Interaction of modulated pulses in the nonlinear Schrödinger equation with periodic potential

Abstract

We consider a cubic nonlinear Schrödinger equation with periodic potential. In a semiclassical scaling the nonlinear interaction of modulated pulses concentrated in one or several Bloch bands is studied. The notion of closed mode systems is introduced which allows for the rigorous derivation of a finite system of amplitude equations describing the macroscopic interaction of these pulses.

Rigorous derivation of nonlinear Dirac equations for wave propagation in honeycomb structures

“…The nonlinear Dirac system (1.6) has been formally derived in [17] and plays the same role as the coupled mode equations derived in [19], or the semi-classical transport equations derived in [5,8]. This becomes even more apparent when we recall that the Bloch waves Φ 1,2 can be written as…”

“…Formal derivation of the Dirac system. In this section, we shall follow the ideas in [8,19], and perform a formal multi-scale expansion of the solution to (1.4) under the Assumption 1. To this end, we seek a solution of the form…”

“…Before we prove the nonlinear stability of our approximation, we recall that it was proved in [19], that the linear Schrödinger group (5.1) S ε (t) := e −iH ε t/ε generated by the periodic Hamiltonian H ε defined in (3.1) satisfies…”

“…The advantage of our scaling is that it yields an asymptotic description for solutions of order O(1) in L ∞ on macroscopic spacetime scales, in contrast to the setting of [1,17], in which the size of the solution varies as ε → 0. Another advantage is that this scaling puts us firmly in the regime of weakly nonlinear, semi-classical NLS with highly oscillatory periodic potentials, which have been extensively studied in, e.g., [5,6,7,8,19], albeit not for the case of honeycomb lattices. From the mathematical point of view, the present work will follow the ideas presented in [19] and adapt them to the current situation.…”

Quadratic-Argument Approach to Nonlinear Schrödinger Equation and Coupled Ones

“…Erdogan and Zharnisky [5] investigated quasi-linear dynamics in the nonlinear Schrödinger equation with periodic boundary condition. Giannoullis, Mielke and Sparber [7] studied the interaction of modulated pulses in the nonlinear Schrödinger equation with periodic potential. Pankov [14] considered the decay of solutions of nonlinear Schrödinger equation.…”