scholarly journals BI-HAMILTONIAN STRUCTURE OF THE SUPERSYMMETRIC NONLINEAR SCHRÖDINGER EQUATION

1995 ◽  
Vol 10 (27) ◽  
pp. 2019-2028 ◽  
Author(s):  
J.C. BRUNELLI ◽  
ASHOK DAS
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Hamiltonian Structure ◽  
Kdv Equation ◽  
Nonlinear Schrodinger Equation ◽  
Hamiltonian Reduction ◽  
Hamiltonian Structures ◽  
Nonlinear Schrödinger ◽  
Supersymmetric Kdv Equation

We show that the supersymmetric nonlinear Schrödinger equation is a bi-Hamiltonian integrable system. We obtain the two Hamiltonian structures of the theory from the ones of the supersymmetric two-boson hierarchy through a field redefinition. We also show how the two Hamiltonian structures of the supersymmetric KdV equation can also be derived from a Hamiltonian reduction of the supersymmetric two-boson hierarchy.

scholarly journals A rigorous derivation of the Hamiltonian structure for the nonlinear Schrödinger equation

2020 ◽  
Vol 365 ◽  
pp. 107054
Author(s):  
Dana Mendelson ◽  
Andrea R. Nahmod ◽  
Nataša Pavlović ◽  
Matthew Rosenzweig ◽  
Gigliola Staffilani
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Hamiltonian Structure ◽  
Nonlinear Schrodinger Equation ◽  
Rigorous Derivation ◽  
Nonlinear Schrödinger

scholarly journals Note on nonlinear Schrödinger equation, KdV equation and 2D topological Yang–Mills–Higgs theory

2019 ◽  
Vol 34 (15) ◽  
pp. 1950074
Author(s):  
Jun Nian
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Kdv Equation ◽  
Nonlinear Schrodinger Equation ◽  
Yang Mills ◽  
Nonlinear Schrödinger ◽  
Small Coupling ◽  
The Kdv Equation ◽  
The Continuum

In this paper, we discuss the relation between the [Formula: see text]D nonlinear Schrödinger equation and the KdV equation. By applying the boson/vortex duality, we can map the classical nonlinear Schrödinger equation into the classical KdV equation in the small coupling limit, which corresponds to the UV regime of the theory. At quantum level, the two theories satisfy the Bethe ansatz equations of the spin-[Formula: see text] [Formula: see text] chain and the [Formula: see text] chain in the continuum limit, respectively. Combining these relations with the dualities discussed previously in the literature, we propose a duality web in the UV regime among the nonlinear Schrödinger equation, the KdV equation and the 2D [Formula: see text] topological Yang–Mills–Higgs theory.

UNSTABLE NONLINEAR SCHRÖDINGER EQUATION — A DRINFEL’D-SOKOLOV APPROACH AND BI-HAMILTONIAN STRUCTURE

1994 ◽  
Vol 09 (24) ◽  
pp. 2189-2195 ◽  
Author(s):  
A. ROY CHOWDHURY ◽  
I. MUKHOPADHYAY
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Hamiltonian Structure ◽  
Independent Set ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Field ◽  
Nonlinear Schrödinger ◽  
Master Symmetry

The unstable nonlinear Schrödinger equation is derived in the formalism of Drinfel’d-Sokolov. It is observed that the required reduction of the independent set of nonlinear field variables is automatic. Bi-Hamiltonian structure is deduced with the help of the master symmetry.

scholarly journals Diverse Soliton Structures for Fractional Nonlinear Schrodinger Equation, KdV Equation and WBBM Equation Adopting a New Technique

2021 ◽  
Author(s):  
Md. Tarikul Islam ◽  
Francisco Gomez ◽  
Md. Ali Akbar
Keyword(s):  
Schrödinger Equation ◽  
Differential Equations ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Kdv Equation ◽  
Nonlinear Schrodinger Equation ◽  
Auxiliary Equation ◽  
Nonlinear Schrödinger ◽  
Wave Solutions ◽  
Fractional Nonlinear Schrödinger Equation

