Exact Solutions and Conservation Laws of a (2+1)-Dimensional Nonlinear KP-BBM Equation

Abstract

In this paper, we study the two-dimensional nonlinear Kadomtsov-Petviashivilli-Benjamin-Bona-Mahony (KP-BBM) equation. This equation is the Benjamin-Bona-Mahony equation formulated in the KP sense. We first obtain exact solutions of this equation using the Lie group analysis and the simplest equation method. The solutions obtained are solitary waves. In addition, the conservation laws for the KP-BBM equation are constructed by using the multiplier method.

1. Introduction

It is well known that the nonlinear evolution equations (NLEEs) are extensively used as models to describe physical phenomena in various fields of sciences, especially in fluid mechanics, solid state physics, plasma physics, plasma waves, and biology. One of the basic physical problems for such models is to obtain their exact solutions. In this respect, during the past few decades, various methods have been developed by researchers to find explicit solutions for the NLEEs. See, for example, Wang et al. [1] and references therein.

The purpose of this paper is to study one such NLEE, namely, the (2+1)-dimensional nonlinear Kadomtsov-Petviashivilli-Benjamin-Bona-Mahony (KP-BBM) equation that is, given by 𝑢_𝑡+𝑢_𝑥−𝛼𝑢²_𝑥−𝛽𝑢_{𝑥𝑥𝑡𝑥}+𝛾𝑢_𝑦𝑦=0. (1) Here, in (1), 𝛼, 𝛽, and 𝛾 are real valued constants.

The solutions of (1) have been studied in various aspects. See, for example, the recent papers [2–5]. Wazwaz [2, 3] used the sine-cosine method, the tanh method and the extended tanh method for finding solitonary solutions of this equation. Abdou [4] used the extended mapping method with symbolic computation to obtain some periodic solutions, solitary wave solution, and triangular wave solution.

In this paper, Lie group analysis [6–9] in conjunction with the simplest equation method [10] is employed to obtain some exact solutions of (1). In addition to this conservation laws will be derived for (1) using the multiplier method [11].

2. Symmetry Analysis

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

2.1. Lie Point Symmetries

A Lie point symmetry of a partial differential equation (PDE) is an invertible transformation of the dependent and independent variables that leaves the equation unchanged. In general, determining all the symmetries of a partial differential equation is a formidable task. However, Sophus Lie observed that if we restrict ourselves to symmetries that depend continuously on a small parameter and that form a group (continuous one-parameter group of transformations), one can linearize the symmetry conditions and end up with an algorithm for calculating continuous symmetries.

The symmetry group of KP-BBM equation (1) will be generated by the vector field of the form 𝑉=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⁾𝑉 to (1), we obtain an overdetermined system of linear partial differential equations. Solving resultant system of linear overdetermined partial differential equations one obtains the following four Lie point symmetries: 𝑉₁=𝜕,𝑉𝜕𝑥₂=𝜕,𝑉𝜕𝑦₃=𝜕,𝑉𝜕𝑡₄𝜕=−𝛼𝑦𝜕𝜕𝑦−2𝛼𝑡+()𝜕𝜕𝑡2𝛼𝑢−1.𝜕𝑢 (3)

2.2. Exact Solutions

One of the main reasons for finding symmetries of a differential equation is to use them for finding exact solutions. In this subsection, we will utilize the symmetries calculated in the previous subsection to deduce exact solutions of (1).

One way to obtain exact solutions of (1) is by reducing it to ordinary differential equations. This can be achieved with the use of Lie point symmetries admitted by (1). It is well known that the reduction of a partial differential equation with respect to 𝑟-dimensional (solvable) subalgebra of its Lie symmetry algebra leads to reducing the number of independent variables by 𝑟.

