scholarly journals Special decompositions and linear superpositions of nonlinear systems: BKP and dispersionless BKP equations

2021 ◽  
Author(s):  
Xiazhi Hao ◽  
Senyue Lou
Keyword(s):  
Nonlinear Systems ◽  
Integrable Systems ◽  
Structure Characteristic ◽  
Linear Superposition ◽  
Bkp Hierarchy ◽  
Bkp Equation ◽  
Classical Integrable

The existence of decomposition solutions of the well-known nonlinear BKP hierarchy is analyzed. It is shown that these decompositions provide simple and interesting relationships between classical integrable systems and the BKP hierarchy. Further, some special decomposition solutions display a rare property: they can be linearly superposed. With the emphasis on the case of the fifth BKP equation, the structure characteristic having linear superposition solutions is analyzed. Finally, we obtain similar superposed solutions in the dispersionless BKP hierarchy.

scholarly journals Bethe Ansatz and Classical Hirota Equation

1997 ◽  
Vol 11 (01n02) ◽  
pp. 75-89 ◽  
Author(s):  
P. Wiegmann
Keyword(s):  
Bethe Ansatz ◽  
Integrable Systems ◽  
Spectral Characteristics ◽  
Quantum Integrable Systems ◽  
Hirota Equation ◽  
S Matrix ◽  
Elliptic Solutions ◽  
Quantum Integrable Models ◽  
Set Up ◽  
Classical Integrable

We discuss an interrelation between quantum integrable models and classical soliton equations with discretized time. It appeared that spectral characteristics of quantum integrable systems may be obtained from entirely classical set up. namely, the eigenvalues of the quantum transfer matrix and the scattering S-matrix itself are identified with a certain τ-functions of the discrete Liouville equation. The Bethe ansatz equations are obtained as dynamics of zeros. For comparison we also present the Bethe ansatz equations for elliptic solutions of the classical discrete Sine-Gordon equation. The paper is based on the recent study of classical integrable structures in quantum integrable systems.1

DEFORMATIONS OF LOOP ALGEBRAS AND CLASSICAL INTEGRABLE SYSTEMS: FINITE-DIMENSIONAL HAMILTONIAN SYSTEMS

2004 ◽  
Vol 16 (07) ◽  
pp. 823-849 ◽  
Author(s):  
T. SKRYPNYK
Keyword(s):  
Lie Algebras ◽  
Hamiltonian Systems ◽  
Spectral Parameter ◽  
Integrable Systems ◽  
Integrable Hamiltonian Systems ◽  
Infinite Dimensional ◽  
Loop Algebras ◽  
Finite Dimensional ◽  
Quasigraded Lie Algebras ◽  
Classical Integrable

We construct a family of infinite-dimensional quasigraded Lie algebras, that could be viewed as deformation of the graded loop algebras and admit Kostant–Adler scheme. Using them we obtain new integrable hamiltonian systems admitting Lax-type representations with the spectral parameter.

scholarly journals COHERENT STATES OF QUANTUM NONLINEAR SYSTEMS

1995 ◽  
Vol 09 (28n29) ◽  
pp. 1839-1844 ◽  
Author(s):  
Z. HABA
Keyword(s):  
Nonlinear Systems ◽  
Wave Packet ◽  
Integrable Systems ◽  
Coherent States ◽  
Quantum Dynamics ◽  
Wave Packets

Quantum dynamics of integrable systems is discussed. Localized wave packets generalizing the conventional coherent states of minimal uncertainty are constructed. The wave packet moves along a certain trajectory and does not change its shape for times of order [Formula: see text].

scholarly journals A QUANTUM BRST–ANTI-BRST APPROACH TO CLASSICAL INTEGRABLE SYSTEMS

2004 ◽  
Vol 19 (09) ◽  
pp. 693-702 ◽  
Author(s):  
MICHAEL CHESTERMAN ◽  
MARCELO B. SILKA
Keyword(s):  
Wave Function ◽  
Quantum Theory ◽  
Inverse Scattering ◽  
Integrable Systems ◽  
Classical Mechanics ◽  
Lax Pair ◽  
Liouville Integrability ◽  
Lax Equation ◽  
Scattering Wave ◽  
Classical Integrable

We reformulate the conditions of Liouville integrability in the language of Gozzi et al.'s quantum BRST–anti-BRST description of classical mechanics. The Das–Okubo geometrical Lax equation is particularly suited for this approach. We find that the Lax pair and inverse scattering wave function appear naturally in certain sectors of the quantum theory.

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