛼𝑣𝑢_𝑥−2𝛼𝑦𝑣𝑢_𝑥𝑦−4𝛼²𝑡𝑢𝑣𝑢_𝑡𝑥−10𝛼²𝑢𝑣𝑢_𝑥−2𝛼²𝑥𝑢𝑣𝑢_𝑥𝑥−4𝛼²𝑦𝑢𝑣𝑢_𝑥𝑦−3𝛼𝑢_𝑥𝑣_𝑥𝑥−2𝛼𝑡𝑢_𝑡𝑥𝑣_𝑥𝑥−𝑥𝛼𝑢_𝑥𝑥𝑣_𝑥𝑥−2𝛼𝑦𝑢_𝑥𝑦𝑣_𝑥𝑥+2𝛼𝑡𝑣_𝑥𝑢_𝑡𝑥𝑥+4𝛼𝑣_𝑥𝑢_𝑥𝑥+𝑥𝛼𝑣_𝑥𝑢_𝑥𝑥𝑥+2𝛼𝑦𝑣_𝑥𝑢_𝑥𝑥𝑦−2𝛼𝑡𝑣𝑢_{𝑡𝑥𝑥𝑥}−5𝛼𝑣𝑢_𝑥𝑥𝑥−2𝛼𝑦𝑣𝑢_{𝑥𝑥𝑥𝑦},𝑇³=−2𝛼𝑦𝑣𝑢_𝑡𝑡+2𝛼𝑦𝑣𝑢_𝑥𝑥+4𝛼²𝑦𝑣𝑢²_𝑥+4𝛼²𝑦𝑣𝑢𝑢_𝑥𝑥+2𝛼𝑦𝑣𝑢_{𝑥𝑥𝑥𝑥}+𝑣_𝑦+2𝛼𝑢𝑣_𝑦+2𝛼𝑡𝑢_𝑡𝑣_𝑦+𝑥𝛼𝑢_𝑥𝑣_𝑦+2𝛼𝑦𝑢_𝑦𝑣_𝑦−4𝛼𝑣𝑢_𝑦−𝑥𝛼𝑣𝑢_𝑥𝑦−2𝛼𝑡𝑣𝑢_𝑡𝑦. (54)

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].