Exact Solutions to KdV6 Equation by Using a New Approach of the Projective Riccati Equation Method

Abstract

We study a new integrable KdV6 equation from the point of view of its exact solutions by using an improved computational method. A new approach to the projective Riccati equations method is implemented and used to construct traveling wave solutions for a new integrable system, which is equivalent to KdV6 equation. Periodic and soliton solutions are formally derived. Finally, some conclusions are given.

1. Introduction

The sixth-order nonlinear wave equation(𝜕³_𝑥+8𝑢_𝑥𝜕_𝑥+4𝑢_𝑥𝑥)(𝑢_𝑡+𝑢_𝑥𝑥𝑥+6𝑢²_𝑥)=0 (1.1) has been recently derived by Karasu-Kalkanl1 et al. [1] as a new integrable particular case of the general sixth-order wave equation𝑢_{𝑥𝑥𝑥𝑥𝑥𝑥}+𝛼𝑢_𝑥𝑢_{𝑥𝑥𝑥𝑥}+𝛽𝑢_𝑥𝑥𝑢_𝑥𝑥𝑥+𝛾𝑢²_𝑥𝑢_𝑥𝑥+𝛿𝑢_𝑡𝑡+𝜌𝑢_{𝑥𝑥𝑥𝑡}+𝜔𝑢_𝑥𝑢_𝑥𝑡+𝜎𝑢_𝑡𝑢_𝑥𝑥=0, (1.2) where, 𝛼, 𝛽, 𝛾, 𝛿, 𝜌, 𝜔, 𝜎 are arbitrary parameters, and 𝑢=𝑢(𝑥,𝑡), is a differentiable function. By means of the change of variable𝑣=𝑢_𝑥,𝑤=𝑢_𝑡+𝑢_𝑥𝑥𝑥+6𝑢²_𝑥, (1.3) equation (1.1) converts to the Korteweg-de Vries equation with a source satisfying a third-order ordinary differential equation (KdV6)𝑣_𝑡+𝑣_𝑥𝑥𝑥+12𝑣𝑣_𝑥−𝑤_𝑥𝑤=0,_𝑥𝑥𝑥+8𝑣𝑤_𝑥+4𝑤𝑣_𝑥=0, (1.4) which is regarded as a nonholonomic deformation of the KdV equation [2]. Setting1𝑣(𝑥,𝑡)=21𝑢(𝑥,−𝑡),𝑤(𝑥,𝑡)=2𝑤(𝑥,𝑡), (1.5) the system (1.4) reduces to [2, 3] 𝑢_𝑡−6𝑢𝑢_𝑥−𝑢_𝑥𝑥𝑥+𝑤_𝑥𝑤=0,_𝑥𝑥𝑥+4𝑢𝑤_𝑥+2𝑢_𝑥𝑤=0. (1.6) A first study on the integrability of (1.6) has been done by Kupershmidt [2]. However, only at the end of the last year, Yao and Zeng [4] have derived the integrability of (1.6). More exactly, they showed that (1.6) is equivalent to the Rosochatius deformations of the KdV equation with self-consistent sources (RD-KdVESCS). This is a remarkable fact because the soliton equations with self-consistent sources (SESCS) have important physical applications. For instance, the KdV equation with self-consistent sources (KdVESCS) describes the interaction of long and short capillary-gravity waves [5]. On the other hand, when 𝑤=0 the system (1.6) reduces to potential KdV equation, so that solutions of the potential KdV equation are solutions to (1.1). Furthermore, solving (1.6) we can obtain new solutions to (1.1). In the soliton theory, several computational methods have been implemented to handle nonlinear evolution equations. Among them are the tanh method [6], generalized tanh method [7, 8], the extended tanh method [9–11], the improved tanh-coth method [12, 13], the Exp-function method [14–16], the projective Riccati equations method [17], the generalized projective Riccati equations method [18–23], the extended hyperbolic function method [24], variational iteration method [25–27], He's polynomials [28], homotopy perturbation method [29–31], and many other methods [32–35], which have been used in a satisfactory way to obtain exact solutions to NLPDEs. Exact solutions to system (1.6) and (1.1) have been obtained using several methods [3, 4, 36–38]. In this paper, we obtain exact solutions to system (1.6). However, our idea is based on a new version of the projective Riccati method which can be considered as a generalized method, from which all other methods can be derived. This paper is organized as follows. In Section 2 we briefly review the new improved projective Riccati equations method. In Section 3 we give the mathematical framework to search exact for solutions to the system (1.6). In Section 4, we mention a new sixth-order KdV system from which novel solutions to (1.6) can be derived. Finally, some conclusions are given.

