Abstract
This paper deals with the existence of traveling wave fronts for a generalized KdV–mKdV equation. We first establish the existence of traveling wave solutions for the equation without delay, and then we prove the existence of traveling wave fronts for the equation with a special local delay convolution kernel and a special nonlocal delay convolution kernel by using geometric singular perturbation theory, Fredholm theory and the linear chain trick.
Highlights
In the past few decades remarkable progresses have been made in understanding the Korteweg–de Vries (KdV) equation, it can be considered as a paradigm in nonlinear science and has many applications in weakly nonlinear and weakly dispersive physical systems
Ut + αU Ux + Uxxx = 0, which was first suggested by Korteweg and de Vries in 1895 [11], they used it as a nonlinear model to study the change of form of long waves advancing in a rectangular channel and given the solitary wave solution
The standard form of the Burgers–KdV equation was first proposed by Johnson in [10] who derived the equation as the governing equation for waves propagating in a liquid-filled elastic tube
Summary
In the past few decades remarkable progresses have been made in understanding the KdV equation, it can be considered as a paradigm in nonlinear science and has many applications in weakly nonlinear and weakly dispersive physical systems. In [13] the G /G-expansion method is introduced to construct more general exact traveling wave solutions to the generalized KdV–mKdV equation with any-order nonlinear terms. The soliton perturbation theory is used in [1] to study the solitons that are governed by the generalized Korteweg–de Vries equation in the presence of perturbation terms. Geometric singular perturbation theory [8] has received a great deal of interest and has been used by many researchers to obtain the existence of traveling waves for different equations, such as [2, 14, 16, 18, 20]. The existence of traveling wave solutions for it are obtained by using geometric singular perturbation theory, Fredholm theory and the linear chain trick
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