Some Explicit Expressions and Interesting Bifurcation Phenomena for Nonlinear Waves in Generalized Zakharov Equations.
In: Abstract & Applied Analysis, 2013, S. 1-19
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Zugriff:
Using bifurcation method of dynamical systems, we investigate the nonlinear waves for the generalized Zakharov equations u tt - c² s u xx = β(|E|²) xx , iE t +αE xx -δ 1 uE+δ 2 |E|²E+δ 3 |E| 4 E = 0, where α,β,δ 1,δ 2,δ 3 , and cs 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. [ABSTRACT FROM AUTHOR]
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Titel: |
Some Explicit Expressions and Interesting Bifurcation Phenomena for Nonlinear Waves in Generalized Zakharov Equations.
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Autor/in / Beteiligte Person: | Li, Shaoyong ; Liu, Rui |
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Zeitschrift: | Abstract & Applied Analysis, 2013, S. 1-19 |
Veröffentlichung: | 2013 |
Medientyp: | academicJournal |
ISSN: | 1085-3375 (print) |
DOI: | 10.1155/2013/869438 |
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