First of all, we utilize the symmetry 𝑉=𝑉₁+𝑉₂+𝑉₃ and reduce the KP-BBM equation (1) to a PDE in two independent variables. It can be seen that the symmetry 𝑉 yields the following three invariants: 𝑓=𝑡−𝑦,𝑔=𝑥−𝑦,𝜃=𝑢. (4) Now treating 𝜃 as the new dependent variable and 𝑓 and 𝑔 as new independent variables, the KP-BBM equation (1) transforms to 𝜃_𝑓𝑔+𝜃_𝑔𝑔−2𝛼𝜃²_𝑔−2𝛼𝜃𝜃_𝑔𝑔−𝛽𝜃_{𝑓𝑔𝑔𝑔}+𝛾𝜃_𝑓𝑓+2𝛾𝜃_𝑓𝑔+𝛾𝜃_𝑔𝑔=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). Equation (5) has the following two translational symmetries: Γ₁=𝜕,Γ𝜕𝑔₂=𝜕.𝜕𝑓 (6) The combination 𝜌Γ₁+Γ₂, where 𝜌≠0 is a constant, of the two symmetries Γ₁ and Γ₂ yields the following two invariants: 𝑟=𝜌𝑓−𝑔𝜌,𝜓=𝜃. (7) This gives rise to a group invariant solution 𝜓=𝜓(𝑟) and consequently using these invariants, (5) is transformed into the fourth-order nonlinear ODE −2𝛼𝜌𝜓^′2−2𝛼𝜌𝜓𝜓^′′+𝜌+𝛾𝜌−𝜌²−2𝛾𝜌²+𝛾𝜌³𝜓^′′+𝛽𝜓^′′′′=0. (8) Integrating the above equation twice and taking the constants of integration to be zero, we obtain a second-order ODE 𝜌−1−𝛾+𝛾𝜌1−𝜌𝜓−𝛼𝜌𝜓²+𝛽𝜓^′′=0. (9) Multiplying (9) by 𝜓^′, integrating once and taking the constant of integration to be zero, we obtain the first-order ODE 12𝜌−1−𝛾+𝛾𝜌1−𝜌𝜓²−13𝛼𝜌𝜓³+12𝛽𝜓^′2=0. (10) One can integrate the above equation by separating the variables. After integrating and reverting back to the original variables, we obtain the following group-invariant solutions of the KP-BBM equation (1): 𝑢𝑥,𝑦,𝑡=𝐴₁sech²12±𝐴₂𝑟−𝐴₃𝐶, (11) where 𝐴₁=12𝛼³,𝐴−1+𝜌−1−𝛾+𝛾𝜌₂=𝜌1−𝜌−1−𝛾+𝛾𝜌𝛽,𝐴₃=3,𝑦−1+𝜌−1−𝛾+𝛾𝜌𝑟=𝑡𝜌−𝑥+1−𝜌𝜌. (12) To find another group-invariant solution of the KP-BBM equation (1), we now make use of the symmetry Γ=𝑎₁𝑉₁+𝑎₂𝑉₂+𝑎₃𝑉₃+𝑎₄𝑉₄  (𝑎_𝑖,𝑖=1,…,4 are constants). The symmetry Γ yields the following three invariants: 𝑓=2𝑎₃𝛼𝑡−𝑎₄2𝑎₃𝛼𝑎₂−𝑎₃𝛼𝑦²𝑎,𝑔=𝑥+¹ln𝑎₂−𝑎₃𝛼𝑦𝑎₃𝛼,𝜃=𝑎²₂𝑢−2𝑎₂𝑎₃𝛼𝑦𝑢+𝑎²₃𝛼²𝑦²𝑢+𝑎₂𝑎₃1𝑦−2𝑎²₃𝛼𝑦². (13) Treating 𝜃 as the new dependent variable and 𝑓 and 𝑔 as new independent variables, the KP-BBM equation (1) transforms to a nonlinear fourth-order PDE in two independent variables, namely, 𝑎²₂𝜃_𝑔𝑔−3𝑎²₂𝑎²₃𝛼𝛾+14𝑎²₃𝛼²𝛾𝑓𝜃_𝑓+𝑎²₁𝛾𝜃_𝑔𝑔−4𝑎₁𝑎₃𝛼𝛾𝑓𝜃_𝑓𝑔−2𝛼𝜃𝜃_𝑔𝑔𝜃_𝑓𝑔−2𝛼𝜃²_𝑔+6𝑎²₃𝛼²𝛾𝜃+4𝑎²₃𝛼²𝛾𝑓²𝜃_𝑓𝑓−5𝑎₁𝑎₃𝛼𝛾𝜃_𝑔−𝛽𝜃_{𝑓𝑔𝑔𝑔}=0. (14) Equation (14) has a single Lie point symmetry, namely, Γ₁=𝜕,𝜕𝑔 (15) and this symmetry yields the two invariants 𝑟=𝑓,𝜓=𝜃, (16) which gives a group-invariant solution 𝜓=𝜓(𝑟) and consequently (14) is then transformed to 4𝛼𝑟²𝜓^′′+14𝛼𝑟𝜓^′+6𝛼𝜓−3𝑎²₂=0, (17) which is a second-order linear ODE. Now, solving this equation and reverting back to the original variables, we obtain the following solutions of the KP-BBM equation (1): 𝑢𝑥,𝑦,𝑡=𝐴₁𝐴₂+𝐶₁𝑟^−3/2+𝐶₂𝑟⁻¹+𝐴₃, (18) where 𝐶₁ and 𝐶₂ are constants of integration and 𝐴₁=1𝑎²₂−2𝑎₂𝑎₃𝛼𝑦+𝑎²₃𝛼²𝑦²,𝐴₂=12𝑎²₃𝛼𝑦²−𝑎₂𝑎₃𝐴𝑦,₃=𝑎²₂,2𝛼𝑟=2𝑎₃𝛼𝑡−𝑎₄2𝑎₃𝛼𝑎₂−𝑎₃𝛼𝑦². (19)

