Stability for nonlinear wave motions damped by time-dependent frictions
Zhe Jiao, Yong Xu, Lijing Zhao
TL;DR
This paper investigates the stability of solutions to nonlinear wave equations with time-dependent damping, establishing conditions for convergence to equilibrium and supporting findings with numerical simulations.
Contribution
It provides a sharp condition on damping coefficients ensuring asymptotic stability without bifurcation or chaos in nonlinear wave systems.
Findings
Solutions are asymptotically stable under specific damping conditions.
Numerical simulations confirm theoretical stability results.
No bifurcation or chaos occurs under the established conditions.
Abstract
We are concerned with the dynamical behavior of solutions to semilinear wave systems with time-varying damping and nonconvex force potential. Our result shows that the dynamical behavior of solution is asymptotically stable without any bifurcation and chaos. And it is a sharp condition on the damping coefficient for the solution to converge to some equilibrium. To illustrate our theoretical results, we provide some numerical simulations for dissipative sine-Gordon equation and dissipative Klein-Gordon equation.
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Taxonomy
TopicsStability and Controllability of Differential Equations · Numerical methods for differential equations · Advanced Mathematical Physics Problems