Exact Explicit Traveling Wave Solution for the Generalized (2+1)-Dimensional Boussinesq Equation

Abstract

The sine-cosine method and the extended tanh method are used to construct exact solitary patterns solution and compactons solutions of the generalized (2+1)-dimensional Boussinesq equation. The compactons solutions and solitary patterns solutions of the generalized (2+1)-dimensional Boussinesq equation are successfully obtained. These solutions may be important and of significance for the explanation of some practical physical problems. It is shown that the sine-cosine and the extended tanh methods provide a powerful mathematical tool for solving great many nonlinear partial differential equations in mathematical physics.

1. Introduction

Studies of various physical structures of nonlinear dispersive equations had attracted much attention in connection with the important problems that arise in scientific applications. Mathematically, these physical structures have been studied by using various analytical methods, such as inverse scattering method [1], Darboux transformation method [2, 3], Hirota bilinear method [4], Lie group method [5], bifurcation method of dynamic systems [6–8], sine-cosine method [9], tanh function method [10–12], Fan-expansion method [13], and homogenous balance method [14]. Practically, there is no unified technique that can be employed to handle all types of nonlinear differential equations.

Recently, by using the sine-cosine method, Taşcan and Bekir [15] studied the following (2+1)-dimensional Boussinesq equation:𝑢_𝑡𝑡−𝑢_𝑥𝑥−𝑢_𝑦𝑦−𝑢²_𝑥𝑥−𝑢_{𝑥𝑥𝑥𝑥}=0.(1.1) More new double periodic and multiple soliton solutions are obtained for the generalized (2+1)-dimensional Boussinesq equation in [16]. Chen et al. [17] study (2+1)-dimensional Boussinesq equation by using the new generalized transformation in homogeneous balance method. Feng et al. [8] have investigated the bifurcations and global dynamic behavior of two variants of (2+1)-dimensional Boussinesq-type equations with positive and negative exponents and obtained the sufficient conditions under which solitary, kink, breaking, and periodic wave solutions appear. A mathematical method is constructed to study two variants of the two-dimensional Boussinesq water equation with positive and negative exponents in [18].

In this paper, we consider the following generalized (2+1)-dimensional Boussinesq equation: (𝑢^𝑛)_𝑡𝑡−(𝑢^𝑛)_𝑥𝑥−(𝑢^𝑛)_𝑦𝑦−𝑎𝑢²_𝑥𝑥−𝑏(𝑢^𝑛)_{𝑥𝑥𝑥𝑥}=0,(1.2) where 𝑎,𝑏 are arbitrary constants and 𝑛 is nonzero integers. Specially, when 𝑎=𝑏=𝑛=1 system (1.2) becomes the (2+1)-dimensional Boussinesq equation.

The sine-cosine method and the tanh method will be mainly used to back up our analysis. The tanh method, developed by Malfliet and Hereman [11, 12] and Wazwaz [19], is a direct and effective algebraic method for handling many nonlinear equations. The sine-cosine method was proved to be powerful in handling nonlinear problems, with genuine nonlinear dispersion, where compactons and solitary patterns solutions are generated. The two methods will be described briefly, where details can be found in [9–12, 19] and the references therein.

Let 𝑢(𝑥,𝑦,𝑡)=𝑢(𝜉),𝜉=𝑥+𝑦−𝑐𝑡, where 𝑐 is the wave speed. Then (1.2) becomes 𝑐²−2(𝑢^𝑛)^−𝑎𝑢²^−𝑏(𝑢^𝑛)^=0,(1.3) where ”′” is the derivative with respect to 𝜉. Integrating (1.3) twice and neglecting constants of integration, we find 𝑐²−2𝑢^𝑛−𝑎𝑢²−𝑏(𝑢^𝑛)^=0.(1.4)

Suppose that 𝑢(𝜉) is a continuous solution of (1.4) for 𝜉∈(−∞,∞) and lim_𝜉→∞𝜙(𝜉)=𝑎,lim_{𝜉→−∞}𝜙(𝜉)=𝑏. Recall that (i) 𝑢(𝑥,𝑡) is called a solitary wave solution if 𝑎=𝑏 and (ii) 𝑢(𝑥,𝑡) is called a kink or antikink solution if 𝑎≠𝑏. It is well known that a limit boundary curve of a smooth family of periodic orbits gives rise to a smooth periodic traveling wave solution, that is called a compacton.

