arXiv subjects

Alexander Tovbis

Publications and source records attributed to Alexander Tovbis.

At least 19 recordsLinked to original sources

arXiv2025

Spectral theory for non-self-adjoint Dirac operators with periodic potentials and inverse scattering transform for the focusing nonlinear Schr\"odinger equation with periodic boundary conditions

We formulate the inverse spectral theory for a non-self-adjoint one-dimensional Dirac operator associated periodic potentials via a Riemann-Hilbert problem approach. We use the resulting formalism to solve the initial value problem for the focusing nonlinear Schr\"odinger equation. We establish a uniqueness theorem for the solutions of the Riemann-Hilbert problem, which provides a new method for obtaining the potential from the spectral data. The formalism applies for both finite- and infinite-genus potentials. As in the defocusing case, the formalism shows that only a single set of Dirichlet eigenvalues is needed in order to uniquely reconstruct the potential of the Dirac operator and the corresponding solution of the focusing NLS equation.

nlin.SI↗

arXiv2025

Spectral theory of soliton gases for the defocusing NLS equation

In this paper we derive Nonlinear Dispersion Relations (NDR) for the defocusing NLS (dark) soliton gas using the idea of thermodynamic limit of quasimomentum and quasienergy differentials on the underlying family of Riemann surfaces. It turns out that the obtained NDR are closely connected with the recently studied NDR for circular soliton gas for the focusing NLS. We find solutions for the kinetic equation for the defocusing NLS soliton condensate, which is defined by the endpoints of the spectral support $\Gamma$ for the NDR. It turns out that, similarly to KdV soliton condensates (\cite{CERT}), the evolution of these endpoints is governed by the defocusing NLS-Whitham equations (\cite{Kodama}). We also study the Riemann problem for step initial data and the kurtosis of genus zero and one defNLS condensates, where we proved that the kurtosis of genus one condensate can not exceed 3/2, whereas for genus zero condensate the kurtosis is always 1.

nlin.PS↗

arXiv2024

Dirichlet energy and focusing NLS condensates of minimal intensity

We consider the family of (poly)continua $\K$ in the upper half-plane ${\mathbb H} $ that contain a preassigned finite {\it anchor} set $E\in\mathbb H$. For a given harmonic external field we define a Dirichlet energy functional $\mathcal I(\mathcal K)$ and show that within each ``connectivity class'' of the family, there exists a minimizing compact $\mathcal K^*$ consisting of critical trajectories of a quadratic differential. In many cases this quadratic differential coincides with the square of the real normalized quasimomentum differential ${\rm d} {\bf p}$ associated with the finite gap solutions of the focusing Nonlinear Schr\"{o}dinger equation (fNLS) defined by a hyperelliptic Riemann surface $\mathfrak R$ branched at the points $E\cup\bar E$. The motivation for this work lies in the problem of soliton condensate of least average intensity such that a given anchor set $E$ belongs to the poly-continuum $\mathcal K$. An fNLS soliton condensate is defined by a compact $\mathcal K\subset{\mathbb H} $ (its spectral support) whereas the average intensity of the condensate is proportional to $\mathcal I(\mathcal K)$. We prove that the spectral support $\mathcal K^*$ provides the fNLS soliton condensate of the least average intensity within a given ``connectivity class''.

math.AP↗

arXiv2024

Approximation of the thermodynamic limit of finite-gap solutions of the focusing NLS hierarchy by multisoliton solutions

In this paper we approximate the thermodynamic limit of finite-gap solutions to any integrable equations in the focusing NLS hierarchy (fNLS, mKdV, ...) with an associated multisoliton solutions using the Riemann-Hilbert Problem approach. Moreover, we show that both the finite-gap and multisoliton solutions are approximated in the thermodynamic limit by a generalization of the primitive potentials introduced by V. Zakharov and his collaborators in the KdV context. Under certain assumptions on the spectral data for the finite gap potentials, we provide error estimates for the approximation on compact subsets of the $(x,t)$-plane.

