Abstract
We use the bifurcation method of dynamical systems to study the periodic wave solutions and their limits for the generalized KP‐BBM equation. A number of explicit periodic wave solutions are obtained. These solutions contain smooth periodic wave solutions and periodic blow‐up solutions. Their limits contain periodic wave solutions, kink wave solutions, unbounded wave solutions, blow‐up wave solutions, and solitary wave solutions.
Highlights
IntroductionThe Benjamin-Bona-Mahony BBM equation 1 , ut ux uux − uxxt 0, 1.1 has been proposed as a model for propagation of long waves where nonlinear dispersion is incorporated.The Kadomtsov-Petviashvili KP equation 2 is given by ut a ux uxxx x uyy 0, 1.2 which is a weekly two-dimensional generalization of the KdV equation in the sense that it accounts for slowly varying transverse perturbations of unidirectional KdV solitons moving along the x-direction.Journal of Applied MathematicsWazwaz 3 presented the Kadomtsov-Petviashvili-Benjamin-Bona-Mahony KPBBM equation ut ux a u2 x − buxxt x ruyy 0, 1.3 and the generalized KP-BBM equation ut ux a u3 x − buxxt x ruyy 0.Wazwaz 3, 4 obtained some solitons solution and periodic solutions of 1.3 by using the sine-cosine method and the extended tanh method
The aim of this paper is to study the traveling wave solutions and their phase portraits for 1.4 by using the bifurcation method and qualitative theory of dynamical systems 6–15
In Proposition 3.3, we prove that the periodic wave solutions, kink wave solutions, blow-up wave solutions, unbounded solutions, and solitary wave solutions can be obtained from the limits of the smooth periodic wave solutions or periodic blow-up solutions
Summary
The Benjamin-Bona-Mahony BBM equation 1 , ut ux uux − uxxt 0, 1.1 has been proposed as a model for propagation of long waves where nonlinear dispersion is incorporated.The Kadomtsov-Petviashvili KP equation 2 is given by ut a ux uxxx x uyy 0, 1.2 which is a weekly two-dimensional generalization of the KdV equation in the sense that it accounts for slowly varying transverse perturbations of unidirectional KdV solitons moving along the x-direction.Journal of Applied MathematicsWazwaz 3 presented the Kadomtsov-Petviashvili-Benjamin-Bona-Mahony KPBBM equation ut ux a u2 x − buxxt x ruyy 0, 1.3 and the generalized KP-BBM equation ut ux a u3 x − buxxt x ruyy 0.Wazwaz 3, 4 obtained some solitons solution and periodic solutions of 1.3 by using the sine-cosine method and the extended tanh method. The Benjamin-Bona-Mahony BBM equation 1 , ut ux uux − uxxt 0, 1.1 has been proposed as a model for propagation of long waves where nonlinear dispersion is incorporated. The Kadomtsov-Petviashvili KP equation 2 is given by ut a ux uxxx x uyy 0, 1.2 which is a weekly two-dimensional generalization of the KdV equation in the sense that it accounts for slowly varying transverse perturbations of unidirectional KdV solitons moving along the x-direction. Wazwaz 3 presented the Kadomtsov-Petviashvili-Benjamin-Bona-Mahony KPBBM equation ut ux a u2 x − buxxt x ruyy 0, 1.3 and the generalized KP-BBM equation ut ux a u3 x − buxxt x ruyy 0. Wazwaz 3, 4 obtained some solitons solution and periodic solutions of 1.3 by using the sine-cosine method and the extended tanh method. Abdou 5 used the extended mapping method with symbolic computation to obtain some periodic solutions of 1.3 , solitary wave solution, and triangular wave solution. Song et al 6 obtained exact solitary wave solutions of 1.3 by using bifurcation method of dynamical systems
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