2. The Method

In the search of the traveling wave solutions to nonlinear partial differential equation of the form𝑃(𝑢,𝑢_𝑥,𝑢_𝑡,𝑢_𝑥𝑥,𝑢_𝑥𝑡,𝑢_𝑡𝑡,…)=0, (2.1) the first step consists in use the wave transformation𝑢(𝑥,𝑡)=𝑣(𝜉),𝜉=𝑥+𝜆𝑡, (2.2) where 𝜆 is a constant. With (2.2), equation (2.1) converts to an ordinary differential equation (ODE) for the function 𝑣(𝜉)𝑃(𝑣,𝑣^′,𝑣^′′,…)=0. (2.3) To find solutions to (2.3), we suppose that 𝑣(𝜉) can be expressed as𝑣(𝜉)=𝐻(𝑓(𝜉),𝑔(𝜉)), (2.4) where 𝐻(𝑓(𝜉),𝑔(𝜉)) is a rational function in the new variables 𝑓(𝜉), 𝑔(𝜉) which are solutions to the system 𝑓^′𝑔(𝜉)=𝜌𝑓(𝜉)𝑔(𝜉),₂(𝜉)=𝑅(𝑓(𝜉)), (2.5) being 𝜌≠0 an arbitrary constant to be determinate and 𝑅(𝑓(𝜉)) a rational function in the variable 𝑓(𝜉). Taking𝑓(𝜉)=𝜙^𝑁(𝜉), (2.6) where 𝜙(𝜉)≠0, and 𝑁≠0, then (2.5) reduces to 𝜙^′𝜌(𝜉)=𝑁𝑔𝜙(𝜉)𝑔(𝜉),₂(𝜉)=𝑅(𝜙^𝑁(𝜉)). (2.7) From (2.7) we obtain(𝜙^′(𝜉))²=𝜌²𝑁²𝜙²(𝜉)𝑅(𝜙^𝑁). (2.8)

Let 𝑁=−1 and 𝑅(𝑓(𝜉))=𝛼+𝛽𝑓(𝜉)+𝛾𝑓(𝜉)², with 𝛼≠0. In this case, (2.8) reduces to(𝜙^′(𝜉))²=𝜌²𝜙(𝜉)²𝑅(𝜙(𝜉)⁻¹)=𝜌²(𝛼𝜙²(𝜉)+𝛽𝜙(𝜉)+𝛾), (2.9) and (2.5) are transformed into 𝑓^′𝑔(𝜉)=𝜌𝑓(𝜉)𝑔(𝜉),₂(𝜉)=𝛼+𝛽𝑓(𝜉)+𝛾𝑓(𝜉)². (2.10) The following are solutions to (2.9): 𝜙₁1(𝜉)=√4𝛼(−2𝛽+(1−Δ)sinh(𝜌𝛼𝜉)+(1+Δ)cosh(𝜌√𝜙𝛼𝜉)),₂1(𝜉)=√4𝛼(−2𝛽−(1−Δ)sinh(𝜌𝛼𝜉)+(1+Δ)cosh(𝜌√𝛼𝜉)). (2.11) Therefore, solutions to (2.10) are given by 𝑓(𝜉)=−4𝛼2𝛽±(1−Δ)sinh(𝜌√𝛼𝜉)−(1+Δ)cosh(𝜌√,√𝛼𝜉)𝑔(𝜉)=𝛼((1+Δ)sinh(𝜌√𝛼𝜉)∓(1−Δ)cosh(𝜌√𝛼𝜉))2𝛽±(1−Δ)sinh(𝜌√𝛼𝜉)−(1+Δ)cosh(𝜌√.𝛼𝜉) (2.12) In all cases Δ=𝛽²−4𝛼𝛾.

