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Abstract
We consider quadratic bundles related to Hermitian symmetric spaces of the type SU(m+n)/S(U(m)×U(n)). We discuss the spectral properties of scattering operator, develop the direct scattering problem associated with it and stress on the effect of reduction on these. By applying a modification of Zakharov-Shabat's dressing procedure we demonstrate how one can obtain reflectionless potentials. That way one is able to generate soliton solutions to the nonlinear evolution equations belonging to the integrable hierarchy associated with quadratic bundles under study.
Highlights
Derivative nonlinear Schrodinger equation (DNLS)iqt + qxx + i(|q|2q)x = 0 (1)is one of classical S-integrable nonlinear evolution equations (NLEE)
DNLS is deeply connected to 2-dimensional Thirring model [13, 15] and the Gerdjikov-Ivanov equation [5, 6], both related to certain reductions of quadratic bundle L operator of generic form: L(λ) = i∂x + U0(x, t) + λU1(x, t) − λ2σ3 (4)
We have formulated the direct scattering problem for quadratic bundles related to Hermitian symmetric spaces of the type A.III in terms Jost solutions, scattering matrix, fundamental analytic solutions etc
Summary
Is one of classical S-integrable nonlinear evolution equations (NLEE). It occurs in plasma physics to describe the propagation of nonlinear Alfven waves with circular polarization [18, 19]. DNLS is deeply connected to 2-dimensional Thirring model [13, 15] and the Gerdjikov-Ivanov equation [5, 6], both related to certain reductions of quadratic bundle L operator of generic form: L(λ) = i∂x + U0(x, t) + λU1(x, t) − λ2σ3. Derivation and study of multicomponent generalizations of classical scalar integrable equations is a trend of current interest [7, 8, 11] in theory of integrable systems. In section we show how one can modify Zakharov-Shabat’s dressing method for the case of quadratic bundles This allows one to generate special types of solutions in an algebraic manner. Last section contains summary of our results and some additional remarks
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