Abstract
The quadratic nonlinear wave equation on a one-dimensional torus with small initial values located in a single Fourier mode is considered. In this situation, the formation of metastable energy strata has recently been described and their long-time stability has been shown. The topic of the present paper is the correct reproduction of these metastable energy strata by a numerical method. For symplectic trigonometric integrators applied to the equation, it is shown that these energy strata are reproduced even on long time intervals in a qualitatively correct way.
Highlights
We consider the nonlinear wave equation∂ttu − ∂xxu + ρu = u2, u = u(x, t) ∈ R, (1)on a one-dimensional torus, x ∈ T = R/(2πZ), with a positive Klein–Gordon parameter ρ
We discretize the nonlinear wave equation and answer the question whether this long-time property of the exact solution is inherited by the numerical method
We consider a spectral collocation in space combined with a symplectic trigonometric integrator in time
Summary
We discretize the nonlinear wave equation and answer the question whether this long-time property of the exact solution is inherited by the numerical method. We consider a spectral collocation in space combined with a symplectic trigonometric integrator in time We show that this numerical method reproduces the energy strata of the exact solution even. Related questions have been studied for symplectic and non-symplectic methods applied to the nonlinear Schrodinger equation In this case, an initial value consisting of a single Fourier mode yields a solution that continues to consist of a single Fourier mode for all times (plane wave solution). The stability in numerical discretizations of these plane wave solutions under small perturbations of the initial value is an old question [22] and has been studied on short time intervals [4, 7, 19, 20, 22] and on long time intervals [10]
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
