Searcharxiv⌕ Search

arXiv subjects

Spyridon Kamvissis

Publications and source records attributed to Spyridon Kamvissis.

At least 19 recordsLinked to original sources
arXiv2026

Semiclassical WKB Problem for the non-self-adjoint Dirac operator

We review some recent rigorous results on the semiclassical behavior ($ε\downarrow0$) of the scattering data of a non-self-adjoint Dirac operator with potential $A\exp\{iS/ε\}$ where both $A$ and $S$ are differentiable functions tending to constants as $x \to \pm \infty$. We have either employed the so-called exact WKB method, or the older WKB theory of Olver. Our analysis is motivated by the need to understand the semiclassical behaviour of the focusing cubic NLS equation with initial data $A\exp\{iS/ε\}$, in view of the well-known fact discovered by Zakharov and Shabat that the spectral analysis of the Dirac operator enables us to obtain the solution of the NLS equation via inverse scattering theory.

math.SP↗
arXiv2025

Infinity of solutions to initial-boundary value problems for linear constant-coefficient evolution PDEs on semi-infinite intervals

In this short communication, we announce an algorithmic procedure for constructing non-uniqueness counter-examples of classical solutions to initial-boundary-value problems for a wide class of linear evolution partial differential equations, of any order and with constant coefficients, formulated in a quarter-plane. Our approach relies on analysis of regularity and asymptotic properties, near the boundary of the spatio-temporal domain, of closed-form integral-representation formulae derived via complex-analytic techniques and rigorous implementation of the modern PDE technique known as Fokas unified transform method. In order to elucidate the novel idea and demonstrate the proposed technique in a self-contained fashion, we explicitly present its application to two concrete examples, namely the heat equation and the linear KdV equation with Dirichlet data. New uniqueness theorems for these two models are also presented herein.

math.AP↗
arXiv2024

Continuous dependence on data for linear evolution PDEs on the quarter-plane

In this note, we announce a systematic analysis of continuous dependence on the data in classical spaces for the initial-boundary-value problem of the diffusion equation on the half-line, with data that are not necessarily compatible at the quadrant corner. This is based on a recent approach to rigorously analyzing integral representations derived via the unified transform method of Fokas. No exotic phenomena were discovered in this case, yet our findings appear to be new in the pertinent literature. These results supplement our previous investigations on existence and (non)uniqueness within the framework of well-posedness. The present detailed exposition elucidates the subtleties involved while also demonstrating a generic technique. Applications of the latter to several other IBVPs and PDEs will be reported elsewhere.

math.AP↗
arXiv2024

A note on uniqueness for linear evolution PDEs posed on the quarter-plane

In this paper, we announce a rigorous approach to establishing uniqueness results, under certain conditions, for initial-boundary-value problems for a class of linear evolution partial differential equations (PDEs) formulated in a quarter-plane. We also effectively propose an algorithm for constructing non-uniqueness counter-examples which do not satisfy the said conditions. Our approach relies crucially on the rigorous analysis of regularity and asymptotic properties of integral representations derived formally via the celebrated Unified Transform Method for each such PDE. For uniqueness, this boundary behavior analysis allows for a careful implementation of an energy-estimate argument on the semi-unbounded domain. Our ideas are elucidated via application of the present technique to two concrete examples, namely the heat equation and the linear KdV equation with Dirichlet data on the positive quadrant, under a particular set of conditions at the domain boundary and at infinity. Importantly, this is facilitated by delicate refinement of previous results concerning the boundary behavior analysis of these two celebrated models. In addition, we announce a uniqueness theorem for the linearized BBM equation, whose proof, in a similar spirit, will appear in a forthcoming paper. Finally, we briefly demonstrate how the general case of oblique Robin data can be recast as a Dirichlet problem. To the best of our knowledge, such well-posedness results appear for the first time in either the classical or the weak sense. Extensions to other classes of equations are underway and will appear elsewhere.

math.AP↗
arXiv2022

Semiclassical WKB Problem for the non-self-adjoint Dirac operator with an analytic rapidly oscillating potential

In this paper we examine the semiclassical behavior of the scattering data of a non-self-adjoint Dirac operator with a rapidly oscillating potential that is complex analytic in some neighborhood of the real line. Some of our results are rigorous and quite general. On the other hand, complete and concrete understanding requires the investigation of the WKB geometry of specific examples. For such detailed computations we use a particular example that has been investigated numerically more than 20 years ago by Bronski and Miller and rely heavily on their numerical computations. Mostly employing the exact WKB method, we provide the complete rigorous uniform semiclassical analysis of the Bohr-Sommerfeld condition for the location of the eigenvalues across unions of analytic arcs as well as the associated norming constants. For the reflection coefficient as well as the eigenvalues near 0 in the spectral plane, we employ instead an older theory that has been developed in great detail by Olver. Our analysis is motivated by the need to understand the semiclassical behaviour of the focusing cubic NLS equation with initial data $A\exp\{iS/ε\}$, in view of the well-known fact discovered by Zakharov and Shabat that the spectral analysis of the Dirac operator enables the solution of the NLS equation via inverse scattering theory.