The paper is organized as follows. In Section 2, the sine-cosine method and the tanh method are briefly discussed. Section 3 represents exact analytical solutions of (1.2) by using the sine-cosine method. Section 4 represents exact analytical solutions of (1.2) by using the extended tanh method. In the last section, we conclude the paper and give some discussions.

2. Analysis of the Two Methods

The sine-cosine method and the extended tanh method have been applied for a wide variety of nonlinear problems. The main features of the two methods will be reviewed briefly.

For both methods, we first use the wave variable 𝜉=𝑥+𝑦−𝑐𝑡 to carry a PDE in two independent variables 𝑃𝑢,𝑢_𝑡,𝑢_𝑥,𝑢_𝑦,𝑢_𝑥𝑥,𝑢_𝑥𝑦,𝑢_𝑦𝑦,…=0(2.1) into an ODE 𝑄𝑢,𝑢^,𝑢^,𝑢^,…=0.(2.2)

Equation (2.2) is then integrated as long as all terms contain derivatives where integration constants are considered zeros.

2.1. The Sine-Cosine Method

The sine-cosine method admits the use of the solution in the form𝑢(𝑥,𝑦,𝑡)=⎧⎪⎨⎪⎩𝜆cos^𝛽(𝜇𝜉),||𝜇𝜉||<𝜋0,otherwi2se,(2.3) or in the form 𝑢(𝑥,𝑦,𝑡)=𝜆sin^𝛽0,(𝜇𝜉),||o𝜇th𝜉e||r<w𝜋is,e,(2.4) where 𝜆,𝜇, and 𝛽 are parameters that will be determined.

We substitute (2.3) or (2.4) into the reduced ordinary differential equation obtained above in (2.2), balance the terms of the cosine functions when (2.3) is used or balance the terms of the sine functions when (2.4) is used, and solve the resulting system of algebraic equations by using the computerized symbolic calculations to obtain all possible values of the parameters 𝜆,𝜇, and 𝛽.

2.2. The Tanh Method and the Extended Tanh Method

The standard tanh method is introduced in [11, 12] where the tanh is used as a new variable, since all derivatives of a tanh are represented by a tanh itself. We use a new independent variable 𝑌=tanh(𝜇𝜉),(2.5) that leads to the change of derivatives: 𝑑𝑑𝑑𝜉=𝜇1−𝑌²𝑑𝑑𝑌₂,𝑑𝜉²=𝜇²1−𝑌²−2𝑌𝑑𝑑𝑌+1−𝑌²𝑑𝑑𝑌²².(2.6)

We then apply the following finite expansion:𝑢(𝜇𝜉)=𝑆(𝑌)=^𝑀_𝑘=0𝑢(𝜇𝜉)=𝑆(𝑌)=_𝑀𝑎_𝑘𝑌^𝑘,(2.7)_𝑘=0𝑎_𝑘𝑌^𝑘+^𝑀_𝑘=1𝑏_𝑘𝑌^−𝑘,(2.8) where 𝑀 is a positive integer that will be determined to derive a closed form analytic solution. However, if 𝑀 is not an integer, then a transformation formula is usually used. Substituting (2.5) and (2.6) into the simplified ODE (2.2) results in an equation in powers of 𝑌. To determine the parameter 𝑀, we usually balance the linear terms of highest order in the resulting equation with the highest order nonlinear terms. With 𝑀 determined, we collect all coefficients of powers of 𝑌 in the resulting equation where these coefficients have to vanish. This will give a system of algebraic equations involving the parameters 𝑎_𝑘(𝑘=0,…,𝑀),𝜇, and 𝑐. Having determined these parameters, knowing that 𝑀 is a positive integer in most cases and using (2.7) or (2.8), we obtain an analytic solution 𝑢(𝑥,𝑡) in a closed form.

