TY - JOUR
T1 - On the derivative nonlinear Schrödinger equation
AU - Hayashi, Nakao
AU - Ozawa, Tohru
N1 - Copyright:
Copyright 2014 Elsevier B.V., All rights reserved.
PY - 1992/2
Y1 - 1992/2
N2 - In this paper we discuss the Cauchy problem for the derivative nonlinear Schrödinger equation: i∂tψ + 2iδ∂x(|;ψ|2ψ) = 0, ψ(0, x) = f{cyrillic}(x), where δ ≠ 0. Under an explicit smallness condition of the initial data, we prove the unique global existence of solutions to this problem in the usual Sobolev spaces, in the weighted Sobolev spaces, and in the Schwartz class. We describe the smoothing effect in detail. Furthermore, for the data decaying exponentially at infinity we prove that the above equation has unique local solutions which are analytic in the space direction.
AB - In this paper we discuss the Cauchy problem for the derivative nonlinear Schrödinger equation: i∂tψ + 2iδ∂x(|;ψ|2ψ) = 0, ψ(0, x) = f{cyrillic}(x), where δ ≠ 0. Under an explicit smallness condition of the initial data, we prove the unique global existence of solutions to this problem in the usual Sobolev spaces, in the weighted Sobolev spaces, and in the Schwartz class. We describe the smoothing effect in detail. Furthermore, for the data decaying exponentially at infinity we prove that the above equation has unique local solutions which are analytic in the space direction.
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U2 - 10.1016/0167-2789(92)90185-P
DO - 10.1016/0167-2789(92)90185-P
M3 - Article
AN - SCOPUS:0002770583
VL - 55
SP - 14
EP - 36
JO - Physica D: Nonlinear Phenomena
JF - Physica D: Nonlinear Phenomena
SN - 0167-2789
IS - 1-2
ER -