arXiv subjects

Nicholas J. Ossi

Publications and source records attributed to Nicholas J. Ossi.

7 recordsLinked to original sources

arXiv2026

Perturbation theory for kinks of the defocusing modified Korteweg-de Vries equation

In this work we develop an integrable perturbation theory for the defocusing modified Korteweg-de Vries kink solution based on the squared eigenfunction expansion associated with the underlying Zakharov-Shabat scattering problem. We derive the completeness relation for the squared eigenfunctions appropriate to the kink background, establish the adjoint structure needed to handle perturbations of both the continuous and discrete spectral components, and obtain explicit evolution equations for the perturbed kink parameters at leading order. The study of the first order correction shows that perturbations generically produce a radiative shelf in front of the kink. We also apply our results to certain physically relevant perturbations and show that the predictions are consistent with direct numerical simulations.

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arXiv2026

Modulation theory for lumps and interactions between lumps and a mean field in the Kadomtsev-Petviashvili equation

A (2+1)-dimensional hyperbolic system of four quasi-linear partial differential equations is derived that describes the modulations of lump solutions of the Kadomtsev-Petviashvili I (KPI) equation in the presence of a mean field. The system is then shown to satisfy the necessary conditions for integrability of hydrodynamic chains. Moreover, a suitable reduction of the resulting modulation system is applied to study the interactions between lumps and a rarefaction wave for the mean field. Precise conditions are derived that describe how the lump parameters change as a result of the interaction, and which in particular determine whether the lump is transmitted through or trapped inside the rarefaction wave. The theoretical predictions are compared to direct numerical simulations of the KPI equation, showing excellent agreement.

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arXiv2026

Integrable perturbation theory for dark solitons of the defocusing nonlinear Schrödinger equation

The goal of this work is to revisit the eigenfunction-expansion-based perturbation theory for the defocusing nonlinear Schrödinger equation a nonzero background, and develop it to correctly predict the slow-time evolution of the dark soliton parameters, as well as the radiation shelf emerging on the soliton sides. Proof of the closure of the squared eigenfunctions is provided, and the complete set of eigenfunctions of the linearization operator is used to expand the first-order perturbation solution. Our closure/completeness relation accounts for the singularities of the scattering data at the branch points of the continuous spectrum, which leads to the correct discrete eigenfunctions. Using the one-soliton closure relation and its correct discrete eigenmodes, the slow-time evolution equations of the soliton parameters are determined. Moreover, the first-order correction integral to the dark soliton is shown to contain a pole due to singularities of the scattering data at the branch points. Analysis of this integral leads to predictions for the shelves, as well as a formula for the slow time evolution of the soliton's phase, which in turn allows one to determine the slow-time dependence of the soliton center. All the results are corroborated by direct numerical simulations, and compared with earlier results.

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arXiv2025

On the integrable six-wave interaction system and its space-time shifted reduction

The multi-dimensional six-wave interaction system is derived in the context of nonlinear optics. Starting from Maxwell's equations, a reduced system of equations governing the dynamics of the electric and polarization fields are obtained. Using a space-time multi-scale asymptotic expansion, a hierarchy of coupled equations describing the spatio-temporal evolution of the perturbed electric and polarization fields are derived. The leading order equation admits a six-wave ansatz satisfying a triad resonance condition. By removing secular terms at next order, a first order in space and time quadratically nonlinear coupled six-wave interaction system is obtained. This resulting system is tied to its integrable counterpart which was postulated by Ablowitz and Haberman in the 1970s. A reduction to a space-time shifted nonlocal three-wave system is presented. The resulting system is solved using the inverse scattering transform, which employs nonlocal symmetries between the associated eigenfunctions and scattering data; soliton solutions are then obtained. Finally, an infinite set of conservation laws for the six-wave system is derived; one is shown to be connected to its Hamiltonian structure.

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arXiv2025

Breather interactions in the integrable discrete Manakov system and trigonometric Yang-Baxter maps

The goal of this work is to obtain a complete characterization of soliton and breather interactions in the integrable discrete Manakov (IDM) system, a vector generalization of the Ablowitz-Ladik model. The IDM system, which in the continuous limit reduces to the Manakov system (i.e., a 2-component vector nonlinear Schrodinger equation), was shown to admit a variety of discrete vector soliton solutions: fundamental solitons, fundamental breathers, and composite breathers. While the interaction of fundamental solitons was studied early on, no results are presently available for other types of soliton-breather and breather-breather interactions. Our study reveals that upon interacting with a fundamental breather, a fundamental soliton becomes a fundamental breather. Conversely, the interaction of two fundamental breathers generically yields two fundamental breathers with polarization shifts, but may also result in a fundamental soliton and a fundamental breather. Composite breathers interact trivially both with each other and with a fundamental soliton or breather. Explicit formulas for the scattering coefficients that characterize fundamental and composite breathers are given. This allows us to interpret the interactions in terms of a refactorization problem and derive the associated Yang-Baxter maps describing the effect of interactions on the polarizations. These give the first examples of parametric Yang-Baxter maps of trigonometric type.

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arXiv2024

On the discrete Kuznetsov-Ma solutions for the defocusing Ablowitz-Ladik equation with large background amplitude

The focus of this work is on a class of solutions of the defocusing Ablowitz-Ladik lattice on an arbitrarily large background which are discrete analogs of the Kuznetsov-Ma (KM) breathers of the focusing nonlinear Schrodinger equation. One such solution was obtained in 2019 as a byproduct of the Inverse Scattering Transform, and it was observed that the solution could be regular for certain choices of the soliton parameters, but its regularity was not analyzed in detail. This work provides a systematic investigation of the conditions on the background and on the spectral parameters that guarantee the KM solution to be non-singular on the lattice for all times. Furthermore, a novel KM-type breather solution is presented which is also regular on the lattice under the same conditions. We also employ Darboux transformations to obtain a multi-KM breather solution, and show that parameters choices exist for which a double KM breather solution is regular on the lattice. We analyze the features of these solutions, including their frequency which, when tending to 0, renders them proximal to rogue waveforms. Finally, numerical results on the stability and spatio-temporal dynamics of the single KM breathers are presented, showcasing the potential destabilization of the obtained states due to the modulational instability of their background.

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arXiv2024

Inverse scattering transform for continuous and discrete space-time shifted integrable equations

Nonlocal integrable partial differential equations possessing a spatial or temporal reflection have constituted an active research area for the past decade. Recently, more general classes of these nonlocal equations have been proposed, wherein the nonlocality appears as a combination of a shift (by a real or a complex parameter) and a reflection. This new shifting parameter manifests itself in the inverse scattering transform (IST) as an additional phase factor in an analogous way to the classical Fourier transform. In this paper, the IST is analyzed in detail for several examples of such systems. Particularly, time, space, and space-time shifted nonlinear Schrödinger (NLS) and space-time shifted modified Korteweg-de Vries (mKdV) equations are studied. Additionally, the semi-discrete IST is developed for the time, space and space-time shifted variants of the Ablowitz-Ladik integrable discretization of the NLS. One soliton solutions are constructed for all continuous and discrete cases.

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