arXiv subjects

Andrey Gelash

Publications and source records attributed to Andrey Gelash.

17 recordsLinked to original sources

arXiv2026

Numerical Direct Scattering Transform for Dark Solitons

We introduce a numerical direct scattering transform scheme for dark solitons of the nonlinear Schrodinger equation, enabling the identification and complete characterization of nonlinear coherent structures in defocusing media with a continuous-wave (CW) background. Our scheme is based on numerically solving the auxiliary Zakharov-Shabat scattering problem with CW boundary conditions and on analytically derived expressions that relate the elements of the transfer matrix to the scattering data for dark solitons and continuous-spectrum waves. To test our approach, we consider two analytically solvable cases of the scattering problem: i) rectangular, and ii) hyperbolic tangent hollows in the CW background, which can contain an arbitrary number of dark solitons, with known scattering data. We revisit the analytical derivations and obtain a complete set of soliton parameters represented by discrete eigenvalues and norming constants. By supplementing the direct scattering transform algorithm with high-precision arithmetic to accurately recover soliton norming constants, we provide a robust method to analyze data from numerical or natural experiments on complex wave fields in optical, hydrodynamical, and other physical systems.

nlin.PS↗

arXiv2026

Eigenvalue-Based Approach to Manipulate and Reconstruct Nonlinear Pulses: Towards Soliton Tomography

Soliton content of nonlinear pulses of different physical nature is universally characterized by a discrete set of eigenvalues. In an ideal channel governed by the nonlinear Schrodinger equation, the eigenvalues do not change along the wave field propagation. Perturbations leave predictable fingerprints on the eigenvalue portrait, which was recently used to manipulate optical fiber solitons in [Phys. Rev. Lett. 134, 193804, 2025]. Here, we develop a theoretical framework to manipulate and reconstruct sech-shaped nonlinear wave fields based on soliton eigenvalue response functions and the corresponding inverse problem. We derive analytical expressions to enable nonlinear manipulation of solitons by applying instant, controllable perturbations. Then we present a concept of perturbation sensing with the key feature of nonlinear propagation of the probe signal over an unknown distance, enabling the extraction of information about the perturbation source hidden within nonlinear media or materials. We introduce an integral equation for the inverse problem of reconstructing the unknown shape of the wave field distortions, when the known observational data is a function of deviations in soliton eigenvalues measured at the end of the nonlinear propagation channel. We evaluate different reconstruction regimes and demonstrate a reliable inverse problem solution in presence of noise, paving the way towards soliton tomography.

nlin.PS↗

arXiv2026

Hybridization of Kerr Solitons in Coupled Microresonators

Recent advances in manufacturing photonic integrated devices enable efficient coupling between high-Q microresonators in both linear and nonlinear regimes, creating a tunable, complex, hybridized optical system. Considering two coupled microresonators with normal and anomalous dispersion and equal free spectral range (FSR), we theoretically predict a novel nonlinear phenomenon: fully coherent hybridization of dissipative Kerr solitons (DKS) and propose a realistic integrated photonic design for its experimental observation. Using the Lugiato-Lefever equations in the supermode basis, we show that the emergent picture of inter-resonator DKS interactions can be understood as the formation of coherent structures in both supermodes generated by an unusual four-wave mixing process. The found hybridized DKS states can exhibit a broad, flat spectral profile near the pumped mode and remarkable oscillatory features in the spectral wings, promising broad applications in the generation and control of optical Kerr frequency combs.

physics.optics↗

arXiv2025

Periodic orbits underlying spatiotemporal chaos in the Lugiato-Lefever model

We obtain and investigate theoretically a broad family of stable and unstable time-periodic orbits-oscillating Turing rolls (OTR)-in the Lugiato-Lefever model of optical cavities. Using the dynamical systems tools developed in fluid dynamics, we access the OTR solution branches in parameter space and elucidate their bifurcation structure. By tracking these exact invariant solutions deeply into the chaotic region of the modulation instability, we connect the main dynamical regimes of the Lugiato-Lefever model: continuous waves, Turing rolls, solitons, and breathers, which completes the classical phase diagram of the optical cavity. We then demonstrate that the OTR periodic orbits play a fundamental role as elementary building blocks in the regime of the intracavity field transition from stable Turing rolls to fully developed turbulent regimes. Depending on the cavity size, we observe that the chaotic intracavity field driven by modulation instability displays either spatiotemporal or purely temporal intermittancy between chaotic dynamics and different families of the OTR solutions, exhibiting locally the distinctive wave patterns and large amplitude peaks. This opens avenues for a theoretical description of optical turbulence within the dynamical systems framework.

