Barbara Prinari

Towards an inverse scattering theory for non-decaying potentials of the heat equation

The resolvent approach is applied to the spectral analysis of the heat equation with non-decaying potentials. The special case of potentials with spectral data obtained by a rational similarity transformation of the spectral data of a generic decaying potential is considered. It is shown that these potentials describe N solitons superimposed by Backlund transformations to a generic background. Dressing operators and Jost solutions are constructed by solving a -problem explicitly in terms of the corresponding objects associated with the original potential. Regularity conditions of the potential in the cases N = 1 and 2 are investigated in detail. The singularities of the resolvent for the case N = 1 are studied, opening the way to a correct definition of the spectral data for a generically perturbed soliton.

Soliton interactions in the vector NLS equation

Collisions of solitons for two coupled and N-coupled NLS equation are investigated from various viewpoints. By suitably employing Manakovs well-known formulae for the polarization shift of interacting vector solitons, it is shown that the multisoliton interaction process is pairwise and the net result of the interaction is independent of the order in which such collisions occur. Further, this is shown to be related to the fact that the map determining the interaction of two solitons with nontrivial internal degrees of freedom (e.g. vector solitons) satisfies the Yang–Baxter relation. The associated matrix factorization problem is discussed in detail. Soliton interactions are also described in terms of linear fractional transformations, and the problem of existence of a solution for a basic three-collision gate, which has recently been introduced, is analysed.

Inverse scattering transform for the integrable discrete nonlinear Schrödinger equation with nonvanishing boundary conditions

The inverse scattering transform for an integrable discretization of the defocusing nonlinear Schrodinger equation with nonvanishing boundary values at infinity is constructed. This problem had been previously studied, and many key results had been established. Here, a suitable transformation of the scattering problem is introduced in order to address the open issue of analyticity of eigenfunctions and scattering data. Moreover, the inverse problem is formulated as a Riemann–Hilbert problem on the unit circle, and a modification of the standard procedure is required in order to deal with the dependence of asymptotics of the eigenfunctions on the potentials. The discrete analog of Gel’fand–Levitan–Marchenko equations is also derived. Finally, soliton solutions and solutions in the small-amplitude limit are obtained and the continuum limit is discussed. (Some figures in this article are in colour only in the electronic version)

Inverse scattering theory of the heat equation for a perturbed one-soliton potential

The inverse scattering theory of the heat equation is developed for a special subclass of potentials nondecaying at space infinity—perturbations of the one-soliton potential by means of decaying two-dimensional functions. Extended resolvent, Green’s functions, and Jost solutions are introduced and their properties are investigated in detail. The singularity structure of the spectral data is given and then the inverse problem is formulated in an exact distributional sense.

The inverse scattering transform for the focusing nonlinear Schrödinger equation with asymmetric boundary conditions

The inverse scattering transform (IST) as a tool to solve the initial-value problem for the focusing nonlinear Schrodinger (NLS) equation with non-zero boundary values ql/r(t)≡Al/re−2iAl/r2t+iθl/r as x → ∓∞ is presented in the fully asymmetric case for both asymptotic amplitudes and phases, i.e., with Al ≠ Ar and θl ≠ θr. The direct problem is shown to be well-defined for NLS solutions q(x, t) such that q(x,t)−ql/r(t)∈L1,1(R∓) with respect to x for all t ⩾ 0, and the corresponding analyticity properties of eigenfunctions and scattering data are established. The inverse scattering problem is formulated both via (left and right) Marchenko integral equations, and as a Riemann-Hilbert problem on a single sheet of the scattering variables λl/r=k2+Al/r2, where k is the usual complex scattering parameter in the IST. The time evolution of the scattering coefficients is then derived, showing that, unlike the case of solutions with equal amplitudes as x → ±∞, here both reflection and transmission coefficients have ...

The Three-Component Defocusing Nonlinear Schrödinger Equation with Nonzero Boundary Conditions

We present a rigorous theory of the inverse scattering transform (IST) for the three-component defocusing nonlinear Schrödinger (NLS) equation with initial conditions approaching constant values with the same amplitude as

Building an extended resolvent of the heat operator via twisting transformations

{x\to\pm\infty}

Dark-bright soliton solutions with nontrivial polarization interactions for the three-component defocusing nonlinear Schrödinger equation with nonzero boundary conditions

x→±∞. The theory combines and extends to a problem with non-zero boundary conditions three fundamental ideas: (i) the tensor approach used by Beals, Deift and Tomei for the n-th order scattering problem, (ii) the triangular decompositions of the scattering matrix used by Novikov, Manakov, Pitaevski and Zakharov for the N-wave interaction equations, and (iii) a generalization of the cross product via the Hodge star duality, which, to the best of our knowledge, is used in the context of the IST for the first time in this work. The combination of the first two ideas allows us to rigorously obtain a fundamental set of analytic eigenfunctions. The third idea allows us to establish the symmetries of the eigenfunctions and scattering data. The results are used to characterize the discrete spectrum and to obtain exact soliton solutions, which describe generalizations of the so-called dark-bright solitons of the two-component NLS equation.