By using the tanh-coth method, we obtained some travelling wave solutions of two well-known nonlinear Sobolev type partial differential equations, namely, the Benney-Luke equation and the higher-order improved Boussinesq equation. We show that the tanh-coth method is a useful, reliable, and concise method to solve these types of equations.

1. Introduction

The term “Sobolev equation” is used in the Russian literature to refer to any equation with spatial derivatives on the highest order time derivative [1]. In other words, they are characterized by having mixed time and space derivatives appearing in the highest-order terms of the equation and were studied by Sobolev [2]. Equations of Sobolev type describe many physical phenomena [3–7]. In recent years considerable attention has been paid to the study of equations of Sobolev type. For more details we refer the reader to [8] and references therein.

The Benney-Luke equation is as follows:
(1.1)utt-uxx+auxxxx-buxxtt+utuxx+2uxuxt=0,
where a and b are positive numbers, such that a-b=σ-1/3 is a Sobolev type equation and studied for a very long time. The dimensionless parameter σ is named the Bond number, which captures the effects of surface tension and gravity force and is a formally valid approximation for describing two-way water wave propagation in the presence of surface tension [9]. In [10] Pego and Quintero studied the propagation of long water waves with small amplitude. They showed that in the presence of a surface tension, the propagation of such waves is governed by (1.1), originally derived by Benney and Luke [11]. There are many studies concerning with this equation. Amongst them the stability analysis [9, 12], Cauchy problem [13–15], existence and analyticity of solutions [16], and travelling wave solutions [17] can be mentioned.

In [18], Schneider and Wayne showed that in the longwave limit the water wave problem without surface tension can be described approximately by two decoupled KdV equations. They considered a class of Boussinesq equation which models the water wave problem with surface tension as follows:
(1.2)-uxxxxtt+uxxtt-utt+uxx+μuxxxx+(u2)xx=0,
where x,t,μ∈ℝ and u(x,t)∈ℝ. Duruk et al. investigated the well posedness of the Cauchy problem
(1.3)-βuxxxxtt+uxxtt-utt+uxx+(g(u))xx=0,x∈ℝ,t>0
and showed that under certain conditions the Cauchy problem is globally well posed [19]. Nevertheless, several types of the improved Boussinesq equation were investigated by many researchers and found exact solutions by using exp-function method [20], modified extended tanh-function method [21], sine-cosine method [22], improved G’/G-expansion method [22], the standard tanh and the extended tanh method [23], and so forth.

The tanh-coth is a powerful and reliable technique for finding exact travelling wave solutions for nonlinear equations. This method has been used extensively, and it was subjected by some modifications using the Riccati equation. The main features of the tanh-coth method will be outlined in the subsequent section, and this method will be applied to the the Benney-Luke and the Higher-order improved Boussinesq equations. The main purpose of this work is to obtain travelling wave solutions of the above-mentioned equations and to show that the tanh-coth method can be easily applied to Sobolev type equations. Throughout the work, Maple is used to deal with the tedious algebraic operations.

2. Outline of the Tanh-Coth Method

Wazwaz has summarized the tanh method in the following manner.

First consider a general form of nonlinear equation
(2.1)P(u,ut,ux,uxx,…)=0.

To find the traveling wave solution of (2.1), the wave variable ξ=x-Vt is introduced, so that
(2.2)u(x,t)=U(μξ).
Based on this one may use the following changes:
(2.3)∂∂t=-Vddξ,∂∂x=μddξ,∂2∂x2=μ2d2dξ2,∂3∂x3=μ3d3dξ3,
and so on for other derivatives. Using (2.3) changes the PDE (2.1) to an ODE as follows:
(2.4)Q(U,U′,U′′,…)=0.

If all terms of the resulting ODE contain derivatives in ξ, then by integrating this equation and by considering the constant of integration to be zero, one obtains a simplified ODE.

A new independent variable
(2.5)Y=tanh(μξ)
is introduced that leads to the change of derivatives:
(2.6)ddξ=μ(1-Y2)ddY,d2dξ2=-2μ2Y(1-Y2)ddY+μ2(1-Y2)2d2dY2,d3dξ3=2μ3(1-Y2)(3Y2-1)ddY-6μ3Y(1-Y2)2d2dY2+μ3(1-Y2)3d3dY3,d4dξ4=-8μ4Y(1-Y2)(3Y2-2)ddY+4μ4(1-Y2)2(9Y2-2)d2dY2-12μ4Y(1-Y2)3d3dY3+μ4(1-Y2)4d4dY4,
where other derivatives can be derived in a similar manner.

The ansatz of the form
(2.7)U(μξ)=S(Y)=∑k=0MakYk+∑k=1MbkY-k
is introduced where M is a positive integer, in most cases, that will be determined. If M is not an integer, then a transformation formula is used to overcome this difficulty. Substituting (2.6) and (2.7) into the ODE, (2.4) yields an equation in powers of Y.

