Time periodic solutions of nonlinear wave equations

1971 ◽  
Vol 5 (2) ◽  
pp. 165-194 ◽  
Author(s):  
Paul H. Rabinowitz
Keyword(s):  
Periodic Solutions ◽  
Nonlinear Wave ◽  
Wave Equations ◽  
Nonlinear Wave Equations ◽  
Time Periodic Solutions ◽  
Time Periodic

Time-periodic solutions to a nonlinear wave equation with periodic or anti-periodic boundary conditions

2008 ◽  
Vol 465 (2103) ◽  
pp. 895-913 ◽  
Author(s):  
Shuguan Ji
Keyword(s):  
Wave Equation ◽  
Boundary Conditions ◽  
Periodic Solutions ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Main Concept ◽  
Time Periodic Solutions ◽  
Time Periodic

This paper is concerned with the existence of time-periodic solutions to the nonlinear wave equation with x -dependent coefficients u ( x ) y tt − ( u ( x ) y x ) x + au ( x ) y +| y | p −2 y = f ( x ,  t ) on (0,  π )× under the periodic or anti-periodic boundary conditions y (0, t )=± y ( π ,  t ), y x (0,  t )=± y x ( π ,  t ) and the time-periodic conditions y ( x ,  t + T )= y ( x ,  t ), y t ( x ,  t + T )= y t ( x ,  t ). Such a model arises from the forced vibrations of a non-homogeneous string and the propagation of seismic waves in non-isotropic media. A main concept is the notion ‘weak solution’ to be given in §2. For T =2 π / k ( k ∈ ), we establish the existence of time-periodic solutions in the weak sense by investigating some important properties of the wave operator with x -dependent coefficients.

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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