New exact traveling and non-traveling wave solutions for (2 + 1)-dimensional Burgers equation
Introduction
It is well known that nonlinear evolution equations are widely used to describe complex phenomena in various fields of science, such as physics, chemistry, biology, and so on. The search for exact solutions of these equations attracted a huge size of research works. Many powerful methods have been presented such as the inverse scattering method [1], Hirota bilinear forms [2], Lie symmetry method [3], F-expansion method [4], extended homoclinic test approach [5], [6] and so on.
Now we consider (2 + 1)-dimensional Burgers equation [7], [8], [9], [10], [11], [12]The Burgers equation have been extensively investigated since it were firstly found by Bateman. In 1940, Burgers [7] got its some special solutions. Recently, some exact solutions for (2 + 1)-dimensional Burgers equation (1.1) have been obtained, for example, Hong et al. [8] obtained the most generalized Painlevé integrable (2 + 1)-dimensional integrable Burgers systems by means of the Painlevé analysis. Wazwaz [9] studied Multiple kink solutions and multiple singular kink solutions by using the Hirota’s bilinear method. Kong and Chen [10] got soliton-like solutions and special soliton-like structures by using the further extended tanh method. Wang et al. [11] found some new and general solutions by use of new Riccati equation rational expansion method. Wang et al. [12] obtained exact traveling and non-traveling wave solutions by use of auxiliary equation method.
In this paper, we mainly study the traveling and non-traveling wave solutions for (2 + 1)-dimensional Burgers equation by the generalized direct ansätz method and different test functions.
Section snippets
The introduction of the method
We consider the (2 + 1)-dimensional Burgers equationwhere .
By using Painlevé analysis we supposewhere f = f(x, y, t) is an unknown real function, u0 is a real constant.
Substituting Eq. (2.2) into Eq. (2.1), we get the bilinear equation
Now we suppose the solution of Eq. (2.3) aswhere a0, a1 and a2 are constants to be determined, P (y, t) is an unknown function
Traveling solutions for Eq. (2.1)
In this section, if P (y, t) = p (y, t) is not fractional function, substituting Eq. (2.4) into Eq. (2.3), we can obtainEq. (3.1) can be changed into
Non-traveling solutions for Eq. (2.1)
In this section, let are not fractional function), then Eqs. (2.4), (2.5) can be rewritten as Example 3 Taking u0 = 0, then the Eqs. (3.2), (3.3), (3.4) can be reduced toLet the test function be
Conclusion
In summary, some families of exact solitary-like wave solution, x- periodic soliton solution, y-periodic soliton solution, doubly periodic solution, rational solution, and new non-traveling wave solution for (2 + 1)-dimensional Burgers equation are obtained by using the generalized direct ansätz method and different test functions, these solutions enrich the structures of exact solutions. As a matter of fact, the method is applicable to many other nonlinear evolution equations in (2 + 1)-dimension.
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