Spectral analysis of Darboux transformations for the focusing NLS hierarchy

Abstract

Abstract. We study Darboux-type transformations associated with the focusing nonlinear Schrödinger equation (NLS − ) and their effect on spectral properties of the underlying Lax operator. The latter is a formally J -selfadjoint (but non-self-adjoint) Dirac-type differential expression of the formAs one of our principal results we prove that under the most general hypothesis q ∈ L 1 loc (R) on q, the maximally defined operatorThe Darboux transformations considered in this paper are the analog of the double commutation procedure familiar in the KdV and Schrödinger operator contexts. As in the corresponding case of Schrödinger operators, the Darboux transformations in question guarantee that the resulting potentials q are locally nonsingular. Moreover, we prove that the construction of Nsoliton NLS − potentials q (N) with respect to a general NLS − background potential q ∈ L 1 loc (R), associated with the Dirac-type operators D q (N) and D(q), respectively, amounts to the insertion of N complex conjugate pairs of L 2 (R) 2 -eigenvalues {z 1 , z 1 , . . . , z N , z N } into the spectrum σ(D(q)) of D(q), leaving the rest of the spectrum (especially, the essential spectrum σe(D(q))) invariant, that is,These results are obtained by establishing the existence of bounded transformation operators which intertwine the background Dirac operator D(q) and the Dirac operator D q (N) obtained after N Darboux-type transformations.

On the GBDT Version of the Bäcklund-Darboux Transformation and its Applications to Linear and Nonlinear Equations and Weyl Theory

“…Bäcklund-Darboux transformations (BDTs) are well-known as a versatile tool in spectral theory as well as for integrable nonlinear equations (see, for instance, [3,7,9,13,14,15,19,25,34,42,43,44,47,49,51,85] and references therein). BDT transforms initial equation or system into another one from the same class and transforms also solutions of the initial equation into solutions of the transformed one.…”

J-self-adjointness of a class of Dirac-type operators

“…In this paper we present an extension of the result in [3], for 2m × 2m matrix-valued Dirac-type differential expressions of the form…”

“…Recently, it has been proved in [3] that the 2 × 2 matrix-valued Lax differential expression (10) corresponding to the scalar focusing NLS hierarchy defines, under the most general hypothesis q ∈ L 1 loc (R) on the potential q, a J -self-adjoint operator in L 2 (R) 2 , where J is the antilinear conjugation, J = σ 1 C, with σ 1 = 0 1 1 0 and C is complex conjugation in C 2 . This is the direct analog of a recently proven fact in [4,Lemma 2.15] that the Dirac-type Lax differential expression in the defocusing nonlinear Schrödinger (NLS) case is always in the limit point case at ±∞.…”

Non-self-adjoint Dirac-type Systems and Related Nonlinear Equations: Wave Functions, Solutions, and Explicit Formulas

“…See also [8,40,42] for the case m 1 = m 2 . The non-self-adjoint case is equally important [1,4,14] and of growing current interest (see, for instance, [3,10,17,19,21] and references therein). The explicit formulas for the wave functions and solutions of the Diractype and nonlinear equations are of special interest.…”