The focusing Nonlinear Schrödinger (NLS) equation is the simplest universal model describing the modulation instability (MI) of quasi monochromatic waves in weakly nonlinear media, the main physical mechanism for the generation of rogue (anomalous) waves (RWs) in Nature. In this paper we investigate the $x$-periodic Cauchy problem for NLS for a generic periodic initial perturbation of the unstable constant background solution, in the case of $N=1,2$ unstable modes. We use matched asymptotic expansion techniques to show that the solution of this problem describes an exact deterministic alternate recurrence of linear and nonlinear stages of MI, and that the nonlinear RW stages are described by the N-breather solution of Akhmediev type, whose parameters, different at each RW appearence, are always given in terms of the initial data through elementary functions. This paper is motivated by a preceeding work of the authors in which a different approach, the finite gap method, was used to investigate periodic Cauchy problems giving rise to RW recurrence.
Rogue waves on the periodic background are considered for the nonlinear Schrodinger (NLS) equation in the focusing case. The two periodic wave solutions are expressed by the Jacobian elliptic functions dn and cn. Both periodic waves are modulationally unstable with respect to long-wave perturbations. Exact solutions for the rogue waves on the periodic background are constructed by using the explicit expressions for the periodic eigenfunctions of the Zakharov-Shabat spectral problem and the Darboux transformations. These exact solutions labeled as rogue periodic waves generalize the classical r…
The infinite families of Peregrine, Akhmediev and Kuznetsov-Ma breather solutions of the focusing Nonlinear Schroedinger (NLS) equation are obtained via a matrix version of the Darboux transformation, with a spectral matrix of the form of a Jordan block. The structure of these solutions is essentially determined by the corresponding solution of the Lyapunov equation. In particular, regularity follows from properties of the Lyapunov equation.
In this paper, by considering the potential application in two mode nonlinear waves in nonlinear fibers under a certain case, we define a coupled nonlinear Schrödinger equation(called Frobenius NLS equation) including its Lax pair. Afterwards, we construct the Darboux transformations of the Frobenius NLS equation. New solutions can be obtained from known seed solutions by the Dardoux transformations. Soliton solutions are generated from trivial seed solutions. Also we derive breather solutions $q,r$ of the Frobenius NLS equation obtained from periodic seed solutions. Interesting enough, we fin…
The derivative nonlinear Schrodinger (DNLS) equation is the canonical model for dynamics of nonlinear waves in plasma physics and optics. We study exact solutions describing rogue waves on the background of periodic standing waves in the DNLS equation. We show that the space-time localization of a rogue wave is only possible if the periodic standing wave is modulationally unstable. If the periodic standing wave is modulationally stable, the rogue wave solutions degenerate into algebraic solitons propagating along the background and interacting with the periodic standing waves. Maximal amplitud…
The the mixed Chen-Lee-Liu derivative nonlinear Schrödinger equation (CLL-NLS) can be considered as simplest model to approximate the dynamics of weakly nonlinear and dispersive waves, taking into account the self-steepnening effect (SSE). The latter effect arises as a higher-order correction of the nonlinear Schrördinger equation (NLS), which is known to describe the dynamics of pulses in nonlinear fiber optics, and constiutes a fundamental part of the generalized NLS. Similar effects are decribed within the framework of the modified NLS, also referred to as the Dysthe equation, in hydrodynam…
The focusing NLS equation is the simplest universal model describing the modulation instability (MI) of quasi monochromatic waves in weakly nonlinear media, considered the main physical mechanism for the appearance of rogue (anomalous) waves (RWs) in Nature. In this paper we study, using the finite gap method, the NLS Cauchy problem for periodic initial perturbations of the unstable background solution of NLS exciting just one of the unstable modes. We distinguish two cases. In the case in which only the corresponding unstable gap is theoretically open, the solution describes an exact determin…
In the present paper, we attempt to construct both the general rogue wave solutions to the fully discrete nonlinear Schrödinger (fd-NLS) equation via the KP-Toda reduction method. First, we deduce the general breather solution of the fd-NLS equation starting from a pair of bilinear equations. We then derive the general rogue wave solution by taking a limit to the breather solution.
