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The new extended Jacobian elliptic function expansion algorithm and its applications in nonlinear mathematical physics equations. (English) Zbl 1196.35193

Summary: More recently we have presented the extended Jacobian elliptic function expansion method and its algorithm to seek more types of doubly periodic solutions. Based on the idea of the method, by studying more relations among all twelve kinds of Jacobian elliptic functions. we further extend the method to be a more general method, which is still called the extended Jacobian elliptic function expansion method for convenience. The new method is more powerful to construct more new exact doubly periodic solutions of nonlinear equations. We choose the (2+1)-dimensional dispersive long-wave system to illustrate our algorithm. As a result, twenty-four families of new doubly periodic solutions are obtained. When the modulus \(m\rightarrow 1\) or \(0\), these doubly periodic solutions degenerate as soliton solutions and trigonometric function solutions. This algorithm can be also applied to other nonlinear equations.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
33E05 Elliptic functions and integrals
35B10 Periodic solutions to PDEs
35Q51 Soliton equations
68W30 Symbolic computation and algebraic computation
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