arXiv subjects

S. Kamvissis

Publications and source records attributed to S. Kamvissis.

4 recordsLinked to original sources

arXiv2026

Lax Pairs: Integrable, Less Integrable and Nonintegrable Systems

Completely integrable finite dimensional Hamiltonian systems are well understood thanks to the work of Liouville and Arnold. On the other hand, the Lax Pair formulation of the KdV equation marks the beginning of the extension of the completely integrable theory to infinite dimensional Hamiltonian systems. Solutions of initial value problems for systems that admit a Lax Pair formulation normally have a tame qualitative behavior if Lax Pairs give rise to an infinite complete set of conserved laws. The situation is different for initial-boundary value problems, even in one space dimension. There are problems where integrability persists and regular (long time asymptotic) behavior can be proven (and we have proven it). There are others where even irregular "fractal-chaotic-looking" behavior can appear. In between there are problems with very explicit solutions that demonstrate the instability of the related Dirichlet-to-Neumann map. In this short article we review an instance of each case. We also make a connection with results from the existing theory of perturbed Lax Pair equations on the real line.

nlin.SI↗

arXiv2026

The Focusing NLS Equation on the Half-Line with Periodic Boundary Conditions

We consider the Dirichlet problem for the focusing NLS equation on the half-line, with given Schwartz initial data and boundary data $q(0,t)$ equal to an exponentially decaying perturbation $u(t)$ of the periodic boundary data $ a e^{2iωt + i ε}$ at $x=0.$ It is known from PDE theory that this problem admits a unique solution (for fixed initial data and fixed $u$). On the other hand, the associated inverse scattering transform formalism involves the Neumann boundary value for $x=0$. Thus the implementation of this formalism requires the understanding of the "Dirichlet-to-Neumann" map which characterises the associated Neumann boundary value. We consider this map in an indirect way: we postulate a certain Riemann-Hilbert problem, on a specified contour but with partially unspecified jump data of some generality, and then prove that the solution of the initial-boundary value problem for the focusing NLS constructed through this Riemann-Hilbert problem satisfies all the required properties: the data $q(x,0)$ are Schwartz and $q(0,t)-a e^{2iωt + i ε}$ is exponentially decaying. More specifically, we focus on the case $-3a^2 < ω< a^2.$ By considering a large class of appropriate scattering data for the t-problem, we provide solutions of the above Dirichlet problem such that the data $q_x(0,t)$ is given by an exponentially decaying perturbation of the function $2iab e^{2iωt + i ε},$ where $ω= a^2-2b^2,~~b>0$.

math.AP↗

arXiv2021

On Soliton Resolution for a Lattice

The soliton resolution conjecture for evolution PDEs of dispersive type states (vaguely) that generic initial data of finite energy give rise asymptotically to a set of receding solitons and a decaying background radiation. In this letter, we investigate a possible extension of this conjecture to discrete lattices of the Fermi-Pasta-Ulam-Tsingou type (rather than PDEs) in two cases: the case of finite energy initial data and a more general case where the initial data are a short range perturbation of a periodic function. In the second case, inspired by rigorous results on the Toda lattice, we suggest that the soliton resolution phenomenon is replaced by something somewhat more complicated: a short range perturbation of a periodic function actually gives rise to different phenomena in different regions. Apart from regions of (asymptotically) pure periodicity and regions of solitons in a periodic background, we also observe "modulated" regions of fast oscillations with slowly varying parameters like amplitude and phase. We have conducted some numerical calculations to investigate if this trichotomy (pure periodicity + solitons + modulated oscillations) persists for any discrete lattices of the Fermi-Pasta-Ulam-Tsingou type. For small perturbations of integrable lattices like the linear harmonic lattice, the Langmuir chain and the Toda lattice, this is true. But in general even chaotic phenomena can occur.

math-ph↗

arXiv2000

Semiclassical Soliton Ensembles for the Focusing Nonlinear Schroedinger Equation

We present a new generalization of the steepest descent method introduced by Deift and Zhou for matrix Riemann-Hilbert problems and use it to study the semiclassical limit of the focusing nonlinear Schroedinger equation with real analytic, even, bell-shaped initial data. We provide explicit strong locally uniform asymptotics for a sequence of exact solutions corresponding to initial data that has been modified in an asymptotically small sense. We call this sequence of exact solutions a semiclassical soliton ensemble. Our asymptotics are valid in regions of the (x,t) plane where a certain scalar complex phase function can be found. We characterize this complex phase function directly by a finite-gap ansatz and also via the critical point theory of a certain functional; the latter provides the correct generalization of the variational principle exploited by Lax and Levermore in their study of the zero-dispersion limit of the Korteweg-de Vries equation.

nlin.SI↗