3. Simplest Equation Method

We now use the simplest equation method, which was introduced by Kudryashov [10] and modified by Vitanov [12], to solve the fourth-order ODE (8). The simplest equations that will be used are the Bernoulli and Riccati equations. Their solutions can be written in elementary functions [13].

Let us consider the solution of (8) in the form 𝜓(𝑟)=^𝑀_𝑖=0𝒜_𝑖(𝐻(𝑟))^𝑖, (20) where 𝐻(𝑟) satisfies the Bernoulli and Riccati equations, 𝑀 is a positive integer that can be determined by balancing procedure as in [12], and 𝐴₀,…,𝐴_𝑀 are parameters to be determined.

3.1. Solutions of (8) Using the Equation of Bernoulli as the Simplest Equation

The balancing procedure [12] yields 𝑀=2 so the solutions of (20) are of the form 𝜓(𝑟)=𝐴₀+𝐴₁𝐻+𝐴₂𝐻². (21) Substituting (21) into (8) and making use of the Bernoulli equation [13] and then equating all coefficients of the functions 𝐻^𝑖 to zero, we obtain an algebraic system of equations in terms of 𝐴₀, 𝐴₁, and 𝐴₂.

Solving the system of algebraic equations, we obtain 𝐴₀=𝑎²𝛽+𝜌+𝛾𝜌−𝜌²−2𝛾𝜌²+𝛾𝜌³,𝐴2𝛼𝜌₁=6𝑎𝑏𝛽𝛼,𝐴₂=6𝑏²𝛽.𝛼𝜌 (22) Therefore, the solution of (1) is given by 𝑢𝑥,𝑦,𝑡=𝐴₀+𝐴₁cosh[𝑎(𝑟+𝐶)]+sinh[𝑎(𝑟+𝐶)]1−𝑏cosh[𝑎(𝑟+𝐶)]−𝑏sinh[𝑎(𝑟+𝐶)]+𝐴₂cosh[𝑎(𝑟+𝐶)]+sinh[𝑎(𝑟+𝐶)]1−𝑏cosh[𝑎(𝑟+𝐶)]−𝑏sinh[𝑎(𝑟+𝐶)]², (23) where 𝑟=𝑡𝜌−𝑥+(1−𝜌)𝑦/𝜌, and 𝐶 is a constant of integration.

3.2. Solutions of (8) Using Riccati Equation as the Simplest Equation

The balancing procedure yields 𝑀=2, so the solutions of (20) are of the form 𝜓(𝑟)=𝐴₀+𝐴₁𝐻+𝐴₂𝐻². (24) Substituting (24) into (8) and making use of the Riccati equation [13], we obtain algebraic equations in terms of 𝐴₀, 𝐴₁, and 𝐴₂ by equating all coefficients of the functions 𝐻^𝑖 to zero.

