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Application of simplest equations of Bernoulli and Riccati kind for obtaining exact traveling-wave solutions for a class of pdes with polynomial nonlinearity. (English) Zbl 1222.35062

Summary: We search for traveling-wave solutions of equations of the form

$\sum _{p=1}^{{N}_{1}}{\alpha }_{p}\frac{{\partial }^{p}Q}{\partial {t}^{p}}+\sum _{q=1}^{{N}_{2}}{\beta }_{q}\frac{{\partial }^{q}Q}{\partial {x}^{q}}+\sum _{m=1}^{M}{\mu }_{m}{Q}^{m}=0$

where ${\alpha }_{p}$, ${\beta }_{q}$ and ${\mu }_{m}$ are parameters. We obtain such solutions by the method of the simplest equation for the cases when the simplest equation is the equation of Bernoulli or the equation of Riccati. We modify the methodology of the simplest equation of Kudryashov as follows. Kudryashov uses the first step of the test for the Painlevé property in order to determine the size of the solution of the studied PDE. We divide the PDEs under consideration into two parts: part A, which contains the derivatives, and part B, which contains the rest of the equation. The application of the ansatz or the extended ansatz of Kudryashov transforms part A and part B into two polynomials. We balance the highest powers of the polynomials for the parts A and B and thus obtain a balance equation which depends on the kind of the simplest equation used. The balance equations are connected to nonlinear algebraic systems of relationships among the parameters of the equations and the parameters of the solution. On the basis of these systems, we obtain numerous solutions of the class of equations considered.

##### MSC:
 35G05 General theory of linear higher-order PDE 35C07 Traveling wave solutions of PDE