Abstract
On the basis of the computer symbolic system Maple and the extended hyperbolic function method, we develop a more mathematically rigorous and systematic procedure for constructing exact solitary wave solutions and exact periodic traveling wave solutions in triangle form of various nonlinear partial differential equations that are with physical backgrounds. Compared with the existing methods, the proposed method gives new and more general solutions. More importantly, the method provides a straightforward and effective algorithm to obtain abundant explicit and exact particular solutions for large nonlinear mathematical physics equations. We apply the presented method to two variant Boussinesq equations and give a series of exact explicit traveling wave solutions that have some more general forms. So consequently, the efficiency and the generality of the proposed method are demonstrated.
Highlights
The nonlinear phenomena are very important in a variety of scientific fields, especially in fluid dynamics, solid-state physics, hydrodynamics, plasma physics, elastic dynamics, acoustics, chemical physics, and nonlinear optics
Nonlinear evolution partial differential equations are widely used as models to describe complex physical phenomena in various fields of sciences, especially in fluid mechanics, solid-state mechanics, atmospheric physics, chemical reaction-diffusion dynamics, ion acoustics, and nonlinear vibration
Many powerful methods have been proposed such as the inverse scattering transformation method 1, the Backlund and Darboux transformation method 2, 3, the Hirota bilinear method 4, the Lie group reduction method 5, the tanh method 6, the tanh-coth method 7, the sine-cosine method 8, 9, the homogeneous balance method 10–12, the Jacobi elliptic function method 13, 14, the extended tanh method 15, 16, the F-expansion method and its extension 17, 18, the Riccati method 19, 20, and extended improved tanh-function method 21, 22
Summary
The nonlinear phenomena are very important in a variety of scientific fields, especially in fluid dynamics, solid-state physics, hydrodynamics, plasma physics, elastic dynamics, acoustics, chemical physics, and nonlinear optics. We present an effective extension to the projective Riccati equation method 19, 20 and extended improved tanh-function method 21, 22 and develop an effective Maple software package “PDESolver” to uniformly construct a series of traveling wave solutions including solitary wave solutions, singular traveling, rational, triangular periodic solitons for general nonlinear evolution equations. Ht Hu x uxxx 0, 1.1 ut Hx uux 0, Ht ux Hu x − αuxxx 0, 1.2 ut Hx uux − 3αuxxt 0, where H x, t , u x, t are the unknown functions depending on the temporal variable t and the spatial variable x These two equations were introduced as models for water waves and called variant Boussinesq equations I and II, respectively 25.
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