Abstract Nonlinear fractional order partial differential equations standing for the numerous dynamical systems relating to nature world are supposed to by unraveled for depicting complex physical phenomena. In this exploration, we concentrate to disentangle the space and time fractional nonlinear Schrodinger equation, Korteweg-De Vries (KdV) equation and the Wazwaz-Benjamin-Bona-Mahony (WBBM) equation bearing the noteworthy significance in accordance to their respective position. A composite wave variable transformation with the assistance of conformable fractional derivative transmutes the declared equations to ordinary differential equations. A successful implementation of the proposed improved auxiliary equation technique collects enormous wave solutions in the form of exponential, rational, trigonometric and hyperbolic functions. The found solutions involving many free parameters under consideration of particular values are figured out which appeared in different shape as kink type, anti-kink type, singular kink type, bell shape, anti-bell shape, singular bell shape, cuspon, peakon, periodic etc. The performance of the proposed scheme shows its potentiality through construction of fresh and further general exact traveling wave solutions of three nonlinear equations. A comparison of the achieved outcomes in this investigation with the results found in the literature ensures the diversity and novelty of ours. Consequently, the improved auxiliary equation technique stands as efficient and concise tool which deserves further use to unravel any other nonlinear evolution equations arise in various physical sciences like applied mathematics, mathematical physics and engineering.

scholarly journals PLANE WAVE STABILITY OF THE SPLIT-STEP FOURIER METHOD FOR THE NONLINEAR SCHRÖDINGER EQUATION

2014 ◽  
Vol 2 ◽  
Author(s):  
ERWAN FAOU ◽  
LUDWIG GAUCKLER ◽  
CHRISTIAN LUBICH
Keyword(s):  
Schrödinger Equation ◽  
Plane Wave ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Fourier Method ◽  
Nonlinear Schrodinger Equation ◽  
Hamiltonian Reduction ◽  
Time Step ◽  
Time Step Size ◽  
Nonlinear Schrödinger

AbstractPlane wave solutions to the cubic nonlinear Schrödinger equation on a torus have recently been shown to behave orbitally stable. Under generic perturbations of the initial data that are small in a high-order Sobolev norm, plane waves are stable over long times that extend to arbitrary negative powers of the smallness parameter. The present paper studies the question as to whether numerical discretizations by the split-step Fourier method inherit such a generic long-time stability property. This can indeed be shown under a condition of linear stability and a nonresonance condition. They can both be verified in the case of a spatially constant plane wave if the time step-size is restricted by a Courant–Friedrichs–Lewy condition (CFL condition). The proof first uses a Hamiltonian reduction and transformation and then modulated Fourier expansions in time. It provides detailed insight into the structure of the numerical solution.

scholarly journals Waves that Appear From Nowhere: Complex Rogue Wave Structures and Their Elementary Particles

2021 ◽  
Vol 8 ◽  
Author(s):  
Nail Akhmediev
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Water Waves ◽  
Schrodinger Equation ◽  
Kdv Equation ◽  
Rogue Waves ◽  
Nonlinear Schrodinger Equation ◽  
Elementary Particles ◽  
Nonlinear Schrödinger ◽  
Wide Range

The nonlinear Schrödinger equation has wide range of applications in physics with spatial scales that vary from microns to kilometres. Consequently, its solutions are also universal and can be applied to water waves, optics, plasma and Bose-Einstein condensate. The most remarkable solution presently known as the Peregrine solution describes waves that appear from nowhere. This solution describes unique events localized both in time and in space. Following the language of mariners they are called “rogue waves”. As thorough mathematical analysis shows, these waves have properties that differ them from any other nonlinear waves known before. Peregrine waves can serve as ‘elementary particles’ in more complex structures that are also exact solutions of the nonlinear Schrödinger equation. These structures lead to specific patterns with various degrees of symmetry. Some of them resemble “atomic like structures”. The number of particles in these structures is not arbitrary but satisfies strict rules. Similar structures may be observed in systems described by other equations of mathematical physics: Hirota equation, Davey-Stewartson equations, Sasa-Satsuma equation, generalized Landau-Lifshitz equation, complex KdV equation and even the coupled Higgs field equations describing nucleons interacting with neutral scalar mesons. This means that the ideas of rogue waves enter nearly all areas of physics including the field of elementary particles.

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