General high-order rogue waves of the NLS-Boussinesq equation are obtained by the KP-hierarchy reduction theory. These rogue waves are expressed with the determinants, whose entries are all algebraic forms. It is found that the fundamental rogue waves can be classified three patterns: four-petals state, dark state, bright state by choosing different parameter $α$ values. As the evolution of the parameter $α$. An interesting phenomena is discovered: the rogue wave changes from four-petals state to dark state, whereafter to the bright state, which are consistent with the change of the correspond…
Rogue wave patterns in the nonlinear Schrödinger (NLS) equation and the derivative NLS equation are analytically studied. It is shown that when the free parameters in the analytical expressions of these rogue waves are large, these waves would exhibit the same patterns, comprising fundamental rogue waves forming clear geometric structures such as triangle, pentagon, heptagon and nonagon, with a possible lower-order rogue wave at its center. These rogue patterns are analytically determined by the root structures of the Yablonskii-Vorob'ev polynomial hierarchy, and their orientations are control…
In this paper, we study the general rogue wave solutions and their patterns in the vector (or $M$-component) nonlinear Schrödinger (NLS) equation. By applying the Kadomtsev-Petviashvili hierarchy reduction method, we derived an explicit solution for the rogue wave expressed by $τ$ functions that are determinants of $K\times K$ block matrices ($K=1,2,\cdots, M$) with an index jump of $M+1$. Patterns of the rogue waves for $M=3,4$ and $K=1$ are thoroughly investigated. We find that when a specific internal parameter is large enough, the wave patterns are linked to the root structures of generali…
Using symmetry analysis we systematically present a higher-dimensional similarity transformation reducing the (3+1)-dimensional inhomogeneous nonlinear Schrodinger (NLS) equation with variable coefficients and parabolic potential to the (1+1)-dimensional NLS equation with constant coefficients. This transformation allows us to relate certain class of localized exact solutions of the (3+1)-dimensional case to the variety of solutions of integrable NLS equation of (1+1)-dimensional case. As an example, we illustrated our technique using two lowest order rational solutions of the NLS equation as …
The generation of rogue waves is investigated via a nonlocal nonlinear Schrodinger (NLS) equation. In this system, modulation instability is suppressed and is usually expected that rogue wave formation would also be limited. On the contrary, a parameter regime is identified where the instability is suppressed but nevertheless the number and amplitude of the rogue events increase, as compared to the standard NLS (which is a limit of the nonlocal system). Furthermore, the nature of these waves is investigated; while no analytical solutions are known to model these events, numerically it is shown…
The higher-order dispersive and nonlinear effects (alias {\it the perturbation terms}) like the third-order dispersion, the self-steepening, and the self-frequency shift play important roles in the study of the ultra-short optical pulse propagation. We consider optical rogue wave solutions and interactions for the generalized higher-order nonlinear Schrödinger (NLS) equation with space- and time-modulated parameters. A proper transformation is presented to reduce the generalized higher-order NLS equation to the integrable Hirota equation with constant coefficients. This transformation allows u…
In this article we present a new method for construction of exact solutions of the Landau-Lifshitz-Gilbert equation (LLG) for ferromagnetic nanowires. The method is based on the established relationship between the LLG and the nonlinear Schrödinger equation (NLS), and is aimed at resolving an old problem: how to produce multiple-rogue wave solutions of NLS using just the Darboux-type transformations. The solutions of this type - known as P-breathers - have been proven to exist by Dubard and Matveev, but their technique heavily relied on using the solutions of yet another nonlinear equation, Ka…
Rogue waves in the nonlocal PT-symmetric nonlinear Schrodinger (NLS) equation are studied by Darboux transformation. Three types of rogue waves are derived, and their explicit expressions in terms of Schur polynomials are presented. These rogue waves show a much wider variety than those in the local NLS equation. For instance, the polynomial degrees of their denominators can be not only $n(n+1)$, but also $n(n-1)+1$ and $n^2$, where $n$ is an arbitrary positive integer. Dynamics of these rogue waves is also examined. It is shown that these rogue waves can be bounded for all space and time, or …
We introduce a mechanism for generating higher order rogue waves (HRWs) of the nonlinear Schrödinger(NLS) equation: the progressive fusion and fission of $n$ degenerate breathers associated with a critical eigenvalue $λ_0$, creates an order $n$ HRW. By adjusting the relative phase of the breathers at the interacting area, it is possible to obtain different types of HRWs. The value $λ_0$ is a zero point of the eigenfunction of the Lax pair of the NLS equation and it corresponds to the limit of the period of the breather tending to infinity. By employing this mechanism we prove two conjectures r…
