Soliton solutions of Burgers equations and perturbed Burgers equation

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Abstract

This paper carries out the integration of Burgers equation by the aid of tanh method. This leads to the complex solutions for the Burgers equation, KdV–Burgers equation, coupled Burgers equation and the generalized time-delayed Burgers equation. Finally, the perturbed Burgers equation in (1+1) dimensions is integrated by the ansatz method.

Introduction

A number of nonlinear phenomena in many branches of sciences such as physical, chemical, economical and biological processes are described by the interplay of reaction and diffusion or by the interaction between convection and diffusion [1], [2], [3], [4], [5], [6], [7], [8], [9], [10]. One of the well known partial differential equations which govern a wide variety of them is the Burgers equation (BE) which provides the simplest nonlinear model of turbulence. The existence of relaxation time or delayed time is an important feature in reaction diffusion and convection diffusion systems. The approximate theory of flow through a shock wave travelling is applied to viscous fluid. Fletcher, using the Hopf–Cole transformation, gave an analytic solution for the system of two dimensional BE.

This paper is to extend the tanh method using complex travelling waves to solve four different types of nonlinear differential equations such as the Burgers, KdV–Burgers, coupled Burgers and the generalized time delayed BE.

Section snippets

Complex tanh method

Consider a general form of nonlinear partial differential equation (PDE)P(u,ux,ut,uxx,)=0.

Assume that the solution is given by u(x, t) = U(z) where z = i(x  λt). Hence, we use the following changest=-iλddz,x=iddz,2x2=-d2dz2,3x3=-id3dz3,and so on. Eq. (1) now becomes an ordinary differential equationQ(U,U,U,)=0.

If all terms of (2) contain derivatives in z then by integrating this equation and taking the constant of integration to be zero, we obtain a simplified ODE. For the tanh method, we

Applications

In this section, the applications of the analytical development, from the last section, will be touched upon. The BE in (1+1) dimensions will be solved. A numerical simulation will also be given. Subsequently, the KdV–BE will also be solved with the complex travelling wave hypothesis. Finally, the coupled BE and time-delayed BE will also be solved in the next two sub-sections. They are all supported by numerical simulations.

Perturbed Burgers equation

In this section the study is going to be focused on the perturbed BE. The solitary wave ansatz method will be adopted to obtain the exact 1-soliton solution of the BE in (1+1) dimensions. The search is going to be for a topological 1-soliton solution. The perturbed BE that is going to be studied in this paper is given by the following form [5]:qt+aqqx+bqxx=αq2qx+βqqxx+γ(qx)2+δqxxx.

Eq. (42) appears in the study of gas dynamics and also in free surface motion of waves in heated fluids. The

Conclusions

In this paper the tanh method has been successfully applied to solve the BE in (1+1) dimensions, KdV–BE, coupled BE and the generalized time-delayed BE. Finally, the perturbed BE in (1+1) dimensions is analyzed and solved by the ansatz method. In this case, an exact topological soliton solution has been obtained together with the constraints that must be valid for the coefficients of the equation along with the perturbation coefficients. In future, these results will be further analyzed via

Acknowledgments

The second author (Marko D. Petković) gratefully acknowledges the support from the research project 144011 of the Serbian Ministry of Science.

The research work of the third author (AB) was fully supported by NSF-CREST Grant No.: HRD-0630388 and this support is genuinely and sincerely appreciated.

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