3. Using the Sine-Cosine Method

Substituting (2.3) into (1.4) yields −𝑎𝜆²cos^2𝛽(𝜇𝜉)+𝑐²−2+𝑏𝑛²𝜇²𝛽²𝜆^𝑛cos^𝑛𝛽(𝜇𝜉)−𝑏𝑛𝜇²𝜆^𝑛𝛽(𝑛𝛽−1)cos^𝑛𝛽−2(𝜇𝜉)=0.(3.1) Equation (3.1) is satisfied only if the following system of algebraic equations holds: 𝑛𝛽≠1,𝑛𝛽−2=2𝛽,𝑐²+𝑏𝑛²𝜇²𝛽²=2,−𝑎𝜆²=𝑏𝑛𝜇²𝜆^𝑛𝛽(𝑛𝛽−1)(3.2) or 𝜆=𝜆≠0,𝑎=0,𝑏≠0,𝑛𝛽−1=0,𝑐²+𝑏𝑛²𝜇²𝛽²=2.(3.3) Solving the system (3.2) and (3.3) gives 𝛽=2𝑛−2,𝜇²=2−𝑐²(𝑛−2)²4𝑏𝑛²,𝜆=2𝑎𝑛𝑐²−2(𝑛+2)^1/𝑛−2𝑏≠0,𝜆=𝜆≠0,𝑎=0,𝛽=1,𝑛≠±2,𝑎𝑏≠0,(3.4)𝑛,𝜇²=2−𝑐²𝑏.(3.5) The results (3.4) and (3.5) can be easily obtained if we also use the sine method (2.4). Combining (3.4) and (3.5) with (2.3) and (2.4), the following compactons solutions are yield: 𝑢₁=⎧⎪⎪⎪⎨⎪⎪⎪⎩±2𝑎𝑛𝑐²−2(𝑛+2)cos²𝑛−22𝑛2−𝑐²𝑏(𝑥+𝑦−𝑐𝑡),^1/𝑛−2||𝑥+𝑦−𝑐𝑡||<𝑛𝜋,𝑛−2𝑏2−𝑐²𝑢0,oth,er𝑛wi=se±,₂2(𝑘+1),𝑘∈𝑍⁺,𝑏2−𝑐²>0,𝑎𝑐²−2>0,=⎧⎪⎪⎪⎨⎪⎪⎪⎩±2𝑎𝑛𝑐²−2(𝑛+2)cos²𝑛−22𝑛2−𝑐²𝑏(𝑥+𝑦−𝑐𝑡)^1/𝑛−2||𝑥+𝑦−𝑐𝑡||<𝑛𝜋,𝑛−2𝑏2−𝑐²𝑢0,,other𝑛wi=se2,₃𝑘+1,𝑘∈𝑍,𝑏2−𝑐²>0=⎧⎪⎪⎪⎨⎪⎪⎪⎩𝜆cos^1/𝑛2−𝑐²||𝑥+𝑦−𝑐𝑡||<𝑏𝜋(𝑥+𝑦−𝑐𝑡),2𝑏2−𝑐²𝑢0,,other𝑏wi2se−,₄𝑐²>0,𝜆≠0,𝑎=0,𝑛≠0,=⎧⎪⎪⎪⎨⎪⎪⎪⎩±2𝑎𝑛𝑐²−2(𝑛+2)sin²𝑛−22𝑛2−𝑐²𝑏(𝑥+𝑦−𝑐𝑡)^1/𝑛−2||𝑥+𝑦−𝑐𝑡||<𝑛𝜋,𝑛−2𝑏2−𝑐²𝑢0,othe,rw𝑛is=e,₅±2(𝑘+1),𝑘∈𝑍⁺,𝑏2−𝑐²>0,𝑎𝑐²−2>0,=⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩±2𝑎𝑛𝑐²−2(𝑛+2)sin²𝑛−22𝑛2−𝑐²𝑏(𝑥+𝑦−𝑐𝑡),^1/𝑛−2||𝑥+𝑦−𝑐𝑡||<𝑛𝜋,𝑛−2𝑏2−𝑐²𝑢0,,otherw𝑛is=e,₆2𝑘+1,𝑘∈𝑍,𝑏2−𝑐²>0,=⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩𝜆sin^1/𝑛2−𝑐²||𝑥+𝑦−𝑐𝑡||<𝑏𝜋(𝑥+𝑦−𝑐𝑡),2𝑏2−𝑐²0,,other𝑏wi2se−𝑐²>0,𝜆≠0,𝑎=0,𝑛≠0,(3.6) However, for 𝑏(2−𝑐²)<0, we obtain the following solitary patterns solutions: 𝑢₇=±2𝑎𝑛𝑐²−2(𝑛+2)cosh²𝑛−22𝑛2−𝑐²−𝑏(𝑥+𝑦−𝑐𝑡)^1/𝑛−2𝑛=±2(𝑘+1),𝑘∈𝑍₊,𝑢,𝑏2−𝑐²<0,𝑎𝑐²−2>0,₈=2𝑎𝑛𝑐²−2(𝑛+2)cosh²𝑛−22𝑛2−𝑐²−𝑏(𝑥+𝑦−𝑐𝑡),^1/𝑛−2𝑛=2𝑘+1,𝑘∈𝑍,𝑏2−𝑐₂,𝑢<0,₉=±−2𝑎𝑛𝑐²−2(𝑛+2)sinh²𝑛−22𝑛2−𝑐²−𝑏(𝑥+𝑦−𝑐𝑡)^1/𝑛−2𝑛=±2(𝑘+1),𝑘∈𝑍₊,𝑢,𝑏2−𝑐²<0,𝑎𝑐²−2<0,₁₀=−2𝑎𝑛𝑐²−2(𝑛+2)sinh²𝑛−22𝑛2−𝑐²−𝑏(𝑥+𝑦−𝑐𝑡)^1/𝑛−2𝑛=2𝑘+1,𝑘∈𝑍,𝑏2−𝑐₂,<0.(3.7)

