KAM Tori for 1D Nonlinear Wave Equations¶with Periodic Boundary Conditions

2000 ◽  
Vol 211 (2) ◽  
pp. 497-525 ◽  
Author(s):  
Luigi Chierchia ◽  
Jiangong You
Keyword(s):  
Boundary Conditions ◽  
Nonlinear Wave ◽  
Wave Equations ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Nonlinear Wave Equations ◽  
Kam Tori

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.

scholarly journals Exact Solution of Linear and Nonlinear Wave Equations by Double Laplace and Iterative Method

2019 ◽  
Vol 11 (5) ◽  
pp. 33
Author(s):  
Adesina K. Adio ◽  
Wumi S. Ajayi ◽  
Olabode O. Bamisile ◽  
Babatunde T. Akanbi
Keyword(s):  
Exact Solution ◽  
Iterative Method ◽  
Boundary Conditions ◽  
Laplace Transform ◽  
Nonlinear Wave ◽  
Wave Equations ◽  
Iteration Process ◽  
Nonlinear Wave Equations ◽  
Double Laplace Transform ◽  
Linear And Nonlinear

A coupling of double Laplace transform with Iterative method is used to solve linear and nonlinear wave equations subject to initial and boundary conditions. The iteration process leads to disappearance of noise terms and exact solution is obtained at first iteration. Through several examples, the convenience and efficiency of the method is demonstrated, showing its usefulness to overcome difficulties associated with some existing techniques.

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