Construction of solutions of the defocusing nonlinear Schrödinger equation with asymptotically time‐periodic boundary values

Abstract

We study the defocusing nonlinear Schrödinger equation in the quarter plane with asymptotically periodic boundary values. We use the unified transform method, also known as the Fokas method, and the Deift‐Zhou nonlinear steepest descent method to construct solutions in a sector close to the boundary whose leading behavior is described by a single exponential plane wave. Furthermore, we compute the subleading terms in the long‐time asymptotics of the constructed solutions.

The defocusing nonlinear Schrödinger equation with step-like oscillatory initial data

“…The proof of Lemma 5.3 does not rely on an explicit expression of r 2 on (E 1 , E 2 ) in order to construct a function r 2,a with the desired properties. This generalizes a related construction in [25], where r 2 was defined on (E 1 , E 2 ) in terms of r by r 2 (k) := −r(k)/(1+r(k 2 )). For consistency, additional assumptions were made in [25], namely that r = ±i on (E 1 , E 2 ) and r(E j ) ∈ {±i}, j = 1, 2.…”

“…This generalizes a related construction in [25], where r 2 was defined on (E 1 , E 2 ) in terms of r by r 2 (k) := −r(k)/(1+r(k 2 )). For consistency, additional assumptions were made in [25], namely that r = ±i on (E 1 , E 2 ) and r(E j ) ∈ {±i}, j = 1, 2. However, numerical experiments have shown that these assumptions are not satisfied in general.…”

“…Our treatments of the left and right sectors will be much briefer. In fact, the proof of the large t asymptotics (2.29) of u in the left sector L is almost identical to the proof of [25,Theorem 1], where a related initial-boundary value problem was considered and the same asymptotic formula was obtained. With the help of Remark 5.5, the proof in [25] can easily be adapted to our situation.…”

“…In fact, the proof of the large t asymptotics (2.29) of u in the left sector L is almost identical to the proof of [25,Theorem 1], where a related initial-boundary value problem was considered and the same asymptotic formula was obtained. With the help of Remark 5.5, the proof in [25] can easily be adapted to our situation. As for the proof of the asymptotic formula (2.31) valid as x → −∞, the proof proceeds along the same lines and the minor modifications that are necessary were explained in [17] for the mKdV equation.…”

“…The construction of analytic approximations is challenging especially near the branch cut associated with the nonzero boundary conditions. We overcome this challenge by generalizing a construction in [25].…”