We study rational solutions of the Boussinesq equation, which is a soliton equation solvable by the inverse scattering method. These rational solutions, which are algebraically decaying and depend on two arbitrary parameters, are expressed in terms of special polynomials that are derived through a bilinear equation, have a similar appearance to rogue-wave solutions of the focusing nonlinear Schrödinger (NLS) equation and have an interesting structure. Further rational solutions of the Kadomtsev-Petviashvili I (KPI) equation are derived in two ways, from rational solutions of the NLS equation a…
Traveling periodic waves of the modified Korteweg-de Vries (mKdV) equation are considered in the focusing case. By using one-fold and two-fold Darboux transformations, we construct explicitly the rogue periodic waves of the mKdV equation expressed by the Jacobian elliptic functions dn and cn respectively. The rogue dn-periodic wave describes propagation of an algebraically decaying soliton over the dn-periodic wave, the latter wave is modulationally stable with respect to long-wave perturbations. The rogue cn-periodic wave represents the outcome of the modulation instability of the cn-periodic…
We propose a novel, analytically tractable, scenario of the rogue wave formation in the framework of the small-dispersion focusing nonlinear Schrödinger (NLS) equation with the initial condition in the form of a rectangular barrier (a "box"). We use the Whitham modulation theory combined with the nonlinear steepest descent for the semi-classical inverse scattering transform, to describe the evolution and interaction of two counter-propagating nonlinear wave trains --- the dispersive dam break flows --- generated in the NLS box problem. We show that the interaction dynamics results in the emerg…
We present explicit forms of general breather (GB), Akhmediev breather (AB), Ma soliton (MS) and rogue wave (RW) solutions of the two component nonlinear Schrödinger (NLS) equation, namely Manakov equation. We derive these solutions through two different routes. In the forward route we first construct a suitable periodic envelope soliton solution to this model from which we derive GB, AB, MS and RW solutions. We then consider the RW solution as the starting point and derive AB, MS and GB in the reverse direction. The second approach has not been illustrated so far for the two component NLS equ…
The Fokas-Lenells (FL) equation arises as a model eqution which describes for nonlinear pulse propagation in optical fibers by retaining terms up to the next leading asymptotic order (in the leading asymptotic order the nonlinear Schrödinger (NLS) equation results). Here we present an explicit analytical representation for the rogue waves of the FL equation. This representation is constructed by deriving an appropriate Darboux transformation (DT) and utilizing a Taylor series expansion of the associated breather solution. when certain higher-order nonlinear effects are considered, the propagat…
In the present paper, we are concerned with the general localized solutions for the complex short pulse equation including soliton, breather and rogue wave solutions. With the aid of a generalized Darboux transformation, we construct the $N$-bright soliton solution in a compact determinant form, then the $N$-breather solution including the Akhmediev breather and a general higher order rogue wave solution. The first- and second-order rogue wave solutions are given explicitly and illustrated by graphs. The asymptotic analysis is performed rigorously for both the $N$-soliton and the $N$-breather …
A study of general rogue waves in some integrable reverse time nonlocal nonlinear equations is presented. Specifically, the reverse time nonlocal nonlinear Schrödinger (NLS) and nonlocal Davey-Stewartson (DS) equations are investigated, which are nonlocal reductions from the AKNS hierarchy. By using Darboux transformation (DT) method, several types of rogue waves are constructed. Especially, a unified binary DT is found for this nonlocal DS system, thus the solution formulas for nonlocal DSI and DSII equation can be written in an uniform expression. Dynamics of these rogue waves is separately …
The one-dimensional focusing nonlinear Schrodinger equation (NLSE) on an unstable condensate background is the fundamental physical model, that can be applied to study the development of modulation instability (MI) and formation of rogue waves. The complete integrability of the NLSE via inverse scattering transform enables the decomposition of the initial conditions into elementary nonlinear modes: breathers and continuous spectrum waves. The small localized condensate perturbations (SLCP) that grow as a result of MI have been of fundamental interest in nonlinear physics for many years. Here, …
The rogue waves in a resonant erbium-doped fibre system governed by a coupled system of the nonlinear Schrödinger equation and the Maxwell-Bloch equation (NLS-MB equations) are given explicitly by a Taylor series expansion about the breather solutions of the normalized slowly varying amplitude of the complex field envelope $E$, polarization $p$ and population inversion $η$. The n-order breather solutions of the three fields are constructed using Darboux transformation (DT) by assuming periodic seed solutions. What is more, the n-order rogue waves are given by determinant forms with $n+3$ free …
