98 research outputs found
A modulation equations approach for numerically solving the moving soliton and radiation solutions of NLS
Based on our previous work for solving the nonlinear Schrodinger equation
with multichannel dynamics that is given by a localized standing wave and
radiation, in this work we deal with the multichannel solution which consists
of a moving soliton and radiation. We apply the modulation theory to give a
system of ODEs coupled to the radiation term for describing the solution, which
is valid for all times. The modulation equations are solved accurately by the
proposed numerical method. The soliton and radiation are captured separately in
the computation, and they are solved on the translated domain that is moving
with them. Thus for a fixed finite physical domain in the lab frame, the
multichannel solution can pass through the boundary naturally, which can not be
done by imposing any existing boundary conditions. We comment on the
differences of this method from the collective coordinates.Comment: 19 pages, 7 figures. To appear on Phys. D. arXiv admin note: text
overlap with arXiv:1404.115
On multichannel solutions of nonlinear Schr\"{o}dinger equations: algorithm, analysis and numerical explorations
We apply the method of modulation equations to numerically solve the NLS with
multichannel dynamics, given by a trapped localized state and radiation. This
approach employs the modulation theory of Soffer-Weinstein, which gives a
system of ODE's coupled to the radiation term, which is valid for all times. We
comment on the differences of this method from the well-known method of
collective coordinates.Comment: 16 pages, 9 figures. Added a comparison to collective coordinates
scheme that includes a radiation ter
A Positivity Criterion for the Wave Equation and Global Existence of Large Solutions
In dimensions one to three, the fundamental solution to the free wave
equation is positive. Therefore, there exists a simple positivity criterion for
solutions. We use this to obtain large global solutions to two well-studied
energy-supercritical semilinear wave equations, as well as some new results in
the subcritical and critical cases.Comment: 26 pages; added some references, fixed some typo
- …