math-ph↗
arXiv2021

Semiclassical WKB problem for the non-self-adjoint Dirac operator with a decaying potential

In this paper we examine the semiclassical behaviour of the scattering data of a non-self-adjoint Dirac operator with a fairly smooth but not necessarily analytic potential decaying at infinity. In particular, using ideas and methods going back to Langer and Olver, we provide the complete rigorous uniform semiclassical analysis of the scattering coefficients away (slightly) from zero, and the rigorous proof of the uniform Bohr-Sommerfeld condition for the location of the eigenvalues. Our analysis is motivated by the potential applications to the focusing cubic NLS equation, in view of the well-known fact discovered by Zakharov and Shabat that the spectral analysis of the Dirac operator is the basis of the solution of the NLS equation via inverse scattering theory. This paper complements and extends a previous work of Fujiié and the second author, which considered a more restricted problem for a strictly analytic potential.

math-ph↗
arXiv2021

Semiclassical WKB Problem for the Non-Self-Adjoint Dirac Operator with a Multi-Humped Decaying Potential

In this paper we continue the study (initiated in arXiv:2003.13584) of the semiclassical behavior of the scattering data of a non-self-adjoint Dirac operator with a real, positive, fairly smooth but not necessarily analytic potential decaying at infinity; in this paper we allow this potential to have several local maxima and minima. We provide the rigorous semiclassical analysis of the Bohr-Sommerfeld condition for the location of the eigenvalues, the norming constants, and the reflection coefficient.

math-ph↗
arXiv2019

Semiclassical WKB problem for the non-self-adjoint Dirac operator with analytic potential

In this paper we examine the semiclassical behaviour of the scattering data of a non-self-adjoint Dirac operator with analytic potential decaying at infinity. In particular, employing the exact WKB method, we provide the complete rigorous uniform semiclassical analysis of the reflection coefficient and the Bohr-Sommerfeld condition for the location of the eigenvalues. Our analysis has some interesting consequences concerning the focusing cubic NLS equation, in view of the well-known fact discovered by Zakharov and Shabat that the spectral analysis of the Dirac operator is the basis of the solution of the NLS equation via inverse scattering theory.

math-ph↗
arXiv2016

On the Dirichlet to Neumann Problem for the 1-dimensional Cubic NLS equation on the Half-Line; Non-zero Initial Data

Initial-boundary value problems for 1-dimensional `completely integrable' equations can be solved via an extension of the inverse scattering method, which is due to Fokas and his collaborators. A crucial feature of this method is that it requires the values of more boundary data than given for a well-posed problem. In the case of cubic NLS, knowledge of the Dirichet data suffices to make the problem well-posed but the Fokas method also requires knowledge of the values of Neumann data. The study of the Dirichlet to Neumann map is thus necessary before the application of the `Fokas transform'. In this paper, we provide a rigorous study of this map for a large class of decaying Dirichlet data. We show that the Neumann data are also sufficiently decaying and hence that the Fokas method can be applied. This is an addendum to a previous paper. We consider the case of non-zero initial data, thus completing the discussion of the previous paper.

math.AP↗
arXiv2016

On the Dirichlet to Neumann Problem for the 1-dimensional Cubic NLS equation on the Half-Line; Zero Initial Data

Initial-boundary value problems for 1-dimensional `completely integrable' equations can be solved via an extension of the inverse scattering method, which is due to Fokas and his collaborators. A crucial feature of this method is that it requires the values of more boundary data than given for a well-posed problem. In the case of cubic NLS, knowledge of the Dirichet data suffices to make the problem well-posed but the Fokas method also requires knowledge of the values of Neumann data. The study of the Dirichlet to Neumann map is thus necessary before the application of the `Fokas transform'. In this paper, we provide a rigorous study of this map for a large class of decaying Dirichlet data. We show that the Neumann data are also sufficiently decaying and hence that the Fokas method can be applied. For simplicity we considered here the case of zero initial data. An addendum will follow, discussing the case of non-zero initial data.

math.AP↗
arXiv2016

Existence and Regularity for an Energy Maximization Problem in Two Dimensions

We consider the variational problem of maximizing the weighted equilibrium Green's energy of a distribution of charges free to move in a subset of the upper half-plane, under a particular external field. We show that this problem admits a solution and that, under some conditions, this solution is an S-curve (in the sense of Gonchar-Rakhmanov). The above problem appears in the theory of the semiclassical limit of the integrable focusing nonlinear Schrödinger equation. In particular, its solution provides a justification of a crucial step in the asymptotic theory of nonlinear steepest descent for the inverse scattering problem of the associated linear non-self-adjoint Zakharov-Shabat operator and the equivalent Riemann-Hilbert factorization problem.