Rogue waves are extraordinarily high and steep isolated waves, which appear suddenly in a calm sea and disappear equally fast. However, though the Rogue waves are localized surface waves, their theoretical models and experimental observations are available mostly in one dimension(1D) with the majority of them admitting only limited and fixed amplitude and modular inclination of the wave. We propose a two-dimensional(2D), exactly solvable Nonlinear Schrödinger equation(NLS), derivable from the basic hydrodynamic equations and endowed with integrable structures. The proposed 2D equation exhibits…
The nonlinear Schrödinger equation (NLSE) stands out as the dispersive nonlinear partial differential equation that plays a prominent role in the modeling and understanding of the wave phenomena relevant to many fields of nonlinear physics. The question of random input problems in the one-dimensional and integrable NLSE enters within the framework of integrable turbulence, and the specific question of the formation of rogue waves (RWs) has been recently extensively studied in this context. The determination of exact analytic solutions of the focusing 1D-NLSE prototyping RW events of statistica…
In this paper, we consider the complex modified Korteweg-de Vries (mKdV) equation as a model of few-cycle optical pulses. Using the Lax pair, we construct a generalized Darboux transformation and systematically generate the first-, second- and third-order rogue wave solutions and analyze the nature of evolution of higher-order rogue waves in detail. Based on detailed numerical and analytical investigations, we classify the higher-order rogue waves with respect to their intrinsic structure, namely, fundamental pattern, triangular pattern, and ring pattern. We also present several new patterns o…
We construct the multi-breather solutions of the focusing nonlinear Schrödinger equation (NLSE) on the background of elliptic functions by the Darboux transformation, and express them in terms of the determinant of theta functions. The dynamics of the breathers in the presence of various kinds of backgrounds such as dn, cn, and non-trivial phase-modulating elliptic solutions are presented, and their behaviors dependent on the effect of backgrounds are elucidated. We also determine the asymptotic behaviors for the multi-breather solutions with different velocities in the limit $t\to\pm\infty$, …
The extreme events are investigated for an $n$-component nonlinear Schrödinger ($n$-NLS) system in the focusing Kerr-like nonlinear media, which appears in many physical fields. We report and discuss the novel multi-parametric families of vector rational rogue wave (RW) solutions featuring the parity-time (PT) symmetry, which are characterized by non-identical boundary conditions for the components, and consistent with the degeneracy of $n$ branches of Benjamin-Feir instability. Explicit examples of PT-symmetric vector RWs are presented. Some parameter constraints can make some components gene…
In this paper, we construct a special kind of breather solution of the nonlinear Schrödinger (NLS) equation, the so-called breather-positon ({\it b-positon} for short), which can be obtained by taking the limit $λ_{j}$ $\rightarrow$ $λ_{1}$ of the Lax pair eigenvalues in the order-$n$ periodic solution which is generated by the $n$-fold Darboux transformation from a special "seed" solution--plane wave. Further, an order-$n$ {\it b-positon} gives an order-$n$ rogue wave under a limit $λ_1\rightarrow λ_0$. Here $λ_0$ is a special eigenvalue in a breather of the NLS equation such that its period …
Considering certain terms of the next asymptotic order beyond the nonlinear Schrodinger equation (NLS) equation, the Fokas-Lenells (FL) equation governed by the FL system arise as a model for nonlinear pulse propagation in optical fibers. The expressions of the q[n] and r[n] in the FL system are generated by n-fold Darboux transformation (DT). Further, a Taylor series expansion about the n-order breather solutions q[n] generated using DT by assuming periodic seed solutions under reduction can generate the n-order rogue waves of the FL equation explicitly with 2n+3 free parameters.
We first report the first- and higher-order vector Peregrine solitons (alias rational rogue waves) for the any multi-component NLS equations based on the loop group theory, an explicit (n + 1)-multiple eigenvalue of a characteristic polynomial of degree (n + 1) related to the condition of Benjamin-Feir instability, and inverse functions. Particularly, these vector rational rogue waves are parity-time symmetric for some parameter constraints. A systematic and effective approach is proposed to study the asymptotic behaviors of these vector rogue waves such that the decompositions of rogue waves …
We derive the two-breather solution of the class I infinitely extended nonlinear Schrodinger equation (NLSE). We present a general form of this multi-parameter solution that includes infinitely many free parameters of the equation and free parameters of the two breather components. Particular cases of this solution include rogue wave triplets, and special cases of breather-to-soliton and rogue wave-to-soliton transformations. The presence of many parameters in the solution allows one to describe wave propagation problems with higher accuracy than with the use of the basic NLSE.
