Periodic solutions of nonlinear wave-equations with a dissipative term

1971 ◽  
Vol 190 (4) ◽  
pp. 313-322 ◽  
Author(s):  
Wolf von Wahl
Keyword(s):  
Periodic Solutions ◽  
Nonlinear Wave ◽  
Wave Equations ◽  
Nonlinear Wave Equations ◽  
Dissipative Term

scholarly journals Global existence and decay estimates for nonlinear wave equations with space-time dependent dissipative term

2015 ◽  
Vol 12 (02) ◽  
pp. 249-276
Author(s):  
Tomonari Watanabe
Keyword(s):  
Global Existence ◽  
Nonlinear Wave ◽  
Wave Equations ◽  
Space Time ◽  
Time Dependent ◽  
Nonlinear Wave Equations ◽  
Energy Estimates ◽  
Space Region ◽  
Decay Estimates ◽  
Dissipative Term

We study the global existence and the derivation of decay estimates for nonlinear wave equations with a space-time dependent dissipative term posed in an exterior domain. The linear dissipative effect may vanish in a compact space region and, moreover, the nonlinear terms need not be in a divergence form. In order to establish higher-order energy estimates, we introduce an argument based on a suitable rescaling. The proposed method is useful to control certain derivatives of the dissipation coefficient.

A REDUCTION mKdV METHOD WITH SYMBOLIC COMPUTATION TO CONSTRUCT NEW DOUBLY-PERIODIC SOLUTIONS FOR NONLINEAR WAVE EQUATIONS

2003 ◽  
Vol 14 (05) ◽  
pp. 661-672 ◽  
Author(s):  
ZHENYA YAN
Keyword(s):  
Periodic Solutions ◽  
Nonlinear Wave ◽  
Symbolic Computation ◽  
Wave Equations ◽  
Kdv Equation ◽  
Soliton Solutions ◽  
Transformation Method ◽  
Mkdv Equation ◽  
Nonlinear Wave Equations ◽  
Cubic Nonlinear Schrödinger Equation

Firstly twenty-four types of doubly-periodic solutions of the reduction mKdV equation are given. Secondly based on the reduction mKdV equation and its solutions, a systemic transformation method (called the reduction mKdV method) is developed to construct new doubly-periodic solutions of nonlinear equations. Thirdly with the aid of symbolic computation, we choose the KdV equation, the coupled variant Boussinesq equation and the cubic nonlinear Schrödinger equation to illustrate our method. As a result many types of solutions are obtained. These show that this method is simple and powerful to obtain more exact solutions including doubly-periodic solutions, soliton solutions and singly-periodic solutions to a wide class of nonlinear wave equations. Finally we further extended the method to a general form.

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