Stability of multi-solitons in the cubic NLS equation

Abstract

We address the stability of multi-solitons for the cubic nonlinear Schrödinger (NLS) equation on the line. By using the dressing transformation and the inverse scattering transform methods, we establish the orbital stability of multi-solitons in the L 2 (R) space when the initial data is in a weighted L 2 (R) space.

L2orbital stability of Dirac solitons in the massive Thirring model

“…Our approach follows the strategy used by Mizumachi and Pelinovsky in [25] for proving L 2 -orbital stability of the NLS solitons. Their result was extended by Contreras and Pelinovsky in [11] to multi-solitons of the NLS equation by using a more general dressing transformation. Furthermore, the recent work [12] of Cuccagna and Pelinovsky shows how an asymptotic stability of the NLS solitons can be deduced by combining the auto-Bäcklund transformation and the nonlinear steepest descent method.…”

Rogue periodic waves of the focusing nonlinear Schrödinger equation

“…The N-fold transformation for the NLS equation was derived and justified in [29] by using the dressing method. Adopting the present notations with N = 1, [29] and σ 1 and σ 3 are standard Pauli matrices, we obtain the onefold transformation in the explicit formũ…”

“…By solving the linear system of the dressing method obtained in [29], we obtain solutions r 1,2 of the linear system (1.2)-(1.3) with λ 1,2 and the new potentialũ, where r 1 , r 2 andũ are defined in the form…”

“…This solution was used in [29] to inspect two-soliton solutions of the NLS equation (1.1). By using the non-periodic solutions of the linear system (1.2)-(1.3) and the Darboux transformations (3.13) and (3.14), we can finally obtain the exact solutions for the rogue periodic waves of the NLS equation (1.1) in the sense of definition (1.11).…”

“…To obtain the twofold Darboux transformation by using the formalism of [29], we set N = 2, λ 1,2 = −iz 1,2 , (p 1,2 , q 1,2 ) = σ 3 σ 1s1,2 and the transformation matrix…”