Spectral analysis of Darboux transformations for the focusing NLS hierarchy

Abstract

We study Darboux-type transformations associated with the focusing nonlinear Schrodinger equation (NLS_) and their effect on spectral properties of the underlying Lax operator. The latter is a formallyJ (but nonself-adjoint) Diract-type differential expression of the form 1 $$M(q) = i\left( {\begin{array}{*{20}c} {\frac{d}{{dx}}} & { - q} { - \bar q} & { - \frac{d}{{dx}}} \end{array} } \right)$$ satisfying $${\mathcal{J}} M(q)\mathcal{J} = M(q)^* $$ , whereJ is defined byJ C, andC denotes the antilinear conjugation map in ℂ2, $${\mathcal{C}}(a,b)^{\rm T} = (\bar a,\bar b)^{\rm T} ,a,b \in $$ ℂ. As one of our principla results, we prove that under the most general hypothesisq ∈ loc 1 (ℝ) onq, the maximally defined operatorD(q) generated byM(q) is actually {itJ}-self-adjoint in inL 2(ℝ)2. Moreover, we establish the existence of Weyl-Titchmarsh-type solutions ψ+(z, ·) ∃L 2 ([R, ∞))2 and ψ−(z, ·) ∈L 2 ((−∞,R]) for allR∈ℝ ofM(q)Ψ ± (z)=zΨ ± (z) forz in the resolvent set ofD(q). The Darboux transformations considered in this paper are the analogue of the double commutation procedure familiar in the KdV and Schrodinger operator contexts. As in the corresponding case of Schrodinger operators, the Darboux transformations in question guarantee that the resulting potentialsq are locally nonsingular. Moreover, we prove that the construction ofN-soliton NLS_potentialsq (N) with respect to a general NLS background potentialq ∃L loc 1 (ℝ), associated with the Dirac-type operatorsD(q (N) ) andD(q), respectively, amounts to the insertio ofN complex conjugate pairs ofL 2({ℝ}2-eigenvalues $$\{ z_1 ,\bar z_1 ,...,z_N ,\bar z_N \} $$ into the spectrum σ(D(q)) ofD(q), leaving the rest of the spectrum (especially, the essential spectrum σe(itD)(q))) invariant, that is, 1 $$\sigma (D(q^{(N)} )) = \sigma (D(q)) \cup \{ z_1 ,\bar z_1 ,...,z_N ,\bar z_N \} ,$$ 1 $$\sigma _e (D(q^{(N)} )) = \sigma _e (D(q))$$ These results are obtained by establishing the existence of bounded transformation operators which intertwine the background Dirac operatorD(q) and the Dirac operatorD(q (N) ) obtained afterN Darboux-type transformations.