267 research outputs found
Closed Form Solutions of Integrable Nonlinear Evolution Equations
2010 Mathematics Subject Classification: 35Q55.In this article we obtain closed form solutions of integrable nonlinear evolution equations associated with the nonsymmetricmatrix Zakharov- Shabat system by means of the inverse scattering transform. These solutions are parametrized by triplets of matrices. Alternatively, the time evolution of the Marchenko integral kernels and direct substitution are employed in deriving these solutions.Research supported by INdAM, MIUR under PRIN grant No. 2008KLJEZ-003, and the Autonomous Region of Sardinia (RAS) under grant CRP3-138, L.R. 7/2007
Quaternion algebra approach to nonlinear Schrödinger equations with nonvanishing boundary conditions
In this article we apply quaternionic linear algebra and quaternionic linear system theory to develop the inverse scattering transform theory for the nonlinear Schrödinger equation with nonvanishing boundary conditions. We also determine its soliton solutions by using triplets of quaternionic matrices
Symmetries for exact solutions to the nonlinear Schr\"odinger equation
A certain symmetry is exploited in expressing exact solutions to the focusing
nonlinear Schr\"odinger equation in terms of a triplet of constant matrices.
Consequently, for any number of bound states with any number of multiplicities
the corresponding soliton solutions are explicitly written in a compact form in
terms of a matrix triplet. Conversely, from such a soliton solution the
corresponding transmission coefficients, bound-state poles, bound-state norming
constants and Jost solutions for the associated Zakharov-Shabat system are
evaluated explicitly. It is also shown that these results hold for the matrix
nonlinear Schr\"odinger equation of any matrix size
The Sturm-Liouville inverse spectral problem with boundary conditions depending on the spectral parameter
We present the complete version including proofs of the results announced in [van der Mee C., Pivovarchik V.: A Sturm-Liouville spectral problem with boundary conditions depending on the spectral parameter. Funct. Anal. Appl. 36 (2002), 315–317 [Funkts. Anal. Prilozh. 36 (2002), 74–77 (Russian)]]. Namely, for the problem of small transversal vibrations of a damped string of nonuniform stiffness with one end fixed we give the description of the spectrum and solve the inverse problem: find the conditions which should be satisfied by a sequence of complex numbers to be the spectrum of a damped string
The continuous classical Heisenberg ferromagnet equation with in-plane asymptotic conditions. I. Direct and inverse scattering theory
We develop the direct and inverse scattering theory of the linear eigenvalue problem associated with the classical Heisenberg continuous equation with in-plane asymptotic conditions. In particular, analyticity of the scattering eigenfunctions and scattering data, and their asymptotic behaviours are derived. The inverse problem is formulated in terms of Marchenko equations, and the reconstruction formula of the potential in terms of eigenfunctions and scattering data is provided
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