The goal of this work is to revisit the squared-eigenfunctions-based perturbation theory of the scalar defocusing nonlinear Schrödinger equation on a nonzero background, and develop it so that it can correctly predict the slow-time evolution of the dark soliton parameters, as well as the radiation shelf emerging on the sides of the soliton. Proof of the completeness of the squared eigenfunctions is provided. Our closure relation accounts for the singularities of the scattering data at the branch points of the continuous spectrum, which leads to the correct discrete eigenfunctions of the closure relation. Using the one-soliton closure relation and its 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 first-order correction integral leads to predictions for the height, velocity and phase gradient of 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.

We investigate the dynamics of dark optical solitons of the defocusing nonlinear Schrödinger (NLS) equation under the influence of small perturbations. Building on the integrable structure of the unperturbed NLS equation, we present two complementary analytical frameworks: a perturbed conservation-law approach and a linearization-based eigenfunction expansion rooted in inverse scattering theory. Both yield consistent evolution equations for the key soliton parameters—velocity, depth, center and phase—but the eigenfunction approach additionally provides an explicit expression for the firstorder correction, from which the structure of the shelf, a plateau-like radiation field that inevitably develops around the soliton due to the nonvanishing boundary conditions, can be extracted. It is found that the existence of the shelf is characterized by a single parameter α, which depends on the form of the perturbation. A central result of this work is an exact algebraic relation between the shelf parameter α and the renormalized soliton energy: the rate of energy change is directly and exclusively coupled to α, while all other macroscopic conserved quantities—momentum, Hamiltonian, and center of energy—are shown to decouple from the shelf at leading order. This energy–shelf relation provides a unified and physically transparent accounting of how perturbation energy is partitioned between the soliton core and the radiation it emits. Our analytical results—which are illustrated through several examples of physically relevant perturbations—are consistent with established results in limiting cases. To our knowledge, the energy–shelf relation and the systematic comparison of the two perturbative frameworks within a single, unified treatment are presented here for the first time.

In this paper we apply Hirota's bilinear method to the scalar and coupled Ablowitz-Ladik systems in the defocusing dispersion regime under the assumption of a background amplitude 0<ρ<1. We first establish, in the scalar case, the correspondence between the Hirota's parameters and the spectral parameters of the inverse scattering transform. Then we show that when the Hirota parameter associated to the discrete eigenvalue is chosen outside the range corresponding to a discrete dark soliton, novel solutions of the Ablowitz-Ladik system emerge. In general, these solutions are singular, but there exists a class of time-periodic solutions for which it is possible to choose the soliton parameters so that the breathers remain regular on the lattice for all times. We also discuss the interactions between a dark soliton and a regular breather, and between two regular breathers. For the coupled Ablowitz-Ladik system, by including in the background discrete, counter-propagating plane waves, we use Hirota's method to derive novel Akhmediev-type (i.e., space-periodic) discrete breathers which are regular for all times. Finally, taking the limit of the discrete Akhmediev breathers as the period approaches infinity (i.e., as the wavenumber approaches zero) we obtain novel rogue wave solutions of the coupled Ablowitz-Ladik system.

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.

The aim of this paper is to apply Hirota's bilinear method to the integrable discrete Manakov system in the focusing dispersion regime in order to construct and analyze soliton and breather solutions. After deriving the general bilinear form of the system, we show how to obtain fundamental solitons, as well as fundamental and composite breathers. We then obtain solutions exhibiting 2 solitons and 2 breathers and combinations of a soliton and a breather, and discuss all “two-body” interactions properties, with particular emphasis on explicit formulas, visualization, and long-time asymptotic behavior, thus rigorously confirming the highly nontrivial interaction properties of these coherent structures.

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.

We present a direct perturbation method to study the dynamics of dark–bright (DB) solitons of the Manakov system under the influence of perturbations. Our methodology combines a multi-scale expansion method, perturbed conservation laws and a boundary-layer approach, which breaks the problem into an inner region, pertinent to the soliton core and an outer region, which evolves independently of the dark–bright soliton. We find that a shelf emerges around the dark soliton component, which propagates with a speed depending on the background intensity. Conserved quantities of the Manakov system are employed to determine the properties of the perturbed solutions. We focus on dissipative perturbations, such as diffusion, as well as linear and nonlinear loss, and show that the effect of the bright (filling) soliton component is to partially stabilize ‘bare’ dark solitons of the scalar case against perturbation-induced dissipation. Our analytical predictions are corroborated by results of direct numerical simulations.

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.

The Maxwell-Bloch system of equations with inhomogeneous broadening is studied, and the local and global well-posedness of the corresponding initial-boundary value problem is established by taking advantage of the integrability of the system and making use of the corresponding inverse scattering transform (IST). A key ingredient in the analysis is the L2{L}^{2}-Sobolev bijectivity of the direct and IST established by Xin Zhou for the focusing Zakharov-Shabat problem.

