Existence, scattering, blow up, and decay of solutions to systems of nonlinear Schroedinger equations
Thesis/Dissertation
·
OSTI ID:6257894
We prove global existence, scattering, and blow up of solutions to systems of nonlinear Schroedinger equations with interactions of a particular type. Thus, we consider iu/sub t/ + ..delta..u = F(u), u(t/sub 0/,chi) = u/sub 0/(chi) where u is a function from R x R/sup n/ to C/sup m/ and F is a vector field on C/sup m/. We also determine the dependence of blow up and decay of solutions on a parameter lambda in nonlinear interactions of the general form F(u) = u absolute value u/sup p-1/ - lambda u absolute value u/sup q-1/ + eta u absolute value u/sup r-1/ for m = 1. We obtain global existence, scattering, and blow up if F is an exact interaction, i.e., an exact vector field on C/sup M/. For global existence, we allow F to be of the form u absolute value u/sup p-1/ + u absolute value u/sup q-1/, 2 less than or equal to p < q < n + 2/n-2 with restriction p > (n + 2)(2n - (n-2)q)/sup -1/. Using different techniques, we remove this restriction for n = 3 or 4 and prove global existence in H/sup 2/. For the case m = 1, these results allow more general interactions than (2) where global existence is proven in H/sup 1/. We also obtain global existence for perturbations of exact interactions of the form i ..beta..(t)u absolute value u/sup p-1/ for ..beta.. an odd function of t. If F is not an exact vector field, we obtain global existence provided that F satisfies a certain type of Lipshitz estimate.
- Research Organization:
- Duke Univ., Durham, NC (USA)
- OSTI ID:
- 6257894
- Country of Publication:
- United States
- Language:
- English
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Related Subjects
657002* -- Theoretical & Mathematical Physics-- Classical & Quantum Mechanics
658000 -- Mathematical Physics-- (-1987)
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
DIFFERENTIAL EQUATIONS
EQUATIONS
NONLINEAR PROBLEMS
NUMERICAL SOLUTION
PARTIAL DIFFERENTIAL EQUATIONS
SCHROEDINGER EQUATION
WAVE EQUATIONS
658000 -- Mathematical Physics-- (-1987)
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
DIFFERENTIAL EQUATIONS
EQUATIONS
NONLINEAR PROBLEMS
NUMERICAL SOLUTION
PARTIAL DIFFERENTIAL EQUATIONS
SCHROEDINGER EQUATION
WAVE EQUATIONS