A sufficient condition for global existence of solutions to a generalized derivative nonlinear Schrödinger equation

Abstract

We give a sufficient condition for global existence of the solutions to a generalized derivative nonlinear Schrödinger equation (gDNLS) by a variational argument. The variational argument is applicable to a cubic derivative nonlinear Schrödinger equation (DNLS). For (DNLS), Wu (2015) proved that the solution with the initial data u 0 is global if u 0 2 L 2 < 4π by the sharp Gagliardo-Nirenberg inequality. The variational argument gives us another proof of the global existence for (DNLS). Moreover, by the variational argument, we can show that the solution to (DNLS) is global if the initial data u 0 satisfies u 0 2 L 2 = 4π and the momentum P(u 0 ) is negative.

Global well-posedness for the derivative non-linear Schrödinger equation

“…4π (or }u 0 } L 2 " ? 4π with negative momentum), there exists a unique solution u P CpR, H 1{2 pRqq [6,7]. A key structural property of DNLS discovered by Kaup and Newell [11] is that it is integrable by inverse scattering: that is, there is a linear spectral problem with upx, tq as potential whose spectral data (consisting of a reflection coefficient, describing the continuous spectrum of the linear problem, together with eigenvalues and norming constants, describing the discrete spectrum of the linear problem) evolve linearly under the flow.…”

“…We may factorize v " pI´w´q´1pI`w`q where (4. 6) pw`, w´qˇˇΣ "ˆˆ0 0ȓ 0˙,ˆ0 r 0 0˙˙, w´ˇˇ˘γ j " 0, w`ˇˇ˘γ j " 0.…”

Long-period limit of exact periodic traveling wave solutions for the derivative nonlinear Schrödinger equation

“…For the case ω > c 2 /4, this global result is essentially proved in [12]. In [15], they mainly proved that there exists (ω, c) satisfying both (1.10) and the condition (1.26) if the initial data u 0 satisfies u 0 2 L 2 < 4π, or u 0 2 L 2 = 4π and P (u 0 ) < 0. This gives a simple alternative proof of Wu's global result.…”

“…The condition of two parameters (ω, c) (1.10) ω > c 2 /4, or ω = c 2 /4 and c > 0 is a necessary and sufficient condition for the existence of non-trivial solutions of (1.8) vanishing at infinity (see Appendix A in [15] or [6]). Guo and Wu [22] proved that the soliton u ω,c is orbitally stable when ω > c 2 /4 and c < 0 by applying the abstract theory of Grillakis, Shatah, and Strauss [19,20].…”

“…Recently, Fukaya, Hayashi and Inui [15] gave a sufficient condition for global existence by using potential well theory; if the initial data u 0 of (1.3) satisfying the following condition that there exists (ω, c) satisfying (1.10) such that…”