3. Exact Solutions to the Integrable KdV6 System

Using the traveling wave transformation𝑢(𝑥,𝑡)=𝑣(𝜉),𝑤(𝑥,𝑡)=𝑤(𝜉),𝜉=𝑥+𝜆𝑡+𝜉₀, (3.1) the system (1.6) reduces to (𝜆𝑣(𝜉)−3𝑣²(𝜉)−𝑣^′′(𝜉)+𝑤(𝜉))^′𝑤=0,_′′′(𝜉)+4𝑣(𝜉)𝑤^′(𝜉)+2𝑣^′(𝜉)𝑤(𝜉)=0. (3.2)(3.3) Integrating (3.2) with respect to 𝜉 and setting the constant of integration to zero we obtain𝜆𝑣(𝜉)−3𝑣²(𝜉)−𝑣^′′𝑤(𝜉)+𝑤(𝜉)=0,_′′′(𝜉)+4𝑣(𝜉)𝑤^′(𝜉)+2𝑣^′(𝜉)𝑤(𝜉)=0. (3.4) Using the idea of the projective Riccati equations method [19–22], we seek solutions to (3.4) as follows:𝑣(𝜉)=𝐻₁(𝑓(𝜉),𝑔(𝜉))=^𝑀₀𝑎_𝑖𝑓^𝑖(𝜉)+^2𝑀_𝑀+1𝑎_𝑖𝑔(𝜉)𝑓^{𝑖−(𝑀+1)}(𝜉),𝑤(𝜉)=𝐻₂(𝑓(𝜉),𝑔(𝜉))=^𝑁₀𝑏_𝑖𝑓^𝑖(𝜉)+^2𝑁_𝑁+1𝑏_𝑖𝑔(𝜉)𝑓^{𝑖−(𝑁+1)}(𝜉), (3.5) where 𝑓(𝜉) and 𝑔(𝜉) satisfy the system given by (2.10) (with 𝜌=1). Substituting (3.5) into (3.4), after balancing we have that𝑀=2, (3.6) and 𝑁is an arbitrary positive constant. By simplicity we take 𝑁=𝑀. Therefore, (3.5) reduce to 𝑣(𝜉)=𝐻₁(𝑓(𝜉),𝑔(𝜉))=²₀𝑎_𝑖𝑓^𝑖(𝜉)+⁴₃𝑎_𝑖𝑔(𝜉)𝑓^𝑖−(3)(𝜉),𝑤(𝜉)=𝐻₂(𝑓(𝜉),𝑔(𝜉))=²₀𝑏_𝑖𝑓^𝑖(𝜉)+⁴₃𝑏_𝑖𝑔(𝜉)𝑓^𝑖−(3)(𝜉). (3.7) Substituting this last two equations into (3.4), using (2.10) we obtain an algebraic system in the unknowns 𝑎₀, 𝑎₁, 𝑎₂, 𝑎₃, 𝑎₄, 𝑏₀, 𝑏₁, 𝑏₂, 𝑏₃, 𝑏₄, 𝜆, 𝛼, 𝛽, and 𝛾. Solving it and using (3.7), (2.12), and (3.1) we have the following set of new nontrivial solutions to KdV6 system (1.6). In all cases, 𝑎₁=𝑎₃=𝑏₁=𝑏₃=𝛽=0𝜆=2𝑏₄∓𝛼²𝑎²₄+9𝑏²₄𝑎₄,𝑏₀=16(−𝛼𝑏₄𝑎₄−6𝑏²₄𝑎²₄±2𝑏₄𝛼²𝑎²₄+9𝑏²₄𝑎²₄𝑏),₂=−𝑎₄𝑏₄,𝑎₀=−𝛼𝑎₄+3𝑏₄∓𝛼²𝑎²₄+9𝑏²₄6𝑎₄,𝑎₂=−𝑎²₄,𝛾=𝑎²₄. (3.8) A combined formal soliton solution is:𝑢₁(𝑥,𝑡)=−𝛼𝑎₄+3𝑏₄∓𝛼²𝑎²₄+9𝑏²₄6𝑎₄+(−𝑎²₄)(−4𝛼±(1+4𝛼𝑎²₄)sinh[√𝛼𝜉]−(1−4𝛼𝑎²₄)cosh[√)𝛼𝜉]²+𝑎₄(−4𝛼±(1+4𝛼𝑎²₄)sinh[√𝛼𝜉]−(1−4𝛼𝑎²₄)cosh[√)√𝛼𝜉]×(𝛼((1−4𝛼𝑎²₄)sinh[√𝛼𝜉]∓(1+4𝛼𝑎²₄)cosh[√𝛼𝜉])±(1+4𝛼𝑎²₄)sinh[√𝛼𝜉]−(1−4𝛼𝑎²₄)cosh[√𝑤𝛼𝜉]),₁1(𝑥,𝑡)=6(−𝛼𝑏₄𝑎₄−6𝑏²₄𝑎²₄±2𝑏₄𝛼²𝑎²₄+9𝑏²₄𝑎²₄)+(−𝑎₄𝑏₄)(−4𝛼±(1+4𝛼𝑎²₄)sinh[√𝛼𝜉]−(1−4𝛼𝑎²₄)cosh[√)𝛼𝜉]²√×(𝛼((1−4𝛼𝑎²₄)sinh[√𝛼𝜉]∓(1+4𝛼𝑎²₄)cosh[√𝛼𝜉])±(1+4𝛼𝑎²₄)sinh[√𝛼𝜉]−(1−4𝛼𝑎²₄)cosh[√)𝛼𝜉]^′ (3.9) where 𝑎₄,𝑏₄,𝛼 are arbitrary constants, and 𝜉=𝑥+𝜆𝑡+𝜉₀.

