Journal of Applied Mathematics and Computation

DOI：http://dx.doi.org/10.26855/jamc.2022.09.011

Date: October 20,2022 Hits: 463

In this paper, we concern the singular traveling waves for a generalized Camassa-Holm equation. Camassa and Holm derived an equation to describe unidirectional propagation of shallow water waves over a flat bottom, which is called Camassa-Holm equation, the Camassa-Holm equation is completely integrable and possesses an infinite number of conservation laws, it has been well investigated in the view of mathematical point and many results were obtained. Furthermore, some authors have studied a generalized Camassa-Holm equation. Special solutions play an important role in the research of partial differential equations, and it can be used to describe and explain many phenomena in physics and engineering and so on. A particular kind of product of distributions is introduced and applied to solve non-smooth solutions of this equation. It is proved that, under certain conditions, the distribution solutions such as singular Dirac delta function and Heaviside function exist for the Camassa-Holm equation.

[1] R. Camassa and D. D. Holm. (1993). An integrable shallow water equation with peaked solitons. Physical Review Letters, 11, 1661-1664, (1993).

[2] A. S. Fokas and B. Fuchssteiner. (1981). Symplectic structures, their Backlund transformations and hereditary symmetries. Physica D: Nonlinear Phenomena, 1, 47-66, (1981).

[3] A. Constantin and J. Escher. (1998). Wave breaking for nonlinear nonlocal shallow water equations. Acta Mathematica, 2, 229-243, 1998.

[4] C. E. Kenig, G. Ponce, and L. Vega. (1993). Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle. Communications on Pure and Applied Mathematics, 4, 527-620, (1993).

[5] G. B. Whitham. (1980). Linear and Nonlinear Waves, John Wiley & Sons, New York, NY, USA, 1980.

[6] M. S. Osman, M. Inc, J. G. Liu, et al. (2020). Different wave structures and stability analysis for the generalized (2+1)-dimensional Camassa-Holm-Kadomtsev-Petviashvili equation [J]. Physica Scripta, 95, 035229, (2020).

[7] P. Silva, I. L. Freire. (2022). Existence, persistence, and continuation of solutions for a generalized 0-Holm-Staley equation [J]. Journal of Differential Equations, 320, 371-398, (2022).

[8] X. Lu, A. Chen, T. Deng. (2019). Orbital Stability of Peakons for a Generalized Camassa-Holm Equation [J]. Journal of Applied Mathematics and Physics, 07(10), 2200-2211, (2019).

[9] S. Albeverio, Z. Brzeniak, A. Daletskii. (2021). Stochastic Camassa-Holm equation with convection type noise [J]. Journal of Differential Equations, 276(2), 404-432, (2021).

[10] S. Zheng, Z. Y. Ouyang, K. L. Wu. (2019). Singular traveling wave solutions for Boussinesq equation with power law nonlinearity and dual dispersion. Advances in Difference Equations, 1, 501, (2019).

[11] L. X. Tian and J. L. Yin. (2004). New compacton solutions and solitary wave solutions of fully nonlinear generalized Camassa-Holm equations. Chaos, Solitons and Fractals, 2, 289-299, (2004).

[12] Z. Liu and T. Qian. (2001). Peakons and their bifurcation in a generalized Camassa-Holm equation. International Journal of Bifurcation and Chaos, 3, article 781-792, (2001).

[13] T. F. Qian and M. Y. Tang. (2001). Peakons and periodic cusp waves in a generalized Camassa-Holm equation. Chaos, Solitons and Fractals, 7, 1347-1360, (2001).

[14] Z. Y. Liu and T. F. Qian. (2002). Peakons of the Camassa-Holm equation. Applied Mathematical Modelling, 3, 473-480, (2002).

[15] L. Tian and X. Song. (2004). New peaked solitary wave solutions of the generalized Camassa-Holm equation. Chaos, Solitons and Fractals, 3, 621-637, (2004).

[16] S. A. Khuri. (2005). New ansaz for obtaining wave solutions of the generalized Camassa-Holm equation. Chaos, Solitons and Fractals, 3, 705-710, (2005).

[17] Z. Y. Yin. (2007). On the Cauchy problem for the generalized Camassa-Holm equation. Nonlinear Analysis: Theory, Methods & Applications, 2, 460-471, (2007).

[18] O. G. Mustafa. (2006). On the Cauchy problem for a generalized Camassa-Holm equation. Nonlinear Analysis. Theory, Methods & Applications, 6, 1382-1399, (2006).

[19] C. O. R. Sarrico. (2003). Distributional Products and Global Solutions for Nonconservative Inviscid Burgers Equation. J. Math. Anal. Appl. 281, 641-656, (2003).

[20] C. O. R. Sarrico. (2012). Products of Distributions and Singular Travelling Waves as Solutions of Advection-Reaction Equations. Russian Journal of Mathematical Physics, 19, 244-255, (2012).

[21] C. O. R. Sarrico. (1988). About a Family of Distributional Products Important in the Applications. Port. Math. 45, 295-316, (1988).

[22] C. O. R. Sarrico. (1995). Distributional Products with Invariance for the Action of Unimodular Groups. Riv. Math. Univ. Parma 4, 79-99, (1995).

[23] C. O. R. Sarrico. (2006). New Solutions for the One-Dimensional Nonconservative Inviscid Burgers Equation. J. Math. Anal. Appl., 317, 496-509, (2006).

[24] C. O. R. Sarrico. (2010). Collision of Delta-Waves in a Turbulent Model Studied via a Distribution Product. Nonlinear Anal., 73, 2868-2875, (2010).

**Singular Travelling Wave Solutions for a Generalized Camassa Holm Equation**

**How to cite this paper:** Zhengyong Ouyang. (2022) Singular Travelling Wave Solutions for a Generalized Camassa Holm Equation. *Journal of Applied Mathematics and Computation*, **6**(**3**), 380-389.

**DOI: http://dx.doi.org/10.26855/jamc.2022.09.011**

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