Bifurcation of peakons and periodic cusp waves for the generalization of the Camassa–Holm equation

2015

Nonlinear Dyn

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In this paper, we study the bifurcations and nonlinear wave solutions for a generalized twocomponent integrable Dullin-Gottwald-Holm system. Through the bifurcations of phase portraits, we not only show the existence of several types of nonlinear wave solutions, including solitary waves, peakons, periodic cusp waves, periodic waves, compacton-like waves and kink-like waves, but also obtain their implicit expressions. Additionally, the numerical simulations of the nonlinear wave solutions are given to show the correctness of our results.

Section: Introductionmentioning

confidence: 99%

Bifurcations and nonlinear wave solutions for the generalized two-component integrable Dullin–Gottwald–Holm system

2015

Nonlinear Dyn

Self Cite

“…In our previous work , we studied the peakons and periodic cusp wave solutions of Eq. when n = 1, m ≥2 and n ≥2, m = n + 1, by exploiting the bifurcation method and qualitative theory of dynamical systems , and setting the integral constant to be zero. The results of are summarized as follows: When n = 1, m = 2, Eq.…”

Section: Introductionmentioning

confidence: 99%

“…when n = 1, m ≥2 and n ≥2, m = n + 1, by exploiting the bifurcation method and qualitative theory of dynamical systems , and setting the integral constant to be zero. The results of are summarized as follows: When n = 1, m = 2, Eq. has peakons

{\overset{u}{false}}_{1} (x, t, c) = c {normale}^{- | x - ct |},

and periodic cusp wave solutions

{\overset{u}{false}}_{2} (ξ, K, c) = {\overset{u}{false}}_{0} ((), ξ - 2 normali {\overset{T}{false}}_{0}, K, c),

where

i = 0, \pm 1, \pm 2, \dots, ξ = x - ct \in ([], (2 normali - 1) {\overset{T}{false}}_{0}, (2 normali + 1) {\overset{T}{false}}_{0})

, and

{\overset{u}{false}}_{0} (ξ, K, c) = \frac{3 (2 K + 1) (3 - 2 K)}{16 K^{2} \overset{α}{false}} {normale}^{\sqrt{\frac{2 K}{3}} | ξ |} + \frac{\overset{α}{false}}{4} {normale}^{- \sqrt{\frac{2 K}{3}} | ξ |} + \frac{3 - 2 K}{4 K<...>}

…”

Section: Introductionmentioning

confidence: 99%

Extension on peakons and periodic cusp waves for the generalization of the Camassa-Holm equation

2014

Math. Meth. Appl. Sci.

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In this paper, we employed the bifurcation method and qualitative theory of dynamical systems to study the peakons and periodic cusp waves of the generalization of the Camassa-Holm equation, which may be viewed as an extension of peaked waves of the same equation. Through the bifurcation phase portraits of traveling wave system, we obtained the explicit peakons and periodic cusp wave solutions. Further, we exploited the numerical simulation to confirm the qualitative analysis, and indeed, the simulation results are in accord with the qualitative analysis. Compared with the previous works, several new nonlinear wave solutions are obtained.

“…In this paper, we employ the bifurcation method and qualitative theory of dynamical systems [11][12][13][14][15][16][17][18][19][20][21] to investigate the nonlinear wave solutions for (1), and we obtain many exact explicit expressions of nonlinear wave solutions for (1). These nonlinear wave solutions contain solitary wave solutions, singular solutions, periodic singular solutions, and kink-shaped solutions, most of which, to our knowledge, are newly obtained.…”

Section: Introductionmentioning

confidence: 99%

New Exact Explicit Nonlinear Wave Solutions for the Broer-Kaup Equation

Journal of Applied Mathematics

2014

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We study the nonlinear wave solutions for the Broer-Kaup equation. Many exact explicit expressions of the nonlinear wave solutions for the equation are obtained by exploiting the bifurcation method and qualitative theory of dynamical systems. These solutions contain solitary wave solutions, singular solutions, periodic singular solutions, and kink-shaped solutions, most of which are new. Some previous results are extended.