On stabilization of small solutions in the nonlinear Dirac equation with a trapping potential

Abstract

We consider a Dirac operator with short range potential and with eigenvalues. We add a nonlinear term and we show that the small standing waves of the corresponding nonlinear Dirac equation (NLD) are attractors for small solutions of the NLD. This extends to the NLD results already known for the Nonlinear Schrödinger Equation (NLS).

Solitary Waves in the Nonlinear Dirac Equation

“…These assumptions are verified in some perturbative context with V = 0. This case is analyzed in [45]. The asymptotic stability approach from [17,40,127] is developed under important restrictions on the types of admissible perturbations.…”

On small energy stabilization in the NLKG with a trapping potential

“…As discussed in [9], the approach in [32] involving guesses on the trajectory of a solution, the turns of the solution away from unstable standing waves and, usually, its final convergence either to 0 or to a stable standing wave, appears a considerably difficult task under our hypothesis (H2), which is much more complex combinatorially than the situation in [32]. In this paper we frame the problem as in [9] and extend to the NLKG equation the NLS result obtained in [9], exactly as in [11] the result of [9] has been extended to Dirac equations. It turns out that the NLKG presents no significant new problems with respect to the NLS.…”

“…Such invariant spaces exist near the origin also for the nonlinear problem (1.1) in the form of topological disks which are tangent to the linear spaces at the origin. In the case of the nonlinear Schrödinger equation, [9], and of the nonlinear Dirac equation, [11], it as been shown that the union of these disks is an attractor for all small energy solutions. We will prove the same result for (1.1).…”

“…In the meantime, h scatters because one can apply Strichartz estimates to (1.17). So the work in all the papers [2,9,11], as well as here, consists in finding a coordinate system where (1.1) has hamiltonian that, up to negligible error terms, is similar to (1.16). In [2] this is simpler because there the attractor set is formed just by the vacuum solution of (1.1).…”

Solutions with Compact Time Spectrum to Nonlinear Klein–Gordon and Schrödinger Equations and the Titchmarsh Theorem for Partial Convolution

“…Let us mention the results on convergence of small solutions to (one-frequency) solitary waves, particularly in the context of the nonlinear Schrödinger equation: in other words, the attractor of small solutions is formed by small amplitude solitary waves. See in particular [TY02,SW04,CM15,CMP16,CT16].…”