Abstract
Based on the sine-Gordon and the sinh-Gordon equations, transformations are introduced in a variable separated ODE method to construct entirely new doubly periodic (Jacobian elliptic function) solutions of some nonlinear differential equations, which are powerful and simple. Distinct sets of exact doubly periodic solutions, that possess distinct physical structures, are obtained for the MKdV-sine-Gordon, the MKdV-sinh-Gordon equations and Born-Infeld equation. When the module m → 1 or 0, the corresponding solitary wave solutions are also found. This approach can be also applied to solve other nonlinear differential equations.
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