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.
We consider the following nonlinear Schrödinger equation with derivative: (1) iut = −uxx − i|u| 2 ux − b|u| 4 u, (t, x) ∈ R × R, b ∈ R. If b = 0, this equation is a gauge equivalent form of the well-known derivative nonlinear Schrödinger (DNLS) equation. The equation (1) for b ≥ 0 has degenerate solitons whose momentum and energy are zero, and if b = 0, they are algebraic solitons. Inspired from the works [29, 8] on instability theory of the L 2-critical generalized KdV equation, we study the instability of degenerate solitons of (1) in a qualitative way, and when b > 0, we obtain a large set of initial data yielding the instability. The arguments except one step in our proof work for the case b = 0 in exactly the same way, and in particular the unstable directions of algebraic solitons are detected. This is a step towards understanding the dynamics around algebraic solitons of the DNLS equation. Contents
We study uniqueness and nondegeneracy of ground states for stationary nonlinear Schrödinger equations with a focusing power-type nonlinearity and an attractive inverse-power potential. We refine the results of Shioji and Watanabe (2016) and apply it to prove the uniqueness and nondegeneracy of ground states for our equations. We also discuss the orbital instability of ground state-standing waves.
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