Furthermore,𝜆=8𝛼²+20𝛼𝑎₀+15𝑎²₀2𝛼+3𝑎₀,𝑏₀=−2(4𝛼²𝑎₀+7𝛼𝑎²₀+3𝑎³₀)2𝛼+3𝑎₀,𝑏₄=0,𝑎₄=0,𝑏₂=−2(4𝛼𝛾𝑎₀+3𝛾𝑎²₀)2𝛼+3𝑎₀,𝑎₂=−2𝛾. (3.10) A soliton solution is given by𝑢₂(𝑥,𝑡)=𝑎₀+(−2𝛾)(−4𝛼±(1+4𝛼𝛾)sinh[√𝛼𝜉]−(1−4𝛼𝛾)cosh[√)𝛼𝜉]²,𝑤₂(𝑥,𝑡)=−2(4𝛼²𝑎₀+7𝛼𝑎²₀+3𝑎³₀)2𝛼+3𝑎₀+(−2(4𝛼𝛾𝑎₀+3𝛾𝑎²₀)2𝛼+3𝑎₀)×(−4𝛼±(1+4𝛼𝛾)sinh[√𝛼𝜉]−(1−4𝛼𝛾)cosh[√)𝛼𝜉]², (3.11) where 𝑎₀,𝛼,𝛾 are arbitrary constants and 𝜉=𝑥+𝜆𝑡+𝜉₀.

