# Universality for the Focusing Nonlinear Schrödinger Equation at the Gradient Catastrophe Point: Rational Breathers and Poles of the Tritronquée Solution to Painlevé I

@article{Bertola2010UniversalityFT, title={Universality for the Focusing Nonlinear Schr{\"o}dinger Equation at the Gradient Catastrophe Point: Rational Breathers and Poles of the Tritronqu{\'e}e Solution to Painlev{\'e} I}, author={Marco Bertola and Alexander Tovbis}, journal={Communications on Pure and Applied Mathematics}, year={2010}, volume={66} }

The semiclassical (zero‐dispersion) limit of solutions $q=q(x,t,\epsilon)$ to the one‐dimensional focusing nonlinear Schrödinger equation (NLS) is studied in a scaling neighborhood D of a point of gradient catastrophe ($x_0,t_0$). We consider a class of solutions, specified in the text, that decay as $|x| \rightarrow \infty$. The neighborhood D contains the region of modulated plane wave (with rapid phase oscillations), as well as the region of fast‐amplitude oscillations (spikes). In this…

## 132 Citations

### Universality Near the Gradient Catastrophe Point in the Semiclassical Sine‐Gordon Equation

- MathematicsCommunications on Pure and Applied Mathematics
- 2021

We study the semiclassical limit of the sine‐Gordon (sG) equation with below threshold pure impulse initial data of Klaus‐Shaw type. The Whitham averaged approximation of this system exhibits a…

### Rogue waves generation through multiphase solutions degeneration for the derivative nonlinear Schrödinger equation

- PhysicsNonlinear Dynamics
- 2019

The generation of rogue waves from the development of modulational instability of the plane waves due to small perturbations or interactions of solitons is usually modeled by some breather solutions,…

### The finite gap method and the analytic description of the exact rogue wave recurrence in the periodic NLS Cauchy problem. 1

- PhysicsNonlinearity
- 2018

The focusing nonlinear Schrödinger (NLS) equation is the simplest universal model describing the modulation instability (MI) of quasi monochromatic waves in weakly nonlinear media; MI is considered…

### The defocusing nonlinear Schr\"odinger equation with step-like oscillatory initial data

- Mathematics
- 2021

We study the Cauchy problem for the defocusing nonlinear Schrödinger (NLS) equation under the assumption that the solution vanishes as x → +∞ and approaches an oscillatory plane wave as x→ −∞. We…

### Effect of a small loss or gain in the periodic nonlinear Schrödinger anomalous wave dynamics.

- PhysicsPhysical review. E
- 2020

This paper constructs the proper analytic model describing quantitatively how the solution evolves after a suitable transient into slowly varying lower dimensional patterns (attractors) on the (x,t) plane, and expects that these attractors together with their generalizations corresponding to more unstable modes will play a basic role in the theory of periodic AWs in nature.

### Large-Order Asymptotics for Multiple-Pole Solitons of the Focusing Nonlinear Schrödinger Equation

- MathematicsJ. Nonlinear Sci.
- 2019

It is proved that in a local scaling the solitons converge to functions satisfying the second member of the Painlevé-III hierarchy in the sense of Sakka, a generalization of a function recently identified by Suleimanov in the context of geometric optics.

### The finite-gap method and the periodic NLS Cauchy problem of anomalous waves for a finite number of unstable modes

- MathematicsRussian Mathematical Surveys
- 2019

The focusing non-linear Schrödinger (NLS) equation is the simplest universal model describing the modulation instability (MI) of quasimonochromatic waves in weakly non-linear media, and MI is…

### Semiclassical limit of the scattering transform for the focusing Nonlinear Schr

- Mathematics
- 2009

The semiclassical limit of the focusing Nonlinear (cubic) Schr\" odinger Equation (NLS) corresponds to the singularly perturbed Zakharov Shabat (ZS) system that defines the direct and inverse…

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

- PhysicsPhysical review. E
- 2020

This work provides the theoretical underpinning for the recently observed remarkable connection of the soliton gas dynamics with the long-term evolution of spontaneous modulational instability.

### Semiclassical dynamics and coherent soliton condensates in self‐focusing nonlinear media with periodic initial conditions

- PhysicsStudies in Applied Mathematics
- 2020

The semiclassical (small dispersion) limit of the focusing nonlinear Schrödinger equation with periodic initial conditions (ICs) is studied analytically and numerically. First, through a…

## References

SHOWING 1-10 OF 43 REFERENCES

### Universality in the Profile of the Semiclassical Limit Solutions to the Focusing Nonlinear Schrödinger Equation at the First Breaking Curve

- Mathematics
- 2009

We consider the semiclassical (zero-dispersion) limit of the one-dimensional focusing Nonlinear Schrodinger equation (NLS) with decaying potentials. If a potential is a simple rapidly oscillating…

### Universality of the Break-up Profile for the KdV Equation in the Small Dispersion Limit Using the Riemann-Hilbert Approach

- Mathematics
- 2008

AbstractWe obtain an asymptotic expansion for the solution of the Cauchy problem for the Korteweg-de Vries (KdV) equation
$$u_t+6uu_x+\epsilon^{2}u_{xxx}=0,\quad u(x,t=0,\epsilon)=u_0(x),$$for…

### On Universality of Critical Behavior in the Focusing Nonlinear Schrödinger Equation, Elliptic Umbilic Catastrophe and the Tritronquée Solution to the Painlevé-I Equation

- MathematicsJ. Nonlinear Sci.
- 2009

It is argued that the critical behavior near the point of “gradient catastrophe” of the solution of the Cauchy problem for the focusing nonlinear Schrödinger equation is approximately described by a particular solution to the Painlevé-I equation.

### The semiclassical limit of focusing NLS for a family of non-analytic initial data

- Mathematics
- 2011

The small dispersion limit of the focusing nonlinear Schro\"odinger equation (NLS) exhibits a rich structure of sharply separated regions exhibiting disparate rapid oscillations at microscopic…

### On the modified nonlinear Schr\"odinger equation in the semiclassical limit: supersonic, subsonic, and transsonic behavior

- Mathematics
- 2011

### The N‐soliton of the focusing nonlinear Schrödinger equation for N large

- Mathematics
- 2005

We present a detailed analysis of the solution of the focusing nonlinear Schrödinger equation with initial condition ψ(x, 0) = N sech(x) in the limit N → ∞. We begin by presenting new and more…

### Semiclassical limit of the scattering transform for the focusing Nonlinear Schr

- Mathematics
- 2009

The semiclassical limit of the focusing Nonlinear (cubic) Schr\" odinger Equation (NLS) corresponds to the singularly perturbed Zakharov Shabat (ZS) system that defines the direct and inverse…

### On the semiclassical limit of the focusing nonlinear Schrödinger equation

- Mathematics, Physics
- 1998

### Semiclassical soliton ensembles for the focusing nonlinear Schrödinger equation

- Mathematics
- 2000

This book represents the first asymptotic analysis, via completely integrable techniques, of the initial value problem for the focusing nonlinear Schrodinger equation in the semiclassical asymptotic…

### Poles of tritronquée solution to the Painlevé I equation and cubic anharmonic oscillator

- Mathematics
- 2010

AbstractThe distribution of poles of zero-parameter solution to Painlevé I, specified by P. Boutroux as intégrale tritronquée, is studied in the complex plane. This solution has regular asymptotics…