Smooth and singular traveling wave solutions for the Serre-Green-Naghdi equations

Lijun Zhang; Yixia Shi; Maoan Han

doi:10.3934/dcdss.2020217

Article Contents

2020, Volume 13, Issue 10: 2917-2926. Doi: 10.3934/dcdss.2020217

This issue Previous Article Space-time kernel based numerical method for generalized Black-Scholes equation Next Article Bifurcations and exact traveling wave solutions of the Zakharov-Rubenchik equation

Smooth and singular traveling wave solutions for the Serre-Green-Naghdi equations

1.
College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, Shandong 266590, China
2.
African Institute for Mathematical Sciences, Muizenberg, Cape Town, South Africa
3.
Department of Mathematics, Shanghai Normal University, Shanghai 200234, China
4.
Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China

^* Corresponding author: Yixia Shi
^* Corresponding author: Yixia Shi

Received: January 31, 2019

Revised: May 31, 2019

Early access: December 2019

Published: July 2020

Part of this work was done during L. Zhang's research visit to African Institute for Mathematical Sciences South Africa

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Abstract

In this paper, we consider the traveling wave solutions of the one-dimensional Serre-Green-Naghdi (SGN) equations which are proposed to model dispersive nonlinear long water waves in a one-layer flow over flat bottom. We decouple the traveling wave system of SGN equations into two ordinary differential equations. By studying the bifurcations and phase portraits of each bifurcation set of one equation, we obtain the exact traveling wave solutions of SGN equations for the variable $ u(x, t) $ which represents average horizontal velocity of water wave. For the compacted orbits intersecting with the singular line in phase plane, we obtained two families of solutions: a family of smooth traveling wave solutions including periodic wave solutions and solitary wave solutions, and a family of compacted singular solutions which have continuous first-order derivative but discontinuous second-order derivative.

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Mathematics Subject Classification: 35C07, 34G20, 34C23.

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Figure 1. The phase portraits of system (9) in each bifurcation set for $ A>0 $. (1) $ c>\frac{B}{A}-\frac{3}{2}\sqrt{Ag} $; (2) $ c = \frac{B}{A}-\frac{3}{2}\sqrt{Ag} $; (3) $ c<\frac{B}{A}-\frac{3}{2}\sqrt{Ag} $

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Figure 2. Phase orbit of (12) and the corresponding bounded traveling wave solutions. (1) a compact orbit of (12) intersecting with the singular line; (2) a smooth periodic traveling wave solution; (3) compacton

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Figure 3. Phase orbit of (12) and the corresponding bounded traveling wave solutions. (1) a compact orbit of (12) intersecting with the singular line; (2) a smooth periodic traveling wave solution; (3) compacton

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