The inverse scattering transform is developed to solve the Maxwell-Bloch system of equations that describes two-level systems with inhomogeneous broadening, in the case of optical pulses that do not vanish at infinity in the future. The direct problem, which is formulated in terms of a suitably-defined uniformization variable, combines features of the formalism with decaying as well as non-decaying fields. The inverse problem is formulated in terms of a 2× 2 matrix Riemann-Hilbert problem. A novel aspect of the problem is that no reflectionless solutions can exist, and solitons are always accompanied by radiation. At the same time, it is also shown that, when the medium is initially in the ground state, the radiative components of the solutions decay upon propagation into the medium, giving rise to an asymptotically reflectionless states. Like what happens when the optical pulse decays rapidly in the distant past and the distant future, a medium that is initially excited decays to the stable ground state as t→∞ and for sufficiently large propagation distances. Finally, the asymptotic state of the medium and certain features of the optical pulse inside the medium are considered, and the emergence of a transition region upon propagation in the medium is briefly discussed.

We present a method to solve numerically the Cauchy problem for the defocusing nonlinear Schrödinger (NLS) equation with a box-type initial condition (IC) having a nontrivial background of amplitude q_o>0 as x→±∞ by implementing numerically the corresponding Inverse Scattering Transform (IST). The Riemann–Hilbert problem associated to the inverse transform is solved numerically by means of appropriate contour deformations in the complex plane following the numerical implementation of the Deift-Zhou nonlinear steepest descent method. In this work, the box parameters are chosen so that there is no discrete spectrum (i.e., no solitons). The numerical method is demonstrated to be accurate within the two asymptotic regimes corresponding to two different regions of the (x,t)-plane depending on whether |x/(2t)| < q_o or |x/(2t)| > q_o, as t →∞.

The complex coupled short‐pulse equation (ccSPE) describes the propagation of ultrashort optical pulses in nonlinear birefringent fibers. The system admits a variety of vector soliton solutions: fundamental solitons, fundamental breathers, composite breathers (generic or nongeneric), as well as so‐called self‐symmetric composite solitons. In this work, we use the dressing method and the Darboux matrices corresponding to the various types of solitons to investigate soliton interactions in the focusing ccSPE. The study combines refactorization problems on generators of certain rational loop groups, and long‐time asymptotics of these generators, as well as the main refactorization theorem for the dressing factors that leads to the Yang–Baxter property for the refactorization map and the vector soliton interactions. Among the results obtained in this paper, we derive explicit formulas for the polarization shift of fundamental solitons that are the analog of the well‐known formulas for the interaction of vector solitons in the Manakov system. Our study also reveals that upon interacting with a fundamental breather, a fundamental soliton becomes a fundamental breather and, conversely, that the interaction of two fundamental breathers generically yields two fundamental breathers with a polarization shifts, but may also result into a fundamental soliton and a fundamental breather. Explicit formulas for the coefficients that characterize the fundamental breathers, as well as for their polarization vectors are obtained. The interactions of other types of solitons are also derived and discussed in detail and illustrated with plots. New Yang–Baxter maps are obtained in the process.

In this topical review paper we provide a survey of classical and more recent results on the IST for one-dimensional scalar, vector and square matrix NLS systems on the line ( -∞

Matter-wave interference mechanisms in one-dimensional Bose-Einstein condensates that allow for the controlled generation of dark soliton trains upon choosing suitable box-type initial configurations are described. First, the direct scattering problem for the defocusing nonlinear Schr\"odinger equation with nonzero boundary conditions and general box-type initial configurations is discussed, and expressions for the discrete spectrum corresponding to the dark soliton excitations generated by the dynamics are obtained. It is found that the size of the initial box directly affects the number, size and velocity of the solitons, while the initial phase determines the parity of the solutions. The analytical results obtained for the untrapped system are compared to those of numerical simulations of the Gross-Pitaevskii equation, both in the absence and in the presence of a harmonic trap. The numerical results bear out the analytical results with excellent agreement.

We consider the mean-field analog of the p-star model for homogeneous random networks, and we compare its behavior with that of the p-star model and its classical mean-field approximation in the thermodynamic regime. We show that the partition function of the mean-field model satisfies a sequence of partial differential equations known as the heat hierarchy, and the models connectance is obtained as a solution of a hierarchy of nonlinear viscous PDEs. In the thermodynamic limit, the leading-order solution develops singularities in the space of parameters that evolve as classical shocks regularized by a viscous term. Shocks are associated with phase transitions and stable states are automatically selected consistently with the Maxwell construction. The case p=3 is studied in detail. Monte Carlo simulations show an excellent agreement between the p-star model and its mean-field analog at the macroscopic level, although significant discrepancies arise when local features are compared.

