scholarly journals Infinitely many periodic solutions for a semilinear wave equation with x-dependent coefficients

2020 ◽  
Vol 26 ◽  
pp. 7
Author(s):  
Hui Wei ◽  
Shuguan Ji
Keyword(s):  
Wave Equation ◽  
Periodic Solutions ◽  
Spectral Properties ◽  
Seismic Waves ◽  
Forced Vibrations ◽  
One Dimensional ◽  
Semilinear Wave Equation ◽  
Spatial Interval ◽  
Rational Multiple ◽  
Infinitely Many Periodic Solutions

This paper is devoted to the study of periodic (in time) solutions to an one-dimensional semilinear wave equation with x-dependent coefficients under various homogeneous boundary conditions. Such a model arises from the forced vibrations of a nonhomogeneous string and propagation of seismic waves in nonisotropic media. By combining variational methods with an approximation argument, we prove that there exist infinitely many periodic solutions whenever the period is a rational multiple of the length of the spatial interval. The proof is essentially based on the spectral properties of the wave operator with x-dependent coefficients.

scholarly journals Existence of multiple periodic solutions to a semilinear wave equation with x-dependent coefficients

2019 ◽  
Vol 150 (5) ◽  
pp. 2586-2606
Author(s):  
Hui Wei ◽  
Shuguan Ji
Keyword(s):  
Wave Equation ◽  
Periodic Solutions ◽  
Seismic Waves ◽  
One Dimensional ◽  
Semilinear Wave Equation ◽  
Point Reduction ◽  
Spatial Interval ◽  
Multiple Periodic Solutions ◽  
Rational Multiple ◽  
Asymptotic Character

AbstractThis paper is concerned with the periodic (in time) solutions to an one-dimensional semilinear wave equation with x-dependent coefficients. Such a model arises from the forced vibrations of a nonhomogeneous string and propagation of seismic waves in nonisotropic media. By combining variational methods with saddle point reduction technique, we obtain the existence of at least three periodic solutions whenever the period is a rational multiple of the length of the spatial interval. Our method is based on a delicate analysis for the asymptotic character of the spectrum of the wave operator with x-dependent coefficients, and the spectral properties play an essential role in the proof.

Existence and Multiplicity of Periodic Solutions for Dirichlet–Neumann Boundary Value Problem of a Variable Coefficient Wave Equation

2016 ◽  
Vol 16 (4) ◽  
Author(s):  
Shuguan Ji ◽  
Yang Gao ◽  
Wenzhuang Zhu
Keyword(s):  
Wave Equation ◽  
Boundary Value Problem ◽  
Periodic Solutions ◽  
Variable Coefficient ◽  
Seismic Waves ◽  
Degree Theory ◽  
Neumann Boundary ◽  
Forced Vibrations ◽  
Neumann Boundary Value Problem ◽  
Existence And Multiplicity

AbstractIn this paper, we consider the periodic solutions of a variable coefficient wave equation which models the forced vibrations of a nonhomogeneous string and the propagation of seismic waves in nonisotropic media. Under Dirichlet–Neumann boundary conditions, we find some important properties for the variable coefficient wave operator. Then, based on these properties, we obtain the existence and multiplicity of periodic solutions by using the Leray–Schauder degree theory.

Quasi-Periodic Solutions of Nonlinear Wave Equation with x-Dependent Coefficients

2015 ◽  
Vol 25 (03) ◽  
pp. 1550043 ◽  
Author(s):  
Yixian Gao ◽  
Weipeng Zhang ◽  
Shuguan Ji
Keyword(s):  
Wave Equation ◽  
Normal Form ◽  
Periodic Solutions ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Seismic Waves ◽  
Birkhoff Normal Form ◽  
General Boundary Conditions ◽  
Kam Theorem ◽  
Infinite Dimensional

This paper is devoted to the study of quasi-periodic solutions of a nonlinear wave equation with x-dependent coefficients. Such a model arises from the forced vibration of a nonhomogeneous string and the propagation of seismic waves in nonisotropic media. Based on the partial Birkhoff normal form and an infinite-dimensional KAM theorem, we can obtain the existence of quasi-periodic solutions for this model under the general boundary conditions.

Time-periodic solutions to the one-dimensional wave equation with periodic or anti-periodic boundary conditions

2007 ◽  
Vol 137 (2) ◽  
pp. 349-371 ◽  
Author(s):  
Shuguan Ji ◽  
Yong Li
Keyword(s):  
Wave Equation ◽  
Weak Solution ◽  
Boundary Conditions ◽  
Periodic Solutions ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
One Dimensional ◽  
Time Periodic Solutions ◽  
Time Periodic ◽  
Dimensional Wave

This paper is devoted to the study of time-periodic solutions to the nonlinear one-dimensional wave equation with x-dependent coefficients u(x)ytt – (u(x)yx)x + g(x,t,y) = f(x,t) on (0,π) × ℝ under the periodic boundary conditions y(0,t) = y(π,t), yx(0,t) = yx(π,t) or anti-periodic boundary conditions y(0, t) = –y(π,t), yx[0,t) = – yx(π,t). Such a model arises from the forced vibrations of a non-homogeneous string and the propagation of seismic waves in non-isotropic media. Our main concept is that of the ‘weak solution’. For T, the rational multiple of π, we prove some important properties of the weak solution operator. Based on these properties, the existence and regularity of weak solutions are obtained.

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