math.CV↗
arXiv2014

Robin boundary condition and shock problem for the focusing nonlinear Schrödinger equation

We consider the initial boundary value (IBV) problem for the focusing nonlinear Schrödinger equation in the quarter plane $x>0,t>0$ in the case of periodic initial data (at $t=0$) and a Robin boundary condition at $x=0$. Our approach is based on the simultaneous spectral analysis of the Lax pair equations combined with symmetry considerations for the corresponding Riemann-Hilbert problems. A connection between the original IBV problem and an associated initial value (IV) problem is established.

nlin.SI↗
arXiv2012

Stability of the periodic Toda lattice under short range perturbations

We consider the stability of the periodic Toda lattice (and slightly more generally of the algebro-geometric finite-gap lattice) under a short range perturbation. We prove that the perturbed lattice asymptotically approaches a modulated lattice. More precisely, let $g$ be the genus of the hyperelliptic curve associated with the unperturbed solution. We show that, apart from the phenomenon of the solitons travelling on the quasi-periodic background, the $n/t$-pane contains $g+2$ areas where the perturbed solution is close to a finite-gap solution in the same isospectral torus. In between there are $g+1$ regions where the perturbed solution is asymptotically close to a modulated lattice which undergoes a continuous phase transition (in the Jacobian variety) and which interpolates between these isospectral solutions. In the special case of the free lattice ($g=0$) the isospectral torus consists of just one point and we recover the known result. Both the solutions in the isospectral torus and the phase transition are explicitly characterized in terms of Abelian integrals on the underlying hyperelliptic curve. Our method relies on the equivalence of the inverse spectral problem to a matrix Riemann--Hilbert problem defined on the hyperelliptic curve and generalizes the so-called nonlinear stationary phase/steepest descent method for Riemann--Hilbert problem deformations to Riemann surfaces.

nlin.SI↗
arXiv2010

Stability of the Periodic Toda Lattice: Higher Order Asymptotics

In a recent paper we have considered the long time asymptotics of the periodic Toda lattice under a short range perturbation and we have proved that the perturbed lattice asymptotically approaches a modulated lattice. In the present paper we capture the higher order asymptotics, at least away from some resonance regions. In particular we prove that the decay rate is $O(t^{-1/2})$. Our proof relies on the asymptotic analysis of the associated Riemann-Hilbert factorization problem, which is here set on a hyperelliptic curve. As in previous studies of the free Toda lattice, the higher order asymptotics arise from "local" Riemann-Hilbert factorization problems on small crosses centered on the stationary phase points. We discover that the analysis of such a local problem can be done in a chart around each stationary phase point and reduces to a Riemann--Hilbert factorization problem on the complex plane. This result can then be pulled back to the hyperelliptic curve.

nlin.SI↗
arXiv2009

Semiclassical Focusing NLS with Barrier Data

We study the semiclassical behavior of the focusing nonlinear Schroedinger equation in 1+1-dimensions under discontinuous "barrier" data and we describe the violent oscillations arising in terms of theta functions. The construction of proofs relies on the analysis of the associated Riemann-Hilbert problem.

math-ph↗
arXiv2008

From Stationary Phase to Steepest Descent

Our aim here is to clarify the distinction between the nonlinear-stationary-phase idea and the nonlinear-steepest-descent idea, stressing the importance of actual steepest-descent contours in some problems. We mostly use the nonlinear Schrödinger equation as our working model, but we also digress to the KdV equation at some point. This is a slightly revised version of a review paper that will appear in the forthcoming volume (Contemporary Mathematics, AMS) honoring Percy Deift.

math-ph↗
arXiv2006

Stability of Periodic Soliton Equations under Short Range Perturbations

We consider the stability of (quasi-)periodic solutions of soliton equations under short range perturbations and give a complete description of the long time asymptotics in this situation. We show that, apart from the phenomenon of the solitons travelling on the quasi-periodic background, the perturbed solution asymptotically approaches a modulated solution. We use the Toda lattice as a model but the same methods and ideas are applicable to all soliton equations in one space dimension. More precisely, let $g$ be the genus of the hyperelliptic Riemann surface associated with the unperturbed solution. We show that the $n/t$-pane contains $g+2$ areas where the perturbed solution is close to a quasi-periodic solution in the same isospectral torus. In between there are $g+1$ regions where the perturbed solution is asymptotically close to a modulated lattice which undergoes a continuous phase transition (in the Jacobian variety) and which interpolates between these isospectral solutions. In the special case of the free solution ($g=0$) the isospectral torus consists of just one point and we recover the classical result. Both the solutions in the isospectral torus and the phase transition are explicitly characterized in terms of Abelian integrals on the underlying hyperelliptic Riemann surface.

nlin.SI↗