The inverse scattering transform (IST) for the defocusing Manakov system is developed with non-zero boundary conditions at infinity comprising non-parallel bound-ary conditions - i.e., asymptotic polarization vectors. The formalism uses a uniformization variable to map two copies of the spectral plane into a single copy of the complex plane, thereby eliminating square root branching. The "adjoint" Lax pair is also used to overcome the problem of non-analyticity of some of the Jost eigenfunctions. The inverse problem is formulated in term of a suitable matrix Riemann-Hilbert problem (RHP). The most significant difference in the IST compared to the case of parallel boundary conditions is the asymptotic behavior of the scattering coefficients, which affects the normal-ization of the eigenfunctions and the sectionally meromorphic matrix in the RHP. When the asymptotic polarization vectors are not orthogonal, two different methods are presented to convert the RHP into a set of linear algebraic-integral equations. When the asymptotic polarization vectors are orthogonal, however, only one of these methods is applicable. Finally, it is shown that, both in the case of orthogonal and non-orthogonal polarization vectors, no reflectionless potentials can exist, which implies that the prob-lem does not admit pure soliton solutions.

In this paper, we develop the Riemann–Hilbert approach to the inverse scattering transform (IST) for the complex coupled short‐pulse equation on the line with zero boundary conditions at space infinity, which is a generalization of recent work on the scalar real short‐pulse equation (SPE) and complex short‐pulse equation (cSPE). As a byproduct of the IST, soliton solutions are also obtained. As is often the case, the zoology of soliton solutions for the coupled system is richer than in the scalar case, and it includes both fundamental solitons (the natural, vector generalization of the scalar case), and fundamental breathers (a superposition of orthogonally polarized fundamental solitons, with the same amplitude and velocity but having different carrier frequencies), as well as composite breathers, which still correspond to a minimal set of discrete eigenvalues but cannot be reduced to a simple superposition of fundamental solitons. Moreover, it is found that the same constraint on the discrete eigenvalues which leads to regular, smooth one‐soliton solutions in the complex SPE, also holds in the coupled case, for both a single fundamental soliton and a single fundamental breather, but not, in general, in the case of a composite breather.

We characterize initial value problems for the defocusing Manakov system (coupled two-component nonlinear Schrödinger equation) with nonzero background and well-defined spatial parity symmetry (i.e., when each of the components of the solution is either even or odd), corresponding to boundary value problems on the half line with Dirichlet or Neumann boundary conditions at the origin. We identify the symmetries of the eigenfunctions arising from the spatial parity of the solution, and we determine the corresponding symmetries of the scattering data (reflection coefficients, discrete spectrum and norming constants). All parity induced symmetries are found to be more complicated than in the scalar (i.e., one-component) case. In particular, we show that the discrete eigenvalues giving rise to dark solitons arise in symmetric quartets, and those giving rise to dark–bright solitons in symmetric octets. We also characterize the differences between the purely even or purely odd case (in which both components are either even or odd functions of x ) and the ‘mixed parity’ cases (in which one component is even while the other is odd). Finally, we show how, in each case, the spatial symmetry yields a constraint on the possible existence of self-symmetric eigenvalues, corresponding to stationary solitons, and we study the resulting behavior of solutions.

We characterize the soliton solutions and their interactions for a system of coupled evolution equations of nonlinear Schrödinger (NLS) type that models the dynamics in one-dimensional repulsive Bose–Einstein condensates with spin one, taking advantage of the representation of such model as a special reduction of a $$2\times 2$$ matrix NLS system. Specifically, we study in detail the case in which solutions tend to a nonzero background at space infinities. First we derive a compact representation for the multi-soliton solutions in the system using the Inverse Scattering Transform (IST). We introduce the notion of canonical form of a solution, corresponding to the case when the background as $$x\rightarrow \infty $$ is proportional to the identity. We show that solutions for which the asymptotic behavior at infinity is not proportional to the identity, referred to as being in non-canonical form, can be reduced to canonical form by unitary transformations that preserve the symmetric nature of the solution (physically corresponding to complex rotations of the quantization axes). Then we give a complete characterization of the two families of one-soliton solutions arising in this problem, corresponding to ferromagnetic and to polar states of the system, and we discuss how the physical parameters of the solitons for each family are related to the spectral data in the IST. We also show that any ferromagnetic one-soliton solution in canonical form can be reduced to a single dark soliton of the scalar NLS equation, and any polar one-soliton solution in canonical form is unitarily equivalent to a pair of oppositely polarized displaced scalar dark solitons up to a rotation of the quantization axes. Finally, we discuss two-soliton interactions and we present a complete classification of the possible scenarios that can arise depending on whether either soliton is of ferromagnetic or polar type.