Non-smooth waves and anti-solitons in the general Degasperis-Procesi model

Georgy Omelyanov

Abstract


We consider  the general Degasperis-Procesi model of shallow water out-flows, which generalizes the list of famous equations:   KdV,  Benjamin-Bona-Mahony, Camassa-Holm,  and Degasperis-Procesi.  Our main object is  the construction of non-smooth  self-similar solutions of this equation. Along with the standard waves (peakons and cuspons) we present a new type of solutions (call them "twins") which is a combination of solitons and cuspons. We demonstrate also the wave-kind dependence on the amplitude for  the waves (solitons, peakons, cuspons, and twins) with positive and negative amplitudes.

Keywords


general Degasperis-Procesi model; soliton; peakon; cuspon; twins

Full Text:

PDF

References


Degasperis A, Procesi M. Asymptotic integrqability. In: Degasperis A, Gaeta G, editors. Symmetry and Perturbation Theory, Singapore: World Sientific; 1999. p. 23-37.

Benjamin T, Bona J, Mahony J. Model equations for long waves in nonlinear dispersive systems. PHILOS T ROY SOC A 1972;272:47-78.

Camassa R, Holm D. An integrable shallow water equation with peaked solitons. PHYS REV LETT 1993;71:1661-64.

Bona J, Pritchard W, Scott L. Solitary-wave interaction. PHYS FLUIDS 1980;23(3):438-41.

Constantin A, Lannes D. The hydrodynamical relevans of the Camassa-Holm and Degasperis-Procesi equations. ARCH RATION MECH AN 2009;192:165-86.

Constantin A, Strauss WA. Stability of peakons. Communications on Pure and Applied Mathematics 2000;53(5):603-10. DOI: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L

Constantin A. On the scattering problem for the Camassa-Holm equation. PHILOS T ROY SOC A 2001;457:953-70.

Lenells J. Traveling wave solutions of the Camassa-Holm

equation. J. Differential Equations 2005;217:393–430.

Popivanov P, Slavova A. Peakons, cuspons, compactons, solitons, kinks and periodic solutions of several third order nonlinear PDE and their Cellular Neural Network realization. FUNCTIONAL DIFFERENTIAL EQUATIONS 2009;16(4):609-26.

Degasperis A, Holm DD, Hone ANW. A new integrable equation with peakon solutions. THEOR MATH PHYS+ 2002;133(2):1463-74.

Lundmark H, Szmigielski J. Multi-peakon solutions of the Degasperi-Procesi equation. RES MEAS AP 2003;19:1241-45.

Matsuno Y. Multisoliton solutions of the Degasperi-Procesi equation and their peakon limit. RES MEAS AP 2005;21:1553-70.

Esher J, Liu Y, Yin Z. Global weak solutions and blow-up structure for the Degasperis-Procesi equation. J FUNCT ANAL 2006;241(2):457-85.

Zhijun Qiao. M-shape peakons, dehisced solitons, cuspons and new 1-peak solitons for the Degasperis-Procesi equation. CHAOS SOLITON FRACT 2008;37(2):501-7.

Zhou J, Tian L. Solitons, peakons, and periodic cuspons of a

generalized Degasperis-Procesi equation. Mathematical Problems in Engineering 2009;2009:1-13, Article ID 249361, doi:10.1155/2009/249361

Zhang G, Qiao Zh. Cuspons and smooth solitons of the Degasperis–Procesi equation under inhomogeneous boundary condition.

Math Phys Anal Geom 2007;10(3):205-25. https://doi.org/10.1007/s11040-007-9027-2

Omel'yanov G. Soliton dynamics for the general Degasperis-Procesi equation. 2017;1-13, http://arxiv.org/abs/1712.04410.

Noyola Rodriguez J, Omel'yanov G. General Degasperis-Procesi equation and its solitary wave solutions. CHAOS SOLITON FRACT 2019;118:41-46.

Gel'fand IM, Shilov GE. Generalized functions. New York: Academic Press; 1964.

Maslov VP, Omel'yanov GA. Asymptotic soliton-form solutions of equations with small dispersion. Russian Math. Serveys 1981;36(3):73-149.

Maslov VP, Omel'yanov GA. Geometric Asymptotics for Nonlinear PDE.

Translations of Mathematical Monographs, v.202. Providence, Rhode Island: AMS; 2001.

Danilov VG, Omel'yanov GA. Weak asymptotics method and the interaction of infinitely narrow delta-solitons.

Nonlinear Analysis: Theory, Methods and Applications 2003;54:773-99.

Danilov VG, Omel'yanov GA, Shelkovich VM. Weak Asymptotics Method and Interaction of Nonlinear Waves. In: Karasev M editor. Asymptotic Methods for Wave and Quantum Problems. AMS Translations, Series 2 "Advances in Mathematical Sciences", 208. Providence, RI: AMS; 2003. p. 33-163.

Omel'yanov G. Multi-soliton collision for essentially nonintegrable equations.

In: Oberguggenberger M, Toft J, Vindas J, et al. editors. Generalized Functions and Fourier Analysis, Series: Operator Theory: Advances and Applications, Vol. 260, Subseries: Advances in Partial Differential Equations. Birkhauser; 2017. p.153-170.

Omel'yanov G. A Perturbation Theory for Nonintegrable Equations with Small Dispersion. In:Lopez Ruiz R. editor. Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals. Intech Open; 2018. p. 14-20.


Refbacks

  • There are currently no refbacks.


Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.