To determine the parameter M, the linear terms of highest order in the resulting equation with the highest order nonlinear terms are balanced. With M determined, one collects the 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 ak and bk, (k=0,…,M), V, and μ. Having determined these parameters, knowing that M is a positive integer in most cases, and using (2.7) one obtains an analytic solution in a closed form.

3. The Benney-Luke Equation

The Benney-Luke equation can be written as
(3.1)utt-uxx+auxxxx-buxxtt+utuxx+2uxuxt=0,
where a and b are positive numbers such that a-b=σ-1/3 (σ is named the Bond number). In order to solve (3.1) by the tanh-coth method, we use the wave transformation u(x,t)=U(μξ) with wave variable ξ=x-Vt; (3.1) takes on the form of an ordinary differential equation as follows:
(3.2)(V2-1)U′′+(a-bV2)U′′′′-3VU′U′′=0.
Balancing the order of U′′′′ with the order of U′U′′ in (3.2) we find M=1. Using the assumptions of the tanh-coth method (2.5)–(2.7) gives the solution in the form
(3.3)U(μξ)=S(Y)=∑k=01akYk+∑k=11bkY-k.
Substituting (3.3) into (3.2), we obtain a system of algebraic equations for a0, a1, b1, and V in the following form:
(3.4)Y10:-24bV2a1μ4+24Va12μ3+24aa1μ4=0,Y9:18Va0a1μ3=0,Y8:-40bV2a1μ4-2V2a1μ2+36Va12μ3-12b1Va1μ3+40aa1μ4+2a1μ2=0,Y7:24Va0a1μ3=0,Y6:-16bV2a1μ4-2V2a1μ2+12Va12μ3-12b1Va1μ3+16aa1μ4+2a1μ2=0,Y5:6Va0a1μ3+6Va0b1μ3=0,Y4:-16bV2b1μ4-2V2b1μ2+12Vb12μ3-12a1Vb1μ3+16ab1μ4+2b1μ2=0,Y3:24Va0b1μ3=0,Y2:-40bV2b1μ4-2V2b1μ2+36Vb12μ3-12a1Vb1μ3+40ab1μ4+2b1μ2=0,Y1:18Va0b1μ3=0,Y0:24Vb12μ3+24ab1μ4-24V2bb1μ4=0.
From the output of the Maple packages we find three sets of solutions:
(3.5)a0=b1=0,a1=(-a+b)μV(2μ2b+1),V=±2μ2a+12μ2b+1,a0=a1=0,b1=(-a+b)μV(2μ2b+1),V=±2μ2a+12μ2b+1,a0=0,a1=b1=(-a+b)μV(8μ2b+1),V=±8μ2a+18μ2b+1,
where μ is left as a free parameter. The travelling wave solutions are as follows:
(3.6)u1(x,t)=2μ2b+12μ2a+1(-a+b)μ(2μ2b+1)tanhμ(x∓2μ2a+12μ2b+1t),u2(x,t)=2μ2b+12μ2a+1(-a+b)μ(2μ2b+1)cothμ(x∓2μ2a+12μ2b+1t),u3(x,t)=8μ2b+18μ2a+1(-a+b)μ(2μ2b+1)(tanhμ(x∓8μ2a+18μ2b+1t)+cothμ(x∓8μ2a+18μ2b+1t)).

4. The Higher-Order Improved Boussinesq Equation

We consider the Higher-order improved Boussinesq equation as follows:
(4.1)-αuxxxxtt+βuxxtt-utt+uxx+(u2)xx=0,
where α and β are arbitrary non zero real constants.