nlin.PS↗

arXiv2025

Copper-impurity-free photonic integrated circuits enable deterministic soliton microcombs

Chip-scale optical frequency combs based on microresonators (microcombs) enable GHz-THz repetition rates, broad bandwidth, compactness, and compatibility with wafer-scale manufacturing. Silicon nitride photonic integrated circuits have become a leading platform due to their low loss, broad transparency, lithographic dispersion control, and commercial 200-mm-wafer foundry access. They have enabled system-level applications in optical communications, LiDAR, frequency synthesis, low-noise microwave generation, and convolutional processing. However, real-world deployment is hindered by the challenge of deterministic soliton microcomb generation, primarily due to thermal instabilities. Although techniques like pulsed pumping, fast scanning, and auxiliary lasers help mitigate these effects, they often add complexity or reduce soliton stability. In this work, we overcome thermal limitations and demonstrate deterministic soliton generation in silicon nitride photonic circuits. We trace the thermal effects to copper impurities within waveguides, originating from residual contaminants in CMOS-grade silicon wafers that are gettered into silicon nitride during fabrication. By developing effective copper removal techniques, we significantly reduce thermal instabilities. This enables soliton generation with arbitrary or slow laser scanning, removing a key barrier to microcomb deployment. Our approach is compatible with front-end-of-line foundry processing, paving the way for broader adoption of soliton microcomb technologies.

physics.optics↗

arXiv2024

Manipulation of strongly interacting solitons in optical fiber experiments

The model underlying physics of guiding light in single-mode fibers -- the one-dimensional nonlinear Schr\"odinger equation (NLSE), reveals a remarkable balance of the fiber dispersion and nonlinearity, leading to the existence of optical solitons. With the Inverse Scattering Transform (IST) method and its perturbation theory extension one can go beyond single-soliton physics and investigate nonlinear dynamics of complex optical pulses driven by soliton interactions. Here, advancing the IST perturbation theory approach, we introduce the eigenvalue response functions, which provide an intuitively clear way to manipulate individual characteristics of solitons even in the case of their entire overlapping, i.e. very strong interactions. The response functions reveal the spatial sensitivity of the multi-soliton pulse concerning its instantaneous perturbations, allowing one to manipulate the velocities and amplitudes of each soliton. Our experimental setup, based on a recirculating optical fiber loop system and homodyne measurement, enables observation of long-distance NLSE dynamics and complete characterization of optical pulses containing several solitons. Adding a localized phase perturbation on a box-shaped wavefield, we change individual soliton characteristics and can detach solitons selectively from the whole pulse. The detached solitons exhibit velocities very close to the theoretical predictions, thereby demonstrating the efficiency and robustness of the response functions approach.

nlin.PS↗

arXiv2023

Bi-solitons on the surface of a deep fluid: an inverse scattering transform perspective based on perturbation theory

We investigate theoretically and numerically the dynamics of long-living oscillating coherent structures - bi-solitons - in the exact and approximate models for waves on the free surface of deep water. We generate numerically the bi-solitons of the approximate Dyachenko-Zakharov equation and fully nonlinear equations propagating without significant loss of energy for hundreds of the structure oscillation periods, which is hundreds of thousands of characteristic periods of the surface waves. To elucidate the long-living bi-soliton complex nature we apply an analytical-numerical approach based on the perturbation theory and the inverse scattering transform (IST) for the one-dimensional focusing nonlinear Schr\"odinger equation model. We observe a periodic energy and momentum exchange between solitons and continuous spectrum radiation resulting in repetitive oscillations of the coherent structure. We find that soliton eigenvalues oscillate on stable trajectories experiencing a slight drift on a scale of hundreds of the structure oscillation periods so that the eigenvalue dynamic is in good agreement with predictions of the IST perturbation theory. Based on the obtained results, we conclude that the IST perturbation theory justifies the existence of the long-living bi-solitons on the surface of deep water which emerge as a result of a balance between their dominant solitonic part and a portion of continuous spectrum radiation.

