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2013 Some Explicit Expressions and Interesting Bifurcation Phenomena for Nonlinear Waves in Generalized Zakharov Equations
Shaoyong Li, Rui Liu
Abstr. Appl. Anal. 2013: 1-19 (2013). DOI: 10.1155/2013/869438

Abstract

Using bifurcation method of dynamical systems, we investigate the nonlinear waves for the generalized Zakharov equations u t t - c s 2 u x x = β ( | E | 2 ) x x , i E t + α E x x - δ 1 u E + δ 2 | E | 2 E + δ 3 | E | 4 E = 0 , where α , β , δ 1 , δ 2 , δ 3 , and c s are real parameters, E = E ( x , t ) is a complex function, and u = u ( x , t ) is a real function. We obtain the following results. (i) Three types of explicit expressions of nonlinear waves are obtained, that is, the fractional expressions, the trigonometric expressions, and the exp-function expressions. (ii) Under different parameter conditions, these expressions represent symmetric and antisymmetric solitary waves, kink and antikink waves, symmetric periodic and periodic-blow-up waves, and 1-blow-up and 2-blow-up waves. We point out that there are two sets of kink waves which are called tall-kink waves and low-kink waves, respectively. (iii) Five kinds of interesting bifurcation phenomena are revealed. The first kind is that the 1-blow-up waves can be bifurcated from the periodic-blow-up and 2-blow-up waves. The second kind is that the 2-blow-up waves can be bifurcated from the periodic-blow-up waves. The third kind is that the symmetric solitary waves can be bifurcated from the symmetric periodic waves. The fourth kind is that the low-kink waves can be bifurcated from four types of nonlinear waves, the symmetric solitary waves, the 1-blow-up waves, the tall-kink waves, and the antisymmetric solitary waves. The fifth kind is that the tall-kink waves can be bifurcated from the symmetric periodic waves. We also show that the exp-function expressions include some results given by pioneers.

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Shaoyong Li. Rui Liu. "Some Explicit Expressions and Interesting Bifurcation Phenomena for Nonlinear Waves in Generalized Zakharov Equations." Abstr. Appl. Anal. 2013 1 - 19, 2013. https://doi.org/10.1155/2013/869438

Information

Published: 2013
First available in Project Euclid: 27 February 2014

zbMATH: 1275.35025
MathSciNet: MR3055940
Digital Object Identifier: 10.1155/2013/869438

Rights: Copyright © 2013 Hindawi

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