Using the wave transformation u(x,t)=U(μξ) with wave variable ξ=x-Vt then by integrating this equation and considering the constant of integration to be zero, we obtain the ODE as follows:
(4.2)-αV2U′′′′+βV2U′′+(1-V2)U+U2=0.
Balancing the first term with the last term in (4.2) we find M=4. Using the assumptions of the tanh-coth method (2.5)–(2.7) gives the solution in the form
(4.3)U(μξ)=S(Y)=∑k=04akYk+∑k=14bkY-k.
Substituting (4.3) into (4.2), we obtain a system of algebraic equations for a0,a1,a2,a3,a4,b1,b2,b3,b4,μ, and V in the following form:
(4.4)Y16:a42-840V2a4αμ4=0,Y15:2a4a3-360V2a3αμ4=0,Y14:2a4a2+a32+20V2a4βμ2+2080V2a4αμ4-120V2a2αμ4=0,Y13:2a4a1+2a3a2+12V2a3βμ2+816V2a3αμ4-24V2a1αμ4=0,Y12:240αV2a2μ4+6βV2a2μ2-1696a4αV2μ4-32a4βV2μ2-a4V2+a22+a4+2a4a0+2a3a1=0,Y11:a3-V2a3+2a4b1+2a3a0+2a1a2-18V2a3βμ2-576V2a3αμ4+2V2a1βμ2+40V2a1αμ4=0,Y10:a2-V2a2+2a4b2+2b1a3+2a0a2+a12+12V2a4βμ2+480V2a4αμ4-8V2a2βμ2-136V2a2αμ4=0,Y9:a1-V2a1+2a4b3+2b2a3+2b1a2+2a0a1+6V2a3βμ2+120V2a3αμ4-2V2a1βμ2-16V2a1αμ4=0,Y8:a0-V2a0+2a4b4+2b3a3+2b1a1+2b2a2+a02-24V2a4αμ4+2V2b2βμ2+16V2b2αμ4-24V2b4αμ4+2V2a2βμ2+16V2a2αμ4+16V2b2αμ4-24V2b4αμ4+2V2a2βμ2+16V2a2αμ4=0,Y7:b1-V2b1+2b4a3+2b1a0+2b2a1+2b3a2-2V2b1βμ2-16V2b1αμ4+6V2b3βμ2+120V2b3αμ4=0,Y6:b2-V2b2+2b2a0+2b3a1+2b4a2+b12-8V2b2βμ2-136V2b2αμ4+12V2b4βμ2+480V2b4αμ4=0,Y5:b3-V2b3+2b1b2+2b3a0+2b4a1+2V2b1βμ2+40V2b1αμ4-18V2b3βμ2-576V2b3αμ4=0,Y4:240αV2b2μ4+6βV2b2μ2-1696b4αV2μ4-32b4βV2μ2-b4V2+b22+b4+2b1b3+2b4a0=0,Y3:2b1b4+2b2b3-24V2b1αμ4+12V2b3βμ2+816V2b3αμ4=0,Y2:2b2b4+b32-120V2b2αμ4+20V2b4βμ2+2080V2b4αμ4=0,Y1:2b3b4-360V2b3αμ4=0,Y0:b42-840V2b4αμ4=0.
Using Maple gives six sets of solutions:
(4.5)a0=a4=105β22(-36β2+169α),a2=-105β2-36β2+169α,a1=a3=b1=b2=b3=b4=0,V=±13-1872μ2β+169,μ=±12613βα,a0=33β22(36β2+169α),a2=-105β236β2+169α,a4=105β22(36β2+169α),a1=a3=b1=b2=b3=b4=0,V=±131872μ2β+169,μ=±12613βα,a0=105β22(-36β2+169α),b2=-105β2-36β2+169α,b4=105β22(-36β2+169α),a1=a2=a3=a4=b1=b3=0,V=±13-1872μ2β+169,μ=±12613βα,a0=33β22(36β2+169α),b2=-105β236β2+169α,b4=105β22(36β2+169α),a1=a2=a3=a4=b1=b3=0,V=±131872μ2β+169,μ=±12613βα,a0=315β216(-36β2+169α),a2=b2=-105β28(-36β2+169α),a4=b4=105β232(-36β2+169α),a1=a3=b1=b3=0,V=±13-7488μ2β+169,μ=±15213βα,a0=-261β216(36β2+169α),a2=b2=-105β28(36β2+169α),a4=b4=105β232(36β2+169α),a1=a3=b1=b3=0,V=±137488μ2β+169,μ=±15213βα,
The travelling wave solutions are as follows:
(4.6)u1(x,t)=105β22(-36β2+169α)-105β2-36β2+169αtanh2μ(x∓13-1872μ2β+169t),u2(x,t)=33β22(36β2+169α)-105β236β2+169αtanh2μ(x∓131872μ2β+169t)+105β22(36β2+169α)tanh4μ(x∓131872μ2β+169t),u3(x,t)=105β22(-36β2+169α)-105β2-36β2+169αcoth2μ(x∓13-1872μ2β+169t)+105β22(-36β2+169α)coth4μ(x∓13-1872μ2β+169t),u4(x,t)=33β22(36β2+169α)-105β236β2+169αcoth2μ(x∓131872μ2β+169t)+105β22(36β2+169α)coth4μ(x∓131872μ2β+169t),u5(x,t)=315β216(-36β2+169α)-105β28(-36β2+169α)tanh2μ(x∓13-7488μ2β+169t)-105β28(-36β2+169α)coth2μ(x∓13-7488μ2β+169t)+105β232(-36β2+169α)tanh4μ(x∓13-7488μ2β+169t)+105β232(-36β2+169α)coth4μ(x∓13-7488μ2β+169t),u6(x,t)=-261β216(36β2+169α)-105β28(36β2+169α)tanh2μ(x∓137488μ2β+169t)-105β28(36β2+169α)coth2μ(x∓137488μ2β+169t)+105β232(36β2+169α)tanh4μ(x∓137488μ2β+169t)+105β232(36β2+169α)coth4μ(x∓137488μ2β+169t).

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