4. Using the Extended Tanh Method

In this section, we employ the extended tanh method to (1.4). Balancing 𝑢² with 𝑢^𝑛−1𝑢^, we find 𝑀=−2𝑛−2.(4.1)

To get a closed-form analytic solution, the parameter 𝑀 should be an integer. A transformation formula 𝑢=𝜙^{−1/𝑛−2}(4.2) should be used to achieve our goal. This in turn transforms (1.4) to 𝑐²−2𝜙²−𝑎𝜙³−2𝑏𝑛²(𝑛−2)²𝜙^²+𝑛𝑏−𝑛2𝜙𝜙^=0.(4.3) Balancing 𝜙𝜙^ with 𝜙³ gives 𝑀=2. The extended tanh method allows us to use the substitution 𝜙(𝜉)=𝑆(𝑌)=𝐴₀+𝐴₁𝑌+𝐴₂𝑌²+𝐵₁𝑌⁻¹+𝐵₂𝑌⁻².(4.4) Substite (4.4) into (4.3), collecte the coefficients of each power of 𝑌, and solve the resulting system of algebraic equations with the help of Maple to find the sets of solutions: 𝐴₁=𝐵₁=𝐴₂=0,𝐵₂=−𝐴₀,𝐴₀=𝐴₀,𝑐²=−2a𝑛𝐴⁰−𝑛−6+2𝑎𝐴₀𝜇𝑛+6₂,=𝑎𝐴⁰(𝑛−2)²2𝑏𝑛(𝑛+6),𝑎𝑏𝐴⁰𝐴(𝑛−2)(𝑛+6)≠0,(4.5)₁=𝐵₁=𝐵₂=0,𝐴₂=−𝐴₀,𝑐²=−2𝑎𝑛𝐴⁰−𝑛−6+2𝑎𝐴₀𝜇𝑛+6₂,=𝑎𝐴⁰(𝑛−2)²2𝑏𝑛(𝑛+6),𝑎𝑏𝐴⁰𝐴(𝑛−2)(𝑛+6)≠0,(4.6)₁=𝐵₁=0,𝐴₀=𝐴₀,𝐴₂=𝐵₂=−12𝐴⁰,𝑐²=−22a𝑛𝐴⁰−𝑛−6+4𝑎𝐴₀𝜇𝑛+6₂,=𝑎𝐴⁰(𝑛−2)²4𝑏𝑛(𝑛+6),𝑎𝑏𝐴⁰(𝑛−2)(𝑛+6)≠0.(4.7)

