What question did this study set out to answer?

This paper aims to outline the global dynamics of threshold odd solutions to the nonlinear Schrodinger equation on the line.

June 20, 2026Open Access

Threshold odd solutions to NLS on the line (Harmonic Analysis and Nonlinear Partial Differential Equations)

Key Points

This paper aims to outline the global dynamics of threshold odd solutions to the nonlinear Schrodinger equation on the line.
Reviewed dynamics of threshold odd solutions in the context of nonlinear Schrodinger equations on the real line.
Defined key conserved quantities: energy and mass associated with the solutions.
Confirmed that the nonlinear Schrodinger equation is locally well-posed in H1(R).
Established that energy E and mass M are conserved during the evolution of solutions.

Abstract

This is a resume of our previous paper 10 for the global dynamics of the threshold odd solutions to the nonlinear Schrodinger equation on the line. § 1. IntroductionWe consider the following nonlinear Schrodinger equation on the real line: where u = u (t, x): I x R 一 C, u° is a given function in H1 (R), and p > 5. It is well known that (NLS) is locally well-posed in H 1 (R). Moreover, the energy E and the mass M, which are defined by E (u) ：=1WIH2 -时L++1and M (u) ：= 11训务, 2 P十丄 are conserved. Here, uf: = 3xu. A blow-up alternative holds, that is, if the forward maximal existence time Tᵃx is finite, then lim^-T -0 |" (圳|か=oo. See e. g. 4. For more general NLS (1. 1) idtu + Au 十 |u|p-1 u = 0, (t, x) e I x Rd

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