Solving the algebraic equations, one obtains 𝐴₀=𝑏²𝛽+𝜌+𝛾𝜌−𝜌²−2𝛾𝜌²+𝛾𝜌³,𝐴2𝛼𝜌₁𝐴=0,₂=22𝑎³𝛽+11𝑎²𝑏𝛽+2𝑎𝑏²𝛽(),𝛼𝜌2𝑎+3𝑏 (25) and hence the solutions of (1) are 𝑢𝑥,𝑦,𝑡=𝐴₀+𝐴₂−𝑏−𝜃2𝑎12𝑎tanh2𝜃(𝑟+𝐶)²,𝑢𝑥,𝑦,𝑡=𝐴₀+𝐴₂−𝑏−𝜃2𝑎12𝑎tanh2+()𝜃𝑟sech𝜃𝑟/2𝐶cosh(𝜃𝑟/2)−(2𝑎/𝜃)sinh(𝜃𝑟/2)², (26) where 𝑟=𝑡𝜌−𝑥+(1−𝜌)𝑦/𝜌, and 𝐶 is a constant of integration.

4. Conservation Laws

In this section, we construct conservation laws for (1). The multiplier method will be used [11]. See also [14].

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, (27) where 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 is given by 𝐷_𝑖=𝜕𝜕𝑥^𝑖+𝑢^𝛼_𝑖𝜕𝜕𝑢^𝛼+𝑢^𝛼_𝑖𝑗𝜕𝜕𝑢^𝛼_𝑗+⋯,𝑖=1,…,𝑛, (28) where the summation convention is used whenever appropriate [9].

The following are known (see, e.g., [9] and the references therein).

The Euler-Lagrange operator, for each 𝛼, is given by 𝛿𝛿𝑢^𝛼=𝜕𝜕𝑢^𝛼+_𝑠≥1(−1)^𝑠𝐷_𝑖1⋯𝐷_𝑖𝑠𝜕𝜕𝑢^𝛼_{𝑖1𝑖2⋯𝑖𝑠},𝛼=1,…,𝑚. (29) The 𝑛-tuple vector 𝑇=(𝑇¹,𝑇²,…,𝑇^𝑛), 𝑇^𝑗∈𝒜, 𝑗=1,…,𝑛, is a conserved vector of (27) if 𝑇^𝑖 satisfies 𝐷_𝑖𝑇^𝑖∣₍₂₇₎=0. (30) Equation (30) defines a local conservation law of system (27).

A multiplier Λ_𝛼(𝑥,𝑢,𝑢₍₁₎,…) has the property that Λ_𝛼𝐸_𝛼=𝐷_𝑖𝑇^𝑖 (31) holds identically. Here, we will consider multipliers of the second order, that is, Λ_𝛼=Λ_𝛼(𝑡,𝑥,𝑢,𝑣,𝑤,𝑢_𝑥,𝑣_𝑥,𝑤_𝑥,𝑢_𝑥𝑥,𝑣_𝑥𝑥,𝑤_𝑥𝑥). The right hand side of (31) is a divergence expression. The determining equation for the multiplier Λ_𝛼 is 𝛿Λ_𝛼𝐸_𝛼𝛿𝑢^𝛼=0. (32) Once the multipliers are obtained, the conserved vectors are calculated via a homotopy formula [11].

