New exact solutions of \((2+1)\)-dimensional Gardner equation via the new sine-Gordon equation expansion method. (English) Zbl 1070.35058
Summary: The \((2+1)\)-dimensional Gardner equation
\[
u_t+ u_{xxx}+ 6\beta uu_x- \tfrac32 \alpha^2 u^2 u_x+ 3\sigma^2 \partial_x^{-1} u_{yy}- 3\alpha \sigma u_x\partial_x^{-1} u_y=0
\]
is investigated using a sine-Gordon equation expansion method, which is presented via a generalized sine-Gordon reduction equation and a new transformation. As a consequence, it is shown that the method is more powerful to obtain many types of new doubly periodic solutions of the \((2+1)\)-dimensional Gardner equation. In particular, solitary wave solutions are also given as simple limits of doubly periodic solutions.
MSC:
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 35-04 | Software, source code, etc. for problems pertaining to partial differential equations |
| 35C05 | Solutions to PDEs in closed form |
| 37K40 | Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems |
Software:
MapleReferences:
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