Long‐time asymptotics of moving boundary problems using an Ehrenpreis‐type representation and its Riemann‐Hilbert nonlinearization

Abstract

AbstractA new method for solving boundary value problems for linear and integrable nonlinear PDEs has recently been introduced. For a linear evolution equation with dispersion relation ω(k), in a time‐dependent concave domain, 𝓁(t) < x < ∞, 𝓁″(t) < 0, this method yields the solution q(x, t) as a line integral in the complex k‐plane, whose integrand has explicit (x, t)‐dependence of the form exp[ikx − iω(k)t]. For a nonlinear integrable evolution equation the situation is similar, but the integrand also involves the solution M(x, t, k) to a Riemann‐Hilbert problem whose jump function also has explicit (x, t)‐dependence of exponential form. These representations for linear and for nonlinear evolution equations are convenient for the study of the long‐time asymptotics, using the stationary phase and the Deift‐Zhou methods, respectively. Here we study the long‐time asymptotics in a time‐dependent concave domain for a linear evolution equation with a spatial derivative of arbitrary order, and for the defocusing nonlinear Schrödinger equation. For completeness we also study the long‐time asymptotics for a linear evolution equation with a spatial derivative of arbitrary order in the fixed domain 0 < x < ∞. © 2003 Wiley Periodicals, Inc.