In the framework of the focusing Nonlinear Schrodinger (NLS) equation we study numerically the nonlinear stage of the modulation instability (MI) of the condensate. As expected, the development of the MI leads to formation of "integrable turbulence" [V.E. Zakharov, Turbulence in integrable systems, Stud. in Appl. Math. 122, no. 3, 219-234, (2009)]. We study the time evolution of it's major characteristics averaged across realizations of initial data - the condensate solution seeded by small random noise with fixed statistical properties. The measured quantities are: (1) wave-action spectrum an…
We give a qualitative explanation of the analog of the Fermi-Pasta-Ulam (FPU) recurrence in a one-dimensional focusing nonlinear Schrodinger equation (NLSE). That recurrence can be considered as a result of the nonlinear development of modulation instability. All known exact localized solitons-type solutions describing propagation on the background of the modulationally unstable condensate show the recurrence to the condensate state after its interaction with solitons. The condensate state locally recovers its original form with the same amplitude but a different phase after soliton leave its …
We study a deformation of the nonlinear Schrödinger equation recently derived in the context of deformation of hierarchies of integrable systems. This systematic method also led to known integrable equations such as the Camassa-Holm equation. Although this new equation has not been shown to be completely integrable, its solitary wave solutions exhibit typical soliton behaviour, including near elastic collisions. We will first focus on standing wave solutions, which can be smooth or peaked, then, with the help of numerical simulations, we will study solitary waves, their interactions and finall…
We visualize the Fermi-Pasta-Ulam-Tsingou (FPUT) recurrence in a classical Heisenberg ferromagnetic (HF) spin chain by exploiting its gauge eq uivalence to the nonlinear Schrödinger equation (NLSE). We discuss two types of spatially periodic breather excitations in the spin chain, that are associated with: (I) Akhmediev breather, and (II) Galilean transformed Akhmediev breather. The recurrence in the former is exact in the sense that the initial and final states are identical. In the later, the spin chain undergoes an additional global rotation during the rec urrence process, which makes the i…
The Peregrine breather is widely discussed as a model for rogue waves in deep water. We present here a detailed numerical study of perturbations of the Peregrine breather as a solution to the nonlinear Schrödinger (NLS) equations. We first address the modulational instability of the constant modulus solution to NLS. Then we study numerically localized and nonlocalized perturbations of the Peregrine breather in the linear and fully nonlinear setting. It is shown that the solution is unstable against all considered perturbations.
The focusing Nonlinear Schrödinger (NLS) equation is the simplest universal model describing the modulation instability (MI) of quasi monochromatic waves in weakly nonlinear media, and MI is considered the main physical mechanism for the appearence of anomalous (rogue) waves (AWs) in nature. In this paper we study, using the finite gap method, the NLS Cauchy problem for generic periodic initial perturbations of the unstable background solution of NLS (what we call the Cauchy problem of the AWs), in the case of a finite number $N$ of unstable modes. We show how the finite gap method adapts to t…
The focusing Nonlinear Schrödinger (NLS) equation is the simplest universal model describing the modulation instability (MI) of $1+1$ dimensional quasi monochromatic waves in weakly nonlinear media, and MI is considered the main physical mechanism for the appearence of anomalous (rogue) waves (AWs) in nature. In analogy with the recently developed analytic theory of periodic AWs of the focusing NLS equation, in this paper we extend these results to a $2+1$ dimensional context, concentrating on the focusing Davey-Stewartson 2 (DS2) equation, an integrable $2+1$ dimensional generalization of the…
In this paper we study the numerical instabilities of the NLS Akhmediev breather, the simplest space periodic, one-mode perturbation of the unstable background, limiting our considerations to the simplest case of one unstable mode. In agreement with recent theoretical findings of the authors, in the situation in which the round-off errors are negligible with respect to the perturbations due to the discrete scheme used in the numerical experiments, the split-step Fourier method (SSFM), the numerical output is well-described by a suitable genus 2 finite-gap solution of NLS. This solution can be …
The focusing Nonlinear Schrödinger (NLS) equation is the simplest universal model describing the modulation instability (MI) of quasi monochromatic waves in weakly nonlinear media, and MI is considered the main physical mechanism for the appearence of anomalous (rogue) waves (AWs) in nature. Using the finite gap method, two of us (PGG and PMS) have recently solved, to leading order and in terms of elementary functions of the initial data, the NLS Cauchy problem for generic periodic initial perturbations of the unstable background solution of NLS (what we call the Cauchy problem of the AWs), in…
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.
