Geometric singular perturbation analysis of Degasperis-Procesi equation with distributed delay

Feifei Cheng; Ji Li

doi:10.3934/dcds.2020305

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2021, Volume 41, Issue 2: 967-985. Doi: 10.3934/dcds.2020305

This issue Previous Article Low Mach number limit for the compressible inertial Qian-Sheng model of liquid crystals: Convergence for classical solutions Next Article

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Laplacian problems with critical double exponential nonlinearities

Geometric singular perturbation analysis of Degasperis-Procesi equation with distributed delay

Feifei Cheng and
Ji Li^,

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China

^* Corresponding author: Ji Li, liji@hust.edu.cn
^* Corresponding author: Ji Li, liji@hust.edu.cn

Received: April 2020

Early access: August 2020

Published: February 2021

The corresponding author is supported by NSFC grant 11771161

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Abstract

In this paper we consider the Degasperis-Procesi equation, which is an approximation to the incompressible Euler equation in shallow water regime. First we provide the existence of solitary wave solutions for the original DP equation and the general theory of geometric singular perturbation. Then we prove the existence of solitary wave solutions for the equation with a special local delay convolution kernel and a special nonlocal delay convolution kernel by using the geometric singular perturbation theory and invariant manifold theory. According to the relationship between solitary wave and homoclinic orbit, the Degasperis-Procesi equation is transformed into the slow-fast system by using the traveling wave transformation. It is proved that the perturbed equation also has a homoclinic orbit, which corresponds to a solitary wave solution of the delayed Degasperis-Procesi equation.

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Mathematics Subject Classification: Primary: 34D15, 74J30, 35Q35; Secondary: 34E10.

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Figure 1. The homoclinic orbit within $ \phi<c-\frac{2}{3}k $

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Figure 2. local delay, k = 1

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Figure 3. noncocal delay, k = 1

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