nlin.PS↗

arXiv2023

Soliton Gas: Theory, Numerics and Experiments

The concept of soliton gas was introduced in 1971 by V. Zakharov as an infinite collection of weakly interacting solitons in the framework of Korteweg-de Vries (KdV) equation. In this theoretical construction of a diluted soliton gas, solitons with random parameters are almost non-overlapping. More recently, the concept has been extended to dense gases in which solitons strongly and continuously interact. The notion of soliton gas is inherently associated with integrable wave systems described by nonlinear partial differential equations like the KdV equation or the one-dimensional nonlinear Schrödinger equation that can be solved using the inverse scattering transform. Over the last few years, the field of soliton gases has received a rapidly growing interest from both the theoretical and experimental points of view. In particular, it has been realized that the soliton gas dynamics underlies some fundamental nonlinear wave phenomena such as spontaneous modulation instability and the formation of rogue waves. The recently discovered deep connections of soliton gas theory with generalized hydrodynamics have broadened the field and opened new fundamental questions related to the soliton gas statistics and thermodynamics. We review the main recent theoretical and experimental results in the field of soliton gas. The key conceptual tools of the field, such as the inverse scattering transform, the thermodynamic limit of finite-gap potentials and the Generalized Gibbs Ensembles are introduced and various open questions and future challenges are discussed.

nlin.SI↗

arXiv2022

Right and left inverse scattering problems formulations for the Zakharov-Shabat system

We consider right and left formulations of the inverse scattering problem for the Zakharov-Shabat system and the corresponding integral Gelfand-Levitan-Marchenko equations. Both formulations are helpful for numerical solving of the inverse scattering problem, which we perform using the previously developed Toeplitz Inner Bordering (TIB) algorithm. First, we establish general relations between the right and left scattering coefficients. Here, along with the known results, we introduce a relation between the left and right norming coefficients for the N-soliton solution. Then we propose an auxiliary kernel of the left Gelfand-Levitan-Marchenko equations, which allows one to solve the right scattering problem numerically. We generalize the TIB algorithm, initially proposed in the left formulation, to the right scattering problem case with the obtained formulas. The test runs of the TIB algorithm illustrate our results reconstructing the various nonsymmetrical potentials from their right scattering data.

nlin.SI↗

arXiv2022

Vector breathers in the Manakov system

We study theoretically the nonlinear interactions of vector breathers propagating on an unstable wavefield background. As a model, we use the two-component extension of the one-dimensional focusing nonlinear Schrodinger equation -- the Manakov system. With the dressing method, we generate the multi-breather solutions to the Manakov model. As shown previously in [D. Kraus, G. Biondini, and G. Kovacic, Nonlinearity 28, 310, 2015] the class of vector breathers is presented by three fundamental types I, II, and III. Their interactions produce a broad family of the two-component (polarized) nonlinear wave patterns. First, we demonstrate that the type I and the types II and III correspond to two different branches of the dispersion law of the Manakov system in the presence of the unstable background. Then we investigate the key interaction scenarios, including collisions of standing and moving breathers and resonance breather transformations. Analysis of the two-breather solution allows us to derive general formulas describing phase and space shifts acquired by breathers in mutual collisions. The found expressions enable us to describe the asymptotic states of the breather interactions and interpret the resonance fusion and decay of breathers as a limiting case of infinite space shift in the case of merging breather eigenvalues. Finally, we demonstrate that only type $\mathrm{I}$ breathers participate in the development of modulation instability from small-amplitude perturbations within the superregular scenario, while the breathers of types $\mathrm{II}$ and $\mathrm{III}$, belonging to the stable branch of the dispersion law, are not involved in this process.

nlin.SI↗

arXiv2021

Lump chains in the KP-I equation

We construct a broad class of solutions of the KP-I equation by using a reduced version of the Grammian form of the $τ$-function. The basic solution is a linear periodic chain of lumps propagating with distinct group and wave velocities. More generally, our solutions are evolving linear arrangements of lump chains, and can be viewed as the KP-I analogues of the family of line-soliton solutions of KP-II. However, the linear arrangements that we construct for KP-I are more general, and allow degenerate configurations such as parallel or superimposed lump chains. We also construct solutions describing interactions between lump chains and individual lumps, and discuss the relationship between the solutions obtained using the reduced and regular Grammian forms.

nlin.SI↗

arXiv2020

Ghost interaction of breathers

Mutual interaction of localized nonlinear waves, e.g. solitons and modulation instability patterns, is a fascinating and intensively-studied topic of nonlinear science. In this research report, we report on the observation of a novel type of breather interaction in telecommunication optical fibers, in which two identical breathers propagate with opposite group velocities. Under certain conditions, neither amplification nor annihilation occur at the collision point and most interestingly, its amplitude is almost equal to another maximum of either oscillating breather. This ghost-like breather interaction dynamics can be fully described by the N-breather solution of the nonlinear Schrödinger equation.