3.1. A New System

A direct calculation shows that (1.1) reduces to𝑢_{𝑥𝑥𝑥𝑥𝑥𝑥}+20𝑢_𝑥𝑢_{𝑥𝑥𝑥𝑥}+40𝑢_𝑥𝑥𝑢_𝑥𝑥𝑥+120𝑢²_𝑥𝑢_𝑥𝑥+𝑢_{𝑥𝑥𝑥𝑡}+4𝑢_𝑥𝑥𝑢_𝑡+8𝑢_𝑥𝑢_𝑥𝑡=0. (3.12) On the other hand, it is easy to see that (3.12) can be written as(𝜕²_𝑥+4𝑢_𝑥𝑥𝜕_𝑥⁻¹+8𝑢_𝑥)(𝑢_𝑥𝑡+𝑢_{𝑥𝑥𝑥𝑥}+12𝑢_𝑥𝑢_𝑥𝑥)=0. (3.13) Using the analogy between KdV equation and MKdV equation and motivated by the structure of (3.13), the authors in [38] have introduced the so-called MKdV6 equation(𝜕³_𝑥+8𝑣²_𝑥𝜕_𝑥+8𝑣_𝑥𝑥𝜕_𝑥⁻¹𝑣_𝑥𝜕_𝑥)(𝑣_𝑡+𝑣_𝑥𝑥𝑥+4𝑣³_𝑥)=0, (3.14) and they showed that(𝜕³_𝑥+8𝑢_𝑥𝜕_𝑥+4𝑢_𝑥𝑥)(𝑢_𝑡+𝑢_𝑥𝑥𝑥+6𝑢²_𝑥)=(2𝑣_𝑥+√22𝑖𝜕_𝑥),(𝜕³_𝑥+8𝑣²_𝑥𝜕_𝑥+8𝑣_𝑥𝑥𝜕_𝑥⁻¹𝑣_𝑥𝜕_𝑥)(𝑣_𝑡+𝑣_𝑥𝑥𝑥+4𝑣³_𝑥)=0, (3.15) where 𝑣²_𝑥+√2/2𝑖𝑣_𝑥𝑥 is the Miura transformation between KdV6 equation (1.1) and MKdV6 equation (3.14). Therefore, solving (3.14), according to (3.15), solutions to (1.1) are obtained. Setting 𝑤_𝑥=𝑣²_𝑥, then the new MKdV6 equation is equivalent to new system𝑣_{𝑥𝑥𝑥𝑥𝑥𝑥}+20𝑣²_𝑥𝑣_{𝑥𝑥𝑥𝑥}+80𝑣_𝑥𝑣_𝑥𝑥𝑣_𝑥𝑥𝑥+20𝑣³_𝑥𝑥+120𝑣⁴_𝑥𝑣_𝑥𝑥+𝑣_{𝑥𝑥𝑥𝑡}+8𝑣²_𝑥𝑣_𝑥𝑡+4𝑣_𝑥𝑥𝑤_𝑡𝑤=0,_𝑥𝑥−2𝑣_𝑥𝑣_𝑥𝑥=0. (3.16) In equivalent form, with 𝑠=𝑣_𝑥, 𝑤=𝑣_𝑡+𝑣_𝑥𝑥𝑥+4𝑣³_𝑥, from (3.14) the following system is derived:𝑠_𝑡+𝑠_𝑥𝑥𝑥+12𝑠²𝑠_𝑥−𝑤_𝑥𝑤=0,_𝑥𝑥𝑥+8𝑠²𝑤_𝑥+8𝑠_𝑥𝑧𝑧=0,_𝑥−𝑠𝑤_𝑥=0. (3.17) We believe that traveling wave solutions to these systems can be obtained using the method used here. By reasons of space, we omit them.

4. Conclusions

In this paper we have derived two new soliton solutions to KdV6 system (1.2) by using a new approach of the improved projective Riccati equations method. The results show that the method is reliable and can be used to handle other NLPDE's. Other methods such as tanh, tanh-coth, and exp-function methods can be derived from the new version of the projective Riccati equation method. Moreover, new methods can be obtained using the exposed ideas in the present paper. Other methods related to the problem of solving nonlinear PDEs exactly may be found in [39, 40].