We obtain the solitary wave solution and the solitary patterns solutions 𝑢₁₁=⎡⎢⎢⎣sinh²(𝑛−2)√𝑎𝐴₀/(2𝑏𝑛(𝑛+6))(𝑥+𝑦−𝑐𝑡)−𝐴₀⎤⎥⎥⎦^1/𝑛−2𝑐,𝐴₀<0,𝑎𝑏𝑛(𝑛+6)<0,₂=−2𝑎𝑛𝐴⁰−𝑛−6+2𝑎𝐴₀𝑛+6>0,𝑛=±2𝑘,𝑘∈𝑍⁺𝑢,𝑛≠2,₁₂=⎡⎢⎢⎣sinh²(𝑛−2)√𝑎𝐴₀/(2𝑏𝑛(𝑛+6))(𝑥+𝑦−𝑐𝑡)−𝐴₀⎤⎥⎥⎦^1/𝑛−2𝑐,𝑎𝑏𝑛(𝑛+6)𝐴₀>0,₂=−2𝑎𝑛𝐴⁰−𝑛−6+2𝑎𝐴₀𝑢𝑛+6>0,𝑛=2𝑘+1,𝑘∈𝑍,₁₃=±⎡⎢⎢⎣cosh²(𝑛−2)√𝑎𝐴₀/(2𝑏𝑛(𝑛+6))(𝑥+𝑦−𝑐𝑡)𝐴₀⎤⎥⎥⎦^1/𝑛−2𝑐,𝐴₀>0,𝑎𝑏𝑛(𝑛+6)<0,₂=−2𝑎𝑛𝐴⁰−𝑛−6+2𝑎𝐴₀𝑛+6>0,𝑛=±2𝑘,𝑘∈𝑍⁺𝑢,𝑛≠2,₁₄=⎡⎢⎢⎣cosh²(𝑛−2)√𝑎𝐴₀/(2𝑏𝑛(𝑛+6))(𝑥+𝑦−𝑐𝑡)𝐴₀⎤⎥⎥⎦^1/𝑛−2𝑐,𝑎𝑏𝑛(𝑛+6)𝐴₀>0,₂=−2𝑎𝑛𝐴⁰−𝑛−6+2𝑎𝐴₀𝑢𝑛+6>0,𝑛=2𝑘+1,𝑘∈𝑍,₁₅=±⎡⎢⎢⎣sinh²(𝑛−2)√𝑎𝐴₀/(𝑏𝑛(𝑛+6))(𝑥+𝑦−𝑐𝑡)−2𝐴₀⎤⎥⎥⎦^1/𝑛−2𝑐,𝐴₀<0,𝑎𝑏𝑛(𝑛+6)<0,₂=−22𝑎𝑛𝐴⁰−𝑛−6+4𝑎𝐴₀𝑛+6>0,𝑛=±2𝑘,𝑘∈𝑍⁺𝑢,𝑛≠2,₁₆=±⎡⎢⎢⎣sinh²(𝑛−2)√𝑎𝐴₀/(𝑏𝑛(𝑛+6))(𝑥+𝑦−𝑐𝑡)−2𝐴₀⎤⎥⎥⎦^1/𝑛−2𝑐,𝑎𝑏𝑛(𝑛+6)𝐴₀>0,₂=−22𝑎𝑛𝐴⁰−𝑛−6+4𝑎𝐴₀𝑛+6>0,𝑛=2𝑘+1,𝑘∈𝑍.(4.8)