4.1. Construction of Conservation Laws for KP-BBM Equation

We now construct conservation laws for the (2+1)-dimensional nonlinear KP-BBM equation (1). The zeroth-order multiplier for (1) is, Λ(𝑡,𝑥,𝑦,𝑢), that is, given by Λ=𝐶₁𝑥+𝐶₁𝐶𝑡𝑥−¹𝑦²2𝛾+𝑦𝐹₂(𝑡)+𝐹₃(𝑡), (33) where 𝐶₁ is a constant, and 𝐹₂(𝑡) and 𝐹₃(𝑡) are arbitrary functions of 𝑡. Corresponding to the above multiplier, we have the following conserved vectors of (33): 𝑇^𝑡₁=18𝛾−4𝛾𝑡𝑢−4𝛾𝑢+2𝛽𝛾𝑡𝑢_𝑥𝑥−2𝛽𝛾𝑡𝑥𝑢_𝑥𝑥𝑥+4𝛾𝑡𝑥𝑢_𝑥+2𝛽𝛾𝑢_𝑥𝑥−2𝛽𝛾𝑥𝑢_𝑥𝑥𝑥+4𝛾𝑥𝑢_𝑥+𝛽𝑦²𝑢_𝑥𝑥𝑥−2𝑦²𝑢_𝑥,𝑇^𝑥₁=18𝛾8𝛼𝑦²𝑢_𝑥𝑢−16𝛼𝛾𝑡𝑥𝑢_𝑥𝑢−16𝛼𝛾𝑥𝑢_𝑥𝑢+8𝛼𝛾𝑡𝑢²+8𝛼𝛾𝑢²−8𝛾𝑡𝑢−4𝛾𝑥𝑢−8𝛾𝑢+4𝛽𝛾𝑡𝑢_𝑡𝑥+4𝛽𝛾𝑢_𝑡𝑥−6𝛽𝛾𝑡𝑥𝑢_𝑡𝑥𝑥−6𝛽𝛾𝑥𝑢_𝑡𝑥𝑥+8𝛾𝑡𝑥𝑢_𝑥+4𝛾𝑡𝑥𝑢_𝑡+4𝛾𝑥𝑢_𝑡+3𝛽𝑦²𝑢_𝑡𝑥𝑥−2𝑦²𝑢_𝑡−4𝛽𝛾𝑢_𝑥+2𝛽𝛾𝑥𝑢_𝑥𝑥+8𝛾𝑥𝑢_𝑥−4𝑦²𝑢_𝑥,𝑇^𝑦₁=122𝑦𝑢+2𝛾𝑡𝑥𝑢_𝑦+2𝛾𝑥𝑢_𝑦−𝑦²𝑢_𝑦;𝑇^𝑡₂=122𝑦𝐹₂(𝑡)𝑢_𝑥−𝛽𝑦𝐹₂(𝑡)𝑢_𝑥𝑥𝑥,𝑇^𝑥₂=14−8𝛼𝑦𝐹₂(𝑡)𝑢_𝑥𝑢−2𝑦𝐹^′₂𝑢−3𝛽𝑦𝐹₂(𝑡)𝑢_𝑡𝑥𝑥+4𝑦𝐹₂(𝑡)𝑢_𝑥+2𝑦𝐹₂(𝑡)𝑢_𝑡+𝛽𝑦𝐹^′₂𝑢_𝑥𝑥,𝑇^𝑦₂=𝛾𝑦𝐹₂(𝑡)𝑢_𝑦−𝛾𝐹₂(𝑡)𝑇𝑢;_𝑡3=142𝐹₃(𝑡)𝑢_𝑥−𝛽𝐹₃(𝑡)𝑢_𝑥𝑥𝑥,𝑇^𝑥₃=14−8𝛼𝐹₃(𝑡)𝑢_𝑥𝑢−2𝐹^′₃𝑢−3𝛽𝐹₃(𝑡)𝑢_𝑡𝑥𝑥+4𝐹₃(𝑡)𝑢_𝑥+2𝐹₃(𝑡)𝑢_𝑡+𝛽𝐹^′₃𝑢_𝑥𝑥,𝑇^𝑦₃=𝛾𝐹₃(𝑡)𝑢_𝑦. (34)

Remark 1. Due to the presence of the arbitrary function in the multiplier, one can obtain an infinitely many conservation laws for the (2+1)-dimensional nonlinear KP-BBM equation.

5. Concluding Remarks

In this paper, we obtained the solutions of the two-dimensional nonlinear Kadomtsov-Petviashivilli-Benjamin-Bona-Mahony (KP-BBM) equation by employing the Lie group analysis and the simplest equation method. The solutions obtained are solitary waves and nontopological solutions. The conservation laws for the underlying equation were also derived by using the multiplier method.