nlin.SI↗

arXiv2024

Breather gas fission from elliptic potentials in self-focusing media

We present an analytical model of integrable turbulence in the focusing nonlinear Schr\"odinger (fNLS) equation, generated by a one-parameter family of finite-band elliptic potentials in the semiclassical limit. We show that the spectrum of these potentials exhibits a thermodynamic band/gap scaling compatible with that of soliton and breather gases depending on the value of the elliptic parameter m of the potential. We then demonstrate that, upon augmenting the potential by a small random noise (which is inevitably present in real physical systems), the solution of the fNLS equation evolves into a fully randomized, spatially homogeneous breather gas, a phenomenon we call breather gas fission. We show that the statistical properties of the breather gas at large times are determined by the spectral density of states generated by the unperturbed initial potential. We analytically compute the kurtosis of the breather gas as a function of the elliptic parameter m, and we show that it is greater than 2 for all non-zero m, implying non-Gaussian statistics. Finally, we verify the theoretical predictions by comparison with direct numerical simulations of the fNLS equation. These results establish a link between semiclassical limits of integrable systems and the statistical characterization of their soliton and breather gases.

nlin.SI↗

arXiv2023

Soliton Condensates for the Focusing Nonlinear Schr\"odinger Equation: a Non-Bound State Case

In this paper, we study the spectral theory of soliton condensates - a special limit of soliton gases - for the focusing NLS (fNLS). In particular, we analyze the kinetic equation for the fNLS circular condensate, which represents the first example of an explicitly solvable fNLS condensate with nontrivial large scale space-time dynamics. Solution of the kinetic equation was obtained by reducing it to Whitham type equations for the endpoints of spectral arcs. We also study the rarefaction and dispersive shock waves for circular condensates, as well as calculate the corresponding average conserved quantities and the kurtosis. We want to note that one of the main objects of the spectral theory - the nonlinear dispersion relations - is introduced in the paper as some special large genus (thermodynamic) limit the Riemann bilinear identities that involve the quasimomentum and the quasienergy meromorphic differentials.

nlin.PS↗

arXiv2023

Non-standard Green energy problems in the complex plane

We consider several non-standard discrete and continuous Green energy problems in the complex plane and study the asymptotic relations between their solutions. In the discrete setting, we consider two problems; one with variable particle positions (within a given compact set) and variable particle masses, the other one with variable masses but prescribed positions. The mass of a particle is allowed to take any value in the range $0\leq m\leq R$, where $R>0$ is a fixed parameter in the problem. The corresponding continuous energy problems are defined on the space of positive measures $μ$ with mass $\|μ\|\leq R$ and supported on the given compact set, with an additional upper constraint that appears as a consequence of the prescribed positions condition. It is proved that the equilibrium constant and equilibrium measure vary continuously as functions of the parameter $R$ (the latter in the weak-star topology). In the unconstrained energy problem we present a greedy algorithm that converges to the equilibrium constant and equilibrium measure. In the discrete energy problems, it is shown that under certain conditions, the optimal values of the particle masses are uniquely determined by the optimal positions or prescribed positions of the particles, depending on the type of problem considered.

math.CA↗

arXiv2023

Partial degeneration of finite gap solutions to the Korteweg-de Vries equation: soliton gas and scattering on elliptic background

We obtain Fredholm type formulas for partial degenerations of Theta functions on (irreducible) nodal curves of arbitrary genus, with emphasis on nodal curves of genus one. An application is the study of "many-soliton" solutions on an elliptic (cnoidal) background standing wave for the Korteweg-de Vries (KdV) equation starting from a formula that is reminiscent of the classical Kay-Moses formula for $N$-solitons. In particular, we represent such a solution as a sum of the following two terms: a ``shifted" elliptic (cnoidal) background wave and a Kay-Moses type determinant containing Jacobi theta functions for the solitonic content, which can be viewed as a collection of solitary disturbances on the cnoidal background. The expressions for the traveling (group) speed of these solitary disturbances, as well as for the interaction kernel describing the scattering of pairs of such solitary disturbances, are obtained explicitly in terms of Jacobi theta functions. We also show that genus $N+1$ finite gap solutions with random initial phases converge in probability to the deterministic cnoidal wave solution as $N$ bands degenerate to a nodal curve of genus one. Finally, we derive the nonlinear dispersion relations and the equation of states for the KdV soliton gas on the residual elliptic background.