nlin.PS↗

arXiv2020

Solitons in a box-shaped wavefield with noise: perturbation theory and statistics

We investigate the fundamental problem of the nonlinear wavefield scattering data corrections in response to a perturbation of initial condition using inverse scattering transform theory. We present a complete theoretical linear perturbation framework to evaluate first-order corrections of the full set of the scattering data within the integrable one-dimensional focusing nonlinear Schrödinger (NLSE) equation. The general scattering data portrait reveals nonlinear coherent structures - solitons - playing the key role in the wavefield evolution. Applying the developed theory to a classic box-shaped wavefield we solve the derived equations analytically for a single Fourier mode acting as a perturbation to the initial condition, thus, leading to the sensitivity closed-form expressions for basic soliton characteristics, i.e. the amplitude, velocity, phase and its position. With the appropriate statistical averaging we model the soliton noise-induced effects resulting in compact relations for standard deviations of soliton parameters. Relying on a concept of a virtual soliton eigenvalue we derive the probability of a soliton emergence or the opposite due to noise and illustrate these theoretical predictions with direct numerical simulations of the NLSE evolution. The presented framework can be generalised to other integrable systems and wavefield patterns.

nlin.PS↗

arXiv2020

Nonlinear spectral synthesis of soliton gas in deep-water surface gravity waves

Soliton gases represent large random soliton ensembles in physical systems that display integrable dynamics at the leading order. Despite significant theoretical developments and observational evidence of ubiquity of soliton gases in fluids and optical media their controlled experimental realization has been missing. We report the first controlled synthesis of a dense soliton gas in deep-water surface gravity waves using the tools of nonlinear spectral theory (inverse scattering transform (IST)) for the one-dional focusing nonlinear Schrödinger equation. The soliton gas is experimentally generated in a one-dimensional water tank where we demonstrate that we can control and measure the density of states, i. e. the probability density function parametrizing the soliton gas in the IST spectral phase space. Nonlinear spectral analysis of the generated hydrodynamic soliton gas reveals that the density of states slowly changes under the influence of perturbative higher-order effects that break the integrability of the wave dynamics.

nlin.PS↗

arXiv2019

Direct scattering transform: catch soliton if you can

Direct scattering transform of nonlinear wave fields with solitons may lead to anomalous numerical errors of soliton phase and position parameters. With the focusing one-dimensional nonlinear Schrödinger equation serving as a model, we investigate this fundamental issue theoretically. Using the dressing method we find the landscape of soliton scattering coefficients in the plane of the complex spectral parameter for multi-soliton wave fields truncated within a finite domain, allowing us to capture the nature of particular numerical errors. They depend on the size of the computational domain $L$ leading to a counterintuitive exponential divergence when increasing $L$ in the presence of a small uncertainty in soliton eigenvalues. In contrast to classical textbooks, we reveal how one of the scattering coefficients loses its analytical properties due to the lack of the wave field compact support in case of $L \to \infty$. Finally, we demonstrate that despite this inherit direct scattering transform feature, the wave fields of arbitrary complexity can be reliably analysed.

nlin.SI↗

arXiv2019

Bound state soliton gas dynamics underlying the noise-induced modulational instability

We investigate theoretically the fundamental phenomenon of the spontaneous, noise-induced modulational instability (MI) of a plane wave. The long-term statistical properties of the noise-induced MI have been previously observed in experiments and in simulations but have not been explained so far. In the framework of inverse scattering transform (IST), we propose a model of the asymptotic stage of the noise-induced MI based on $N$-soliton solutions ($N$-SS) of the integrable focusing one-dimensional nonlinear Schrödinger equation (1D-NLSE). These $N$-SS are bound states of strongly interacting solitons having a specific distribution of the IST eigenvalues together with random phases. We use a special approach to construct ensembles of multi-soliton solutions with statistically large number of solitons $N\sim100$. Our investigation demonstrates complete agreement in spectral (Fourier) and statistical properties between the long-term evolution of the condensate perturbed by noise and the constructed multi-soliton bound states. Our results can be generalised to a broad class of integrable turbulence problems in the cases when the wave field dynamics is strongly nonlinear and driven by solitons.

nlin.SI↗

arXiv2011

Soliton on Unstable Condensate

We construct new exact solutions of the focusing Nonlinear Schrödinger equation (NLSE). This is a soliton propagating on an unstable condensate. The Kuznetsov and Akhmediev solitons as well as the Peregrine instanton are particular cases of this new solution. We discuss applications of this new solution to the description of freak (rogue) waves in the ocean and in optical fibers.

nlin.SI↗