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The General Projective Riccati Equations Method and Exact Solutions for a Class of Nonlinear Partial Differential Equations
並列摘要
By using a simple transformation technique, we have shown that the Hamiltonian amplitude equation, the nonlinear wave equation, the coupled Klein-Gordon-Zakharov (CKGZ) equation, the generalized Davey-Stewartson (GDS) equation, the DS equation, and the generalized Zakharov equation can be reduced to the same family of elliptic-like equations. Then, by using the general projective Riccati equation method, many kinds of exact solutions of this family of elliptic-like equations are obtained in a unified way. These solutions include new solitary wave, periodic, and rational solutions. To our knowledge, this is the first time that the general projective equation method has been used for solving a system of coupled equations (in terms of two unknowns u and v). Thus the most important achievement of this work lies in the fact that we have succeeded in making proper transformations in order to obtain an implicit relation between the two unknowns, such as u = f(Ø) and v = g(Ø); then, once this relation is in hand, the system is decoupled and the equation in Ø is solved by the general projective Riccati equation method.
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延伸閱讀
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