Resonance and long-time existence for the quadratically nonlinear Schroedinger equation

Thesis/Dissertation · Mon Jan 01 04:00:00 EST 1990

OSTI ID:5654551

Cohn, S D

This problem is motivated by the theory of nonlinear dispersive waves. Here, self-interacting waves are modeled by solutions of a cubic Shroedinger equation, even though certain of the original problems (e.g. water waves) led to models in which the nonlinearity is quadratic. Such a change can be made because the quadratic terms are formally nonresonant. In finite dimensional systems nonresonant nonlinearities can be removed by putting the system in Poincare normal form. Recently Shatah adapted this technique to the infinite dimensional case to prove global existence for the quadratic Klein-Gordon equation in three space dimensions. The main result of this thesis is a long time existence theorem for the solution of the initial value problem for the quadratic Schroedinger equation in one space dimension. Formal calculations with uniform wave solutions e{sup ikx}e{sup {minus}ik{sup 2}t} of the homogeneous Schroedinger equation suggest that {bar {nu}}{sup 2}{sub x} is the simplest nonresonant quadratic nonlinearity for which the initial value problem is well posed. Let T(v) be the L{sup 2}{sub s} existence time for the solution v. Using a method developed by Klainerman and Ponce, the author establishes the lower bound T({nu}) {ge} C{sub {var epsilon}}{sup {minus}}4/3, where {var epsilon} is the size of the initial data g in a certain Sobolev space.

• Research Organization: New York Univ., NY (United States)
• OSTI ID: 5654551
• Country of Publication: United States
• Language: English

Related Subjects

99 GENERAL AND MISCELLANEOUS
990200* -- Mathematics & Computers
CALCULATION METHODS
DIFFERENTIAL EQUATIONS
EQUATIONS
KLEIN-GORDON EQUATION
LIMITING VALUES
MATHEMATICAL MODELS
MATHEMATICAL SPACE
NONLINEAR PROBLEMS
PARTIAL DIFFERENTIAL EQUATIONS
QUADRATURES
SCHROEDINGER EQUATION
SPACE
WAVE EQUATIONS