math-ph↗

arXiv2022

Elliptic finite-band potentials of a non-self-adjoint Dirac operator

We present an explicit two-parameter family of finite-band Jacobi elliptic potentials for a non-self-adjoint Dirac operator which connects two previously known limiting cases in which the elliptic parameter is zero or one. A full characterization of the spectrum is obtained by relating the periodic and antiperiodic eigenvalue problems for the Dirac operator to corresponding eigenvalue problems for tridiagonal operators acting on Fourier coefficients in a weighted Hilbert space and to appropriate connection problems for Heun's equation. In turn, these problems are related to four non-self-adjoint unbounded tridiagonal operators, all of which nonetheless have only real eigenvalues. For certain parameter values, the corresponding elliptic potentials generate finite-genus solutions for all the positive and negative flows of the focusing nonlinear Schr\"odinger hierarchy.

math.SP↗

arXiv2022

Recent developments in spectral theory of the focusing NLS soliton and breather gases: the thermodynamic limit of average densities, fluxes and certain meromorphic differentials; periodic gases

In this paper we consider soliton and breather gases for one dimensional integrable focusing Nonlinear Schrödinger Equation (fNLS). We derive average densities and fluxes for such gases by studying the thermodynamic limit of the fNLS finite gap solutions. Thermodynamic limits of quasimomentum, quasienergy and their connections with the corresponding $g$-functions were also established. We then introduce the notion of periodic fNLS gases and calculate for them the average densities, fluxes and thermodynamic limits of meromorphic differentials. Certain accuracy estimates of the obtained results are also included. Our results constitute another step towards the mathematical foundation for the spectral theory of fNLS soliton and breather gases that appeared in work of G. El and A. Tovbis, Phys. Rev. E, 2020.

nlin.PS↗

arXiv2022

Openness of Regular Regimes of Complex Random Matrix Models

Consider the general complex polynomial external field $$ V(z)=\frac{z^{k}}{k}+\sum_{j=1}^{k-1} \frac{t_j z^j}{j}, \qquad t_j \in \mathbb{C}, \quad k \in \mathbb{N}. $$ Fix an equivalence class $\mathcal{T}$ of admissible contours whose members approach $\infty$ in two different directions and consider the associated max-min energy problem. When $k=2p$, $p \in \mathbb{N}$, and $\mathcal{T}$ contains the real axis, we show that the set of parameters $t_1, \cdots, t_{2p-1}$ which gives rise to a regular $q$-cut max-min (equilibrium) measure, $1 \leq q \leq 2p-1 $, is an open set in $\mathbb{C}^{2p-1}$. We use the implicit function theorem to prove that the endpoint equations are solvable in a small enough neighborhood of a regular $q$-cut point. We also establish the real-analyticity of the real and imaginary parts of the end-points for all $q$-cut regimes, $1 \leq q \leq 2p-1$, with respect to the real and imaginary parts of the complex parameters in the external field. Our choice of even $k$ and the equivalence class $\mathcal{T} \ni \mathbb{R}$ of admissible contours is only for the simplicity of exposition and our proof extends to all possible choices in an analogous way.

math.CA↗

arXiv2021

Prediction and manipulation of hydrodynamic rogue waves via nonlinear spectral engineering

