Well-posedness for a generalized derivative nonlinear Schrödinger equation

Abstract

We study the Cauchy problem for a generalized derivative nonlinear Schrödinger equation with the Dirichlet boundary condition. We establish the local well-posedness results in the Sobolev spaces H 1 and H 2 . Solutions are constructed as a limit of approximate solutions by a method independent of a compactness argument. We also discuss the global existence of solutions in the energy space H 1 .

On a Class of Solutions to the Generalized Derivative Schrödinger Equations

“…In [34] Santos showed the existence and uniqueness of solution u ∈ L ∞ ((0, T ); H 3/2 (R)∩ x −1 H 1/2 (R)) for sufficient small initial data in the case 1 < α < 2, and local well-posedness in H 1/2 (R) for small data when α > 2. In [13] Hayashi and Ozawa considered the gDNLS equation in a bounded interval with a Dirichlet condition and established local results in H 2 for α ≥ 1 and H 1 for α ≥ 2. In [8] Fukaya, Hayashi and Inui showed global result for initial data in H 1 (R), for any α ≥ 2, with initial data satisfying u 0 2 2 ≤ 4π.…”

Nonlinear Schrödinger approach to European option pricing

“…Equation (35) with function V(S) representing di erent atomic potentials will be solved numerically using a Runge-Kutta fourth-order method.…”

“…The details of this method can be found in the Appendix. Let us write the second order equation (35) in the standard form (56) of the Runge-Kutta method:…”

Self-similar solutions to the derivative nonlinear Schrödinger equation

“…The motivation for their analysis combined with numerical calculations was to predict the profile of the blowup of the solution in the case where σ > 1. For related subjects, we also refer the reader to [5].…”