112 research outputs found
Initial Value Problems for Integrable Systems on a Semi-Strip
Two important cases, where boundary conditions and solutions of the
well-known integrable equations on a semi-strip are uniquely determined by the
initial conditions, are rigorously studied in detail. First, the case of
rectangular matrix solutions of the defocusing nonlinear Schr\"odinger equation
with quasi-analytic boundary conditions is dealt with. (The result is new even
for a scalar nonlinear Schr\"odinger equation.) Next, a special case of the
nonlinear optics (-wave) equation is considered.Comment: Boundary conditions are recovered from the initial ones. The paper
supplements in this respect our previous article arXiv:1403.8111, where
initial conditions are recovered from the boundary condition
Skew-self-adjoint discrete and continuous Dirac type systems: inverse problems and Borg-Marchenko theorems
New formulas on the inverse problem for the continuous skew-self-adjoint
Dirac type system are obtained. For the discrete skew-self-adjoint Dirac type
system the solution of a general type inverse spectral problem is also derived
in terms of the Weyl functions. The description of the Weyl functions on the
interval is given. Borg-Marchenko type uniqueness theorems are derived for both
discrete and continuous non-self-adjoint systems too
Nonisospectral integrable nonlinear equations with external potentials and their GBDT solutions
Auxiliary systems for matrix nonisospectral equations, including coupled NLS
with external potential and KdV with variable coefficients, were introduced.
Explicit solutions of nonisospectral equations were constructed using the GBDT
version of the B\"acklund-Darboux transformation
Second harmonic generation: Goursat problem on the semi-strip and explicit solutions
A rigorous and complete solution of the initial-boundary-value (Goursat)
problem for second harmonic generation (and its matrix analog) on the
semi-strip is given in terms of the Weyl functions. A wide class of the
explicit solutions and their Weyl functions is obtained also.Comment: 20 page
General-type discrete self-adjoint Dirac systems: explicit solutions of direct and inverse problems, asymptotics of Verblunsky-type coefficients and stability of solving inverse problem
We consider discrete self-adjoint Dirac systems determined by the potentials
(sequences) such that the matrices are positive definite and
-unitary, where is a diagonal matrix and has entries
and entries () on the main diagonal. We construct
systems with rational Weyl functions and explicitly solve inverse problem to
recover systems from the contractive rational Weyl functions. Moreover, we
study the stability of this procedure. The matrices (in the potentials)
are so called Halmos extensions of the Verblunsky-type coefficients .
We show that in the case of the contractive rational Weyl functions the
coefficients tend to zero and the matrices tend to the indentity
matrix .Comment: This paper is a generalization and further development of the topics
discussed in arXiv:math/0703369, arXiv:1206.2915, arXiv:1508.07954,
arXiv:1510.0079
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