Peregrine soliton (PS) is widely regarded as a prototype nonlinear structure capturing properties of rogue waves that emerge in the nonlinear propagation of unidirectional wave trains. As an exact breather solution of the one-dimensional focusing nonlinear Schrödinger equation with nonzero boundary conditions, the PS can be viewed as a soliton on finite background, i.e. a nonlinear superposition of a soliton and a monochromatic wave. A recent mathematical work showed that both nonzero boundary conditions and solitonic content are not pre-requisites for the PS occurrence. Instead, it has been demonstrated that PS can emerge locally, as an asymptotic structure arising from the propagation of an arbitrary large decaying pulse, independently of its solitonic content. This mathematical discovery has changed the widely accepted paradigm of the solitonic nature of rogue waves by enabling the PS to emerge from a partially radiative or even completely solitonless initial data. In this work, we realize the mathematically predicted universal mechanism of the local PS emergence in a water tank experiment with a particular aim to control the point of the PS occurrence in space-time by imposing an appropriately chosen initial chirp. By employing the inverse scattering transform for the synthesis of the initial data, we are able to engineer a localized wave packet with a prescribed solitonic and radiative content. This enabled us to control the position of the emergence of the rogue wave by adjusting the inverse scattering spectrum. The proposed method of the nonlinear spectral engineering is found to be robust to higher-order nonlinear effects inevitable in realistic wave propagation conditions.

nlin.PS↗

arXiv2021

Numerical spectral synthesis of breather gas for the focusing nonlinear Schrödinger equation

We numerically realize breather gas for the focusing nonlinear Schrödinger equation. This is done by building a random ensemble of N $\sim$ 50 breathers via the Darboux transform recursive scheme in high precision arithmetics. Three types of breather gases are synthesized according to the three prototypical spectral configurations corresponding the Akhmediev, Kuznetsov-Ma and Peregrine breathers as elementary quasi-particles of the respective gases. The interaction properties of the constructed breather gases are investigated by propagating through them a "trial" generic breather (Tajiri-Watanabe) and comparing the mean propagation velocity with the predictions of the recently developed spectral kinetic theory (El and Tovbis, PRE 2020).

nlin.PS↗

arXiv2021

On minimal energy solutions to certain classes of integral equations related to soliton gases for integrable systems

We prove existence, uniqueness and non-negativity of solutions of certain integral equations describing the density of states $u(z)$ in the spectral theory of soliton gases for the one dimensional integrable focusing Nonlinear Schrödinger Equation (fNLS) and for the Korteweg de Vries (KdV) equation. Our proofs are based on ideas and methods of potential theory. In particular, we show that the minimizing (positive) measure for certain energy functional is absolutely continuous and its density $u(z)\geq 0$ solves the required integral equation. In a similar fashion we show that $v(z)$, the temporal analog of $u(z)$, is the difference of densities of two absolutely continuous measures. Together, integral equations for $u,v$ represent nonlinear dispersion relation for the fNLS soliton gas. We also discuss smoothness and other properties of the obtained solutions. Finally, we obtain exact solutions of the above integral equations in the case of a KdV condensate and a bound state fNLS condensate. Our results is a first step towards a mathematical foundation for the spectral theory of soliton and breather gases, which appeared in work of El and Tovbis, Phys. Rev. E, 2020. It is expected that the presented ideas and methods will be useful for studying similar classes of integral equation describing, for example, breather gases for the fNLS, as well as soliton gases of various integrable systems.

math-ph↗

arXiv2020

On the spectrum of the periodic focusing Zakharov-Shabat operator

The spectrum of the focusing Zakharov-Shabat operator on the circle is studied, and its explicit dependence on the presence of a semiclassical parameter is also considered. Several new results are obtained. In particular: (i) it is proved that the resolvent set is comprised of two connected components, (ii) new bounds on the location of the Floquet and Dirichlet spectra are obtained, some of which depend explicitly on the value of the semiclassical parameter, (iii) it is proved that the spectrum localizes to a "cross" in the spectral plane in the semiclassical limit. The results are illustrated by discussing several examples in which the spectrum is computed analytically or numerically.

