PEAKON AND CUSPON SOLUTIONS OF A GENERALIZED CAMASSA-HOLM-NOVIKOV EQUATION

Lijun Zhang; Yue Wang; C M Khalique; Yuzhen Bai

doi:10.11948/2018.1938

2018 Volume 8 Issue 6

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Lijun Zhang, Yue Wang, C M Khalique, Yuzhen Bai. PEAKON AND CUSPON SOLUTIONS OF A GENERALIZED CAMASSA-HOLM-NOVIKOV EQUATION[J]. Journal of Applied Analysis & Computation, 2018, 8(6): 1938-1958. doi: 10.11948/2018.1938

Citation:

Lijun Zhang, Yue Wang, C M Khalique, Yuzhen Bai. PEAKON AND CUSPON SOLUTIONS OF A GENERALIZED CAMASSA-HOLM-NOVIKOV EQUATION[J]. Journal of Applied Analysis & Computation, 2018, 8(6): 1938-1958. doi: 10.11948/2018.1938

PEAKON AND CUSPON SOLUTIONS OF A GENERALIZED CAMASSA-HOLM-NOVIKOV EQUATION

1 College of Mathematics and Systems Science, Shandong University of Science and Technology Qingdao, Shandong, 266590, China;
2 Department of Mathematics, School of Science, Zhejiang Sci-Tech University, Hangzhou, Zhejiang, 310018, China;
3 Department of Mathematical Sciences, North-West University, Mafikeng Campus, Private Bag X 2046, Mmabatho, 2735, South Africa;
4 School of Mathematical Sciences, Qufu Normal University, Qufu, 273165, China

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Abstract

Abstract

The bounded traveling wave solutions of a generalized CamassaHolm-Novikov equation with p=2 and p=3 are derived via the dynamical system approach. The singular wave solutions including peakons and cuspons are obtained by the bifurcation analysis of the corresponding singular dynamical system and the orbits intersecting with or approaching the singular lines. The results show that the generalized Camassa-Holm-Novikov equation with p=2 and p=3 both admit smooth solitary wave, smooth periodic wave solutions, solitary peakons, periodic peakons, solitary cuspons and periodic cuspons as well. It is worth pointing out that the Novikov equation has no bounded traveling wave solutions with negative wave speed, but has a family of new periodic cuspons which are distinguished with the normal periodic cuspons for their discontinuous first-order derivatives at both maximum and minimum.
- Generalized Camassa-Holm-Novikov equation /
- dynamical system approach /
- bifurcation /
- singular wave solutions
MSC: 34C23;34C60;35B65;35Q35

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References

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