Abstract
The Hirota bilinear method is used to handle the variant Boussinesq equations. Multiple soliton solutions and multiple singular soliton solutions are formally established. It is shown that the Hirota bilinear method may provide us with a straightforward and effective mathematic tool for generating multiple soliton solutions of nonlinear wave equations in fluid mechanics.
Highlights
Many phenomena in physics, biology, chemistry, mechanics, etc. are described by nonlinear partial differential equations
During the past several decades, many powerful and efficient methods have been proposed to obtain the exact solutions of nonlinear wave equations, such as inverse scattering method [ ], Darboux and Bäcklund transformations [, ], the Hirota bilinear method [ ], the tanh method [ ], the extended tanh method [ ], the sine-cosine method [ ], the homogeneous balance method [ ], the homotopy perturbation method [, ], the F-expansion method [ ], the Exp-function expansion method [, ], the (G /G)-expansion method [, ], the Kudryashov method [ ], the mapping method [ ], the extended mapping method [ ], and so on
We focus on the variant Boussinesq equations, which was derived by Sachs [ ] in as a model for water waves: Ht + (Hu)x + uxxx =, ( )
Summary
Biology, chemistry, mechanics, etc. are described by nonlinear partial differential equations. During the past several decades, many powerful and efficient methods have been proposed to obtain the exact solutions of nonlinear wave equations, such as inverse scattering method [ ], Darboux and Bäcklund transformations [ , ], the Hirota bilinear method [ ], the tanh method [ ], the extended tanh method [ ], the sine-cosine method [ ], the homogeneous balance method [ ], the homotopy perturbation method [ , ], the F-expansion method [ ], the Exp-function expansion method [ , ], the (G /G)-expansion method [ , ], the Kudryashov method [ ], the mapping method [ ], the extended mapping method [ ], and so on. The Hirota bilinear method is rather heuristic and possesses significant features that make it be ideal for the determination of multiple soliton solutions for a wide class of the nonlinear wave equations [ – ]. When the Hirota bilinear method is used, computer symbolic systems such as Maple and Mathematica allow us to perform complicated and tedious calculations
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