math.SP↗

arXiv2020

On the spectral properties of the Hilbert transform operator on multi-intervals

Let $J,E\subset\mathbb R$ be two multi-intervals with non-intersecting interiors. Consider the following operator $$A:\, L^2( J )\to L^2(E),\ (Af)(x) = \frac 1π\int_{ J } \frac {f(y)\text{d} y}{x-y},$$ and let $A^\dagger$ be its adjoint. We introduce a self-adjoint operator $\mathscr K$ acting on $L^2(E)\oplus L^2(J)$, whose off-diagonal blocks consist of $A$ and $A^\dagger$. In this paper we study the spectral properties of $\mathscr K$ and the operators $A^\dagger A$ and $A A^\dagger$. Our main tool is to obtain the resolvent of $\mathscr K$, which is denoted by $\mathscr R$, using an appropriate Riemann-Hilbert problem, and then compute the jump and poles of $\mathscr R$ in the spectral parameter $λ$. We show that the spectrum of $\mathscr K$ has an absolutely continuous component $[0,1]$ if and only if $J$ and $E$ have common endpoints, and its multiplicity equals to their number. If there are no common endpoints, the spectrum of $\mathscr K$ consists only of eigenvalues and $0$. If there are common endpoints, then $\mathscr K$ may have eigenvalues imbedded in the continuous spectrum, each of them has a finite multiplicity, and the eigenvalues may accumulate only at $0$. In all cases, $\mathscr K$ does not have a singular continuous spectrum. The spectral properties of $A^\dagger A$ and $A A^\dagger$, which are very similar to those of $\mathscr K$, are obtained as well.

math.FA↗

arXiv2020

Spectral theory of soliton and breather gases for the focusing nonlinear Schrödinger equation

Solitons and breathers are localized solutions of integrable systems that can be viewed as "particles'' of complex statistical objects called soliton and breather gases. In view of the growing evidence of their ubiquity in fluids and nonlinear optical media these "integrable'' gases present fundamental interest for nonlinear physics. We develop analytical theory of breather and soliton gases by considering a special, thermodynamic type limit of the wavenumber-frequency relations for multi-phase (finite-gap) solutions of the focusing nonlinear Schrödinger equation. This limit is defined by the locus and the critical scaling of the band spectrum of the associated Zakharov-Shabat operator and yields the nonlinear dispersion relations for a spatially homogeneous breather or soliton gas, depending on the presence or absence of the "background'' Stokes mode. The key quantity of interest is the density of states defining, in principle, all spectral and statistical properties of a soliton (breather) gas. The balance of terms in the nonlinear dispersion relations determines the nature of the gas: from an ideal gas of well separated, non-interacting breathers (solitons) to a special limiting state, which we term breather (soliton) condensate, and whose properties are entirely determined by the pairwise interactions between breathes (solitons). For a non-homogeneous breather gas, we derive a full set of kinetic equations describing slow evolution of the density of states and of its carrier wave counterpart. The kinetic equation for soliton gas is recovered by collapsing the Stokes spectral band. A number of concrete examples of breather and soliton gases are considered, demonstrating efficacy of the developed general theory with broad implications for nonlinear optics, superfluids and oceanography.

nlin.PS↗

arXiv2019

Diagonalization of the finite Hilbert transform on two adjacent intervals: the Riemann-Hilbert approach

In this paper we study the spectra of bounded self-adjoint linear operators that are related to finite Hilbert transforms $\mathcal{H}_L:L^2([b_L,0])\to L^2([0,b_R])$ and $\mathcal{H}_R:L^2([0,b_R])\to L^2([b_L,0])$. These operators arise when one studies the interior problem of tomography. The diagonalization of $\mathcal{H}_R,\mathcal{H}_L$ has been previously obtained, but only asymptotically when $b_L\neq-b_R$. We implement a novel approach based on the method of matrix Riemann-Hilbert problems (RHP) which diagonalizes $\mathcal{H}_R,\mathcal{H}_L$ explicitly. We also find the asymptotics of the solution to a related RHP and obtain error estimates.

math.CA↗