The integrable nonlinear Schrödinger equation (NLSE) is a fundamental model of nonlinear science which also has important consequences in engineering. The powerful framework of the periodic inverse scattering transform (IST) provides a description of the nonlinear phenomena modulational instability and Fermi-Pasta-Ulam-Tsingou (FPUT) recurrence in terms of exact solutions. It associates the complex nonlinear dynamics with invariant nonlinear spectral degrees of freedom that may be used to encode information. While optical fiber is an ideal testing ground of its predictions, maintaining integra…
The Peregrine soliton is often considered as a prototype of the rogue waves. After recent advances in the semi-classical limit of the 1-D focusing Nonlinear Schrödinger (NLS) equation this conjecture can be seen from another perspective. In the present paper, connecting deterministic and statistical approaches, we numerically demonstrate the effect of the universal local appearance of Peregrine solitons on the evolution of statistical properties of random waves. Evidences of this effect are found in recent experimental studies in the contexts of fiber optics and hydrodynamics. The present appr…
We study numerically the properties of (statistically) homogeneous soliton gas depending on soliton density (proportional to number of solitons per unit length) and soliton velocities, in the framework of the focusing one-dimensional Nonlinear Schr{ö}dinger (NLS) equation. In order to model such gas we use N-soliton solutions (N-SS) with $N\sim 100$, which we generate with specific implementation of the dressing method combined with 100-digits arithmetics. We examine the major statistical characteristics, in particular the kinetic and potential energies, the kurtosis, the wave-action spectrum …
Solitons on a finite a background, also called breathers, are solutions of the focusing nonlinear Schrödinger equation, which play a pivotal role in the description of rogue waves and modulation instability. The breather family includes Akhmediev breathers (AB), Kuznetsov-Ma (KM), and Peregrine solitons (PS), which have been successfully exploited to describe several physical effects. These families of solutions are actually only particular cases of a more general three-parameter class of solutions originally derived by Akhmediev, Eleonskii and Kulagin [Theor. Math. Phys. {\bf 72}, 809--818 (1…
Recently, an integrable system of coupled (2+1)-dimensional nonlinear Schrodinger (NLS) equations was introduced by Fokas (eq. (18) in Nonlinearity 29}, 319324 (2016)). Following this pattern, two integrable equations [eqs.2 and 3] with specific parity-time symmetry are introduced here, under different reduction conditions. For eq. 2, two kinds of periodic solutions are obtained analytically by means of the Hirota's bilinear method. In the long-wave limit, the two periodic solutions go over into rogue waves (RWs) and semi-rational solutions, respectively. The RWs have a line shape, while the s…
Asymmetric integrable turbulence and rogue wave statistics for the derivative nonlinear Schrödinger equation
University of Chinese Academy of Sciences（中国科学院大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Electronic Science and Technology of China（电子科技大学）；Zhongyuan University of Technology（中原工学院）
We study the asymptotics of recurrence coefficients for monic orthogonal polynomials p_n(z) with the quartic exponential weight exp [-N (1/2 z^2 + t/4 z^4)], where t is complex. Our goals are: A) to describe the regions of different asymptotic behaviour (different genera) globally in t; B) to identify all the critical points, and; C) to study in details the asymptotics in a full neighborhood near of critical points (double scaling limit), including at and near the poles of Painleve' I solutions y(v) that are known to provide the leading correction term in this limit. Our results are: A) We fou…
We study numerically the integrable turbulence in the framework of the one-dimensional nonlinear Schrodinger equation (1D-NLSE) of the focusing type using a new approach called the "growing of turbulence". In this approach, we add a small linear pumping term to the equation and start evolution from statistically homogeneous Gaussian noise. After reaching a certain level of average intensity, we switch off the pumping and examine the resulting integrable turbulence. For sufficiently small initial noise and pumping coefficient, and also for not very wide simulation box (basin length), we observe…
