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Extended Jacobian elliptic function algorithm with symbolic computation to construct new doubly-periodic solutions of nonlinear differential equations. (English) Zbl 1196.33017

Summary: With the aid of computerized symbolic computation, the extended Jacobian elliptic function expansion method and its algorithm are presented by using some relations among ten Jacobian elliptic functions and are very powerful to construct more new exact doubly-periodic solutions of nonlinear differential equations in mathematical physics. The new \((2+1)\)-dimensional complex nonlinear evolution equations is chosen to illustrate our algorithm such that sixteen families of new doubly-periodic solutions are obtained. When the modulus \(m\rightarrow 1\) or 0, these doubly-periodic solutions degenerate as solitonic solutions including bright solitons, dark solitons, new solitons as well as trigonometric function solutions.

MSC:

33E05 Elliptic functions and integrals
33F05 Numerical approximation and evaluation of special functions
35B10 Periodic solutions to PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
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