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Abstract
AbstractIn this paper, we employ the complex method to obtain all meromorphic solutions of an auxiliary ordinary differential equation at first, and then find all meromorphic general solutions of in combination the Newell-Whitehead equation, the NLS equation, and the Fisher equation with degree three. Our result shows that all rational and simply periodic exact solutions of the combined the Newell-Whitehead equation, NLS equation, and Fisher equation with degree three are solitary wave solutions, and the method is simpler than other methods.MSC:30D35, 34A05.
Highlights
Introduction and main resultsNonlinear partial differential equations (NLPDEs) are widely used as models to describe many important dynamical systems in various fields of sciences, in fluid mechanics, solid state physics, plasma physics, and nonlinear optics
Many methods have been developed by mathematicians and physicists to find special solutions of NLPDEs, such as the inverse scattering method [ ], the Darboux transformation method [ ], the Hirota bilinear method [ ], the Lie group method [ ], the bifurcation method of dynamic systems [ – ], the sine-cosine method [ ], the tanh-function method [, ], the Fanexpansion method [ ], and the homogeneous balance method [ ]
It is shown that the complex method provides a powerful mathematical tool for solving a great many nonlinear partial differential equations in mathematical physics
Summary
Introduction and main resultsNonlinear partial differential equations (NLPDEs) are widely used as models to describe many important dynamical systems in various fields of sciences, in fluid mechanics, solid state physics, plasma physics, and nonlinear optics. Yuan et al [ ] derived all traveling wave exact solutions by using the complex method for a type of ordinary differential equations (ODEs): Aw + Bw + Cw + D = , ( ) We employ the complex method to obtain first all meromorphic solutions of Eq ( ) below, Aw + Bw + Cw + Dw = , ( )
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