On the Semiclassical Limit of the General Modified NLS Equation

Abstract

We study the semiclassical limit of the so-called general modified nonlinear Schrodinger equation for initial data with Sobolev regularity, before shocks appear in the limit system. The strict hyperbolicity and genuine nonlinearity are proved for the dispersion limit of the cubic nonlinear case. The limiting transition from the MNLS equation to the NLS equation is also discussed. ᮊ

WKB analysis of generalized derivative nonlinear Schrödinger equations without hyperbolicity

“…In the case g = 0, the semi-classical limit for (1.1) was considered in [9] (case f = 0) and [8] (with f, g ∈ C ∞ (R + ; R)), by considering (1.7). However, in the case where g = 0, hyperbolicity is not a property that one has for free.…”

Shock waves, chiral solitons and semiclassical limit of one-dimensional anyons

“…This was proved rigorously by Jin et al [13] in one dimension using the inverse scattering technique. For derivative nonlinear Schr€ o odinger equation (DNLS for short), the limit behavior is described by the modified Euler equation [5,6,17,21].…”

“…The semiclassical limit of the JNLS equation (1.1a,b) can be discussed in the same strategy as GrenierÕs [8] for NLS equation (see also [5,6,17] for DNLS equations and [19,20] for Schr€ o odinger-Poisson system). Similar to the derivative nonlinear Schr€ o odinger equation [5][6][7]23], the k term in JNLS equation (1.1a,b) is not invariant under Galilean transformation, and it flips sign under parity (x !…”

“…Similar to the derivative nonlinear Schr€ o odinger equation [5][6][7]23], the k term in JNLS equation (1.1a,b) is not invariant under Galilean transformation, and it flips sign under parity (x ! Àx), i.e., the JNLS equation does not possess parity and Galilean invariance and therefore the canonical momentum is not conservative [15][16][17]21,22].…”

“…Although JNLS equation is similar to the DNLS equation, they are quit different from each other. For DNLS equation, even the MNLS equation [6,7,17], the associated Riemann invariant forms are given by simple and beautiful formulas. However, for JNLS equation we need to solve a fourth order equation, the associated Riemann form are ugly.…”