We can obtain the following compactons solutions for (1.4): 𝑢₁₇=±⎡⎢⎢⎣sin²(𝑛−2)√−𝑎𝐴₀/(2𝑏𝑛(𝑛+6))(𝑥+𝑦−𝑐𝑡)𝐴₀⎤⎥⎥⎦^1/𝑛−2𝑐,𝐴₀>0,𝑎𝑏𝑛(𝑛+6)<0,₂=−2𝑎𝑛𝐴⁰−𝑛−6+2𝑎𝐴₀𝑛+6>0,𝑛=±2𝑘,𝑘∈𝑍⁺𝑢,𝑛≠2,₁₈=⎡⎢⎢⎣sin²(𝑛−2)√−𝑎𝐴₀/(2𝑏𝑛(𝑛+6))(𝑥+𝑦−𝑐𝑡)𝐴₀⎤⎥⎥⎦^1/𝑛−2𝑐,𝐴₀>0,𝑎𝑏𝑛(𝑛+6)<0,₂=−2𝑎𝑛𝐴⁰−𝑛−6+2𝑎𝐴₀𝑢𝑛+6>0,𝑛=2𝑘+1,𝑘∈𝑍,₁₉=±⎡⎢⎢⎣cos²(𝑛−2)√−𝑎𝐴₀/(2𝑏𝑛(𝑛+6))(𝑥+𝑦−𝑐𝑡)𝐴₀⎤⎥⎥⎦^1/𝑛−2𝑐,𝐴₀>0,𝑎𝑏𝑛(𝑛+6)<0,₂=−2𝑎𝑛𝐴⁰−𝑛−6+2𝑎𝐴₀𝑛+6>0,𝑛=±2𝑘,𝑘∈𝑍⁺𝑢,𝑛≠2,₂₀=⎡⎢⎢⎣cos²(𝑛−2)√−𝑎𝐴₀/(2𝑏𝑛(𝑛+6))(𝑥+𝑦−𝑐𝑡)𝐴₀⎤⎥⎥⎦^1/𝑛−2𝑐,𝐴₀>0,𝑎𝑏𝑛(𝑛+6)<0,₂=−2𝑎𝑛𝐴⁰−𝑛−6+2𝑎𝐴₀𝑢𝑛+6>0,𝑛=2𝑘+1,𝑘∈𝑍,₂₁=±⎡⎢⎢⎣sin²(𝑛−2)√−𝑎𝐴₀/(𝑏𝑛(𝑛+6))(𝑥+𝑦−𝑐𝑡)2𝐴₀⎤⎥⎥⎦^1/𝑛−2𝑐,𝐴₀>0,𝑎𝑏𝑛(𝑛+6)<0,₂=−22𝑎𝑛𝐴⁰−𝑛−6+4𝑎𝐴₀𝑛+6>0,𝑛=±2𝑘,𝑘∈𝑍⁺𝑢,𝑛≠2,₂₂=±⎡⎢⎢⎣sin²(𝑛−2)√−𝑎𝐴₀/(𝑏𝑛(𝑛+6))(𝑥+𝑦−𝑐𝑡)2𝐴₀⎤⎥⎥⎦^1/𝑛−2𝑐,𝐴₀>0,𝑎𝑏𝑛(𝑛+6)<0,₂=−22𝑎𝑛𝐴⁰−𝑛−6+4𝑎𝐴₀𝑛+6>0,𝑛=2𝑘+1,𝑘∈𝑍.(4.9)

5. Discussion

The sine-cosine method and the extended tanh method were used to investigate the generalized (2+1)-dimensional Boussinesq equation. The study revealed compactons solutions and solitary patterns solutions for some examined variants. The study emphasized the fact that the two methods are reliable in handling nonlinear problems. The obtained results clearly demonstrate the efficiency of the two methods used in this work. Moreover, the methods are capable of greatly minimizing the size of computational work. This emphasizes the fact that the two methods are applicable to a wide variety of nonlinear problems.

Acknowledgments

The authors wish to thank the anonymous reviewers for their helpful comments and suggestions. This work is supported by the National Natural Science Foundation of China (no. 11061010).