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 Zh. Vychisl. Mat. Mat. Fiz., 2019, Volume 59, Number 6, paper published in the English version journal (Mi zvmmf10987)

Abundant dynamical behaviors of bounded traveling wave solutions to generalized $\theta$-equation

Zhenshu Wen

Fujian Province University Key Laboratory of Computational Science, School of Mathematical Sciences, Huaqiao University, Quanzhou, P.R. China

Abstract: We study existence and dynamics of bounded traveling wave solutions to generalized $\theta$-equation from the perspective of dynamical systems. We obtain bifurcation of traveling wave solutions for the equation, prove the existence of several types of bounded traveling wave solutions, including solitary wave solutions, periodic wave solutions, peakons, periodic cusp waves, compactons and kink-like (antikink-like) waves, and derive some of their exact expressions. Most importantly, we confirm abundant dynamical behaviors of the traveling wave solutions to the equation, which are summarized as follows: (1) We confirm that three types of orbits give rise to solitary wave solutions, that is, the homoclinic orbit passing the singular point, the composed homoclinic orbit which is comprised of three heteroclinic orbits of the associated system, and the composed homoclinic orbit which is comprised of two heteroclinic orbits and tangent to the singular line at the singular point of the associated system. (2) We confirm that four types of orbits correspond to periodic wave solutions, that is, the periodic orbit surrounding a center, the periodic orbit surrounding two connected homoclinic orbits, the composed periodic orbit which is comprised of two heteroclinic orbits of the associated system, and the homoclinic orbit of the associated system which is tangent to the singular line at the singular point of the associated system. (3) We confirm that two types of orbits correspond to periodic cusp waves, that is, the semiellipse orbit surrounding a center, and the semiellipse-like orbit surrounding two connected homoclinic orbits. (4) We confirm that two families of periodic orbits, which surround two connected homoclinic orbits and are comprised of two heteroclinic orbits of associated system, respectively, and the composed homoclinic orbit, which is comprised of two heteroclinic orbits and tangent to the singular line at the singular point of associated system, have envelope.

Key words: generalized $\theta$-equation, bifurcation, existence, dynamics, bounded traveling wave solutions.

 Funding Agency Grant Number National Natural Science Foundation of China 11701191 China Scholarship Council, Program for Innovative Research Team in Science and Technology in Fujian Province University Quanzhou High-Level Talents Support Plan under Grant 2017ZT012 This research is partially supported by the National Natural Science Foundation of China (no. 11701191), China Scholarship Council, Program for Innovative Research Team in Science and Technology in Fujian Province University, and Quanzhou High-Level Talents Support Plan under Grant 2017ZT012.

English version:
Computational Mathematics and Mathematical Physics, 2019, 59:6, 926–935

Bibliographic databases:

Revised: 26.12.2018
Accepted:08.02.2019
Language:

Citation: Zhenshu Wen, “Abundant dynamical behaviors of bounded traveling wave solutions to generalized <nobr>$\theta$</nobr>-equation”, Comput. Math. Math. Phys., 59:6 (2019), 926–935

Citation in format AMSBIB
\Bibitem{Wen19} \by Zhenshu~Wen \paper Abundant dynamical behaviors of bounded traveling wave solutions to generalized <nobr>$\theta$</nobr>-equation \jour Comput. Math. Math. Phys. \yr 2019 \vol 59 \issue 6 \pages 926--935 \mathnet{http://mi.mathnet.ru/zvmmf10987} \crossref{https://doi.org/10.1134/S0965542519060150} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000473489900007} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85068577656}