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Acta Physica Polonica B

Separability of Two-oscillator Mixed States from the Subspace with Fixed Total Occupation Quantum Number

Buang Ann Tay

Published Jan 1, 2014• 0 citations
Publisher

Abstract

This paper obtains the exact 1-soliton solution to the chiral nonlinear Schrödinger's equation.There are three types of integration architectures that are implemented in this paper.They are the functional variable approach, first integral method as well as the ansatz method.These soliton solutions are obtained.There are constraint conditions that also fall out which must remain valid in order for the solitons and other solutions to exist.

DOI
10.5506/aphyspolb.45.867

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This paper obtains the exact 1-soliton solution to the chiral nonlinear Schrödinger's equation.There are three types of integration architectures that are implemented in this paper.They are the functional variable approach, first integral method as well as the ansatz method.These soliton solutions are obtained.There are constraint conditions that also fall out which must remain valid in order for the solitons and other solutions to exist.
We return to the notion of optical solitons as these were first introduced—namely as exact soliton and multi-soliton solutions of the semi-classical self-induced transparency (SIT) equations. The SIT equations, even in their most general inhomogeneously broadened forms, are infinite dimensional, completely integrable Hamiltonian systems, and thus display much mathematical structure, yet their exact soliton solutions display well the nonlinear (self-induced) transparency features of the physical phenomenon. The mathematical feature of complete integrability is handed down from the SIT equations to the nonlinear Schrödinger equations and from this point of view the nonlinear Schrödinger (NLS) equations equations are studied first of all at classical level and then at quantum level. The ‘quantum soliton’ of the quantum attractive NLS model is defined and its properties investigated in detail, particularly in a comparison with the particular properties ascribed to the ‘quantum soliton’ as described in a recently published article (Abraham 1999 Phys. World (February) 21) with this title; as suggested in that article, the quantum NLS soliton is seen as a qubit for quantum information purposes. The covariant form of the quantum attractive NLS model is the quantum sine-Gordon model and this is briefly studied in the context of quantum information also; new results for large area Josephson junctions are reported in this context. The quantum soliton of the one dimensional attractive NLS model has apparently been observed in 2002 in a Bose condensed metal vapour ( 7 Li, (Khaykovich et al 2002 Science 296 1290)) and a detailed calculation suggests that the condensate undergoes a ‘collapse’ from three to one space dimensions thus forming an exact soliton solution of the one dimensional quantum attractive NLS model.
The Wigner function, a kind of wavelet transform is studied for one- and two-soliton solutions to the nonlinear Schrödinger equation. The exact expression of the Wigner transform for the one-soliton solution is obtained. The features of soliton-soliton interaction of different two-soliton states, the states of coherence analogous to that of two-localized wave packets in quantum mechanics, are displayed. Suitability of this transform for analyzing “ soliton ” in open systems is also discussed.
Multicomponent nonlinear Schrödinger equations constitute fundamental models for coherent matter waves in multicomponent Bose--Einstein condensates, spinor quantum fluids, and vector nonlinear optical systems. We develop a vector formulation of the Hirota bilinear formalism for the completely integrable Manakov system that treats the coupled nonlinear Schrödinger equations directly at the vector level rather than through the conventional component-wise decomposition. This framework naturally retains the intrinsic multicomponent representation of the model while providing compact analytical expressions for exact vector soliton solutions. Within this approach, we systematically construct bright, dark, and mixed one-, two-, and three-soliton solutions and show how the underlying vector structure provides a unified description of their nonlinear interactions. In particular, the proposed formalism makes the coupling between the different components explicit while preserving the geometric organization of the vector system throughout the bilinearization procedure. Beyond its analytical simplicity, the framework offers a natural perspective for the study of coherent multicomponent nonlinear excitations and provides a foundation for extending vector Hirota methods to other classes of exact solutions, including rogue waves, periodic waves, and rational solutions.
This paper addresses the singular 1-soliton solution of the K(m, n) equation that is being considered with generalized evolution.The ansatz method will be used to extract the singular 1-soliton solution of this equation.A couple of constraint conditions will fall out in order for the singular soliton solutions to exist.Subsequently, the special cases of this equation will be studied.They are the KdV and the mKdV equations where the extended (G /G)-expansion method will be employed to extract a few nonlinear wave solutions.