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Integrability, bilinearization, Bäcklund transformations and solutions for a generalized variable-coefficient Gardner equation with an external-force term in a fluid or plasma
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Integrability, bilinearization, Bäcklund transformations and solutions for a generalized variable-coefficient Gardner equation with an external-force term in a fluid or plasma
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Journal Article
Integrability, bilinearization, Bäcklund transformations and solutions for a generalized variable-coefficient Gardner equation with an external-force term in a fluid or plasma
2024
Overview
A generalized variable-coefficient Gardner equation with an external-force term is investigated which can model the propagation and interaction of some nonlinear waves in a fluid or plasma. Via the Hirota bilinear method, we have obtained some bilinear forms using a dependent variable transformation under certain constraints. The bilinear Bäcklund transformations, Lax pair and Lax-type Bäcklund transformations have also been constructed via the bilinear forms. By virtue of the extended variable-coefficient homogeneous balance method and truncated Painlevé expansion, we have obtained the same Bäcklund transformations and three different kinds of the analytical solutions. Additionally, profiles of some types of the obtained solutions are illustrated graphically. Graphic analysis on those waves shows that: (i) For the soliton-like and rational waves, the characteristic lines and velocities are related to the dispersive, dissipative, perturbed and external-force coefficients; the backgrounds are related to the perturbed and external-force coefficients; (ii) For the periodic waves, the velocities are related to the dispersive, dissipative, perturbed and external-force coefficients; the amplitudes are related to the dispersive, perturbed and external-force coefficients, while the backgrounds are related to the perturbed and external-force coefficients.
Publisher
Springer Netherlands,Springer Nature B.V
