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Journal of Physics A Mathematical and General

On a Bethe-ansatz approach to the derivative nonlinear Schrodinger equation

S Sen, Asesh Roy Chowdhury

Published Nov 11, 1987• 7 citations
Publisher

Abstract

The authors continue their analysis of the extended derivative nonlinear Schrodinger equation with the help of the Bethe-ansatz technique. In the previous analysis by the QISM approach it was not possible to consider the reduction of the extended system to the derivative NLSE, as the theory then becomes non-ultralocal. Therefore, they have applied the approach of the Bethe ansatz to both the extended DNLSE and the usual DNLSE, and show that it is possible to construct multiparticle quantum states in both the cases.

DOI
10.1088/0305-4470/20/16/010

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The authors continue their analysis of the extended derivative nonlinear Schrodinger equation with the help of the Bethe-ansatz technique. In the previous analysis by the QISM approach it was not possible to consider the reduction of the extended system to the derivative NLSE, as the theory then becomes non-ultralocal. Therefore, they have applied the approach of the Bethe ansatz to both the extended DNLSE and the usual DNLSE, and show that it is possible to construct multiparticle quantum states in both the cases.
We have reformulated the Bethe Ansatz approach to a Derivative Nonlinear Schrödinger equation, known to be a non-ultralocal system. In this present version we have shown that an approach exactly similar to the usual NLS case is possible. Furthermore the Yang-Baxter equation and factorization is shown to be valid. Afterwards the R (μ, v ) matrix has been constructed from the Bethe Ansatz result (which is impossible otherwise, as the Faddeev approach is not applicable) and the commutation rules for the scattering data is deduced. These are then utilised to construct the algebraic Bethe ansatz eigen states. Next the periodicity condition is invoked and the eigenvalues are determined via some algebraic equations which are also seen to be reproduced via the algebraic approach. Lastly we consider the possibility of exceptional states in the algebraic Bethe states.
We have formulated the quantum inverse problem for an alternative version of the nonlinear Schr\"odinger equation (NLS), which is a variant of the derivative NLS (DNLS) equation. While the DNLS equation is nonultralocal and cannot be formulated as a problem of the quantum inverse scattering method, our equation is ultralocal, and hence it is possible to construct the quantum R matrix and the commutation rules for the scattering data. Next we have constructed the algebraic Bethe ansatz, from which the eigenvalue equation for the excited states is derived. This eigenvalue equation admits stringlike configurations, and the density of states corresponding to real and complex eigenvalues is explicitly derived in the thermodynamic limit.
Nonlocal nonlinear Schrodinger model is quantised and exactly solved using the canonical framework. It is found that the usual canonical quantisation of the model leads to a theory with pathological inner product. This problem is resolved by constructing another inner product over the vector space of the theory. The resultant theory is found to be identical to that of nonrelativistic bosons with delta function interaction potential, devoid of any nonlocality. The exact eigenstates are found using the Bethe ansatz technique.
The generalisation of the quantum inverse scattering method is used to derive the Bethe ansatz equation of a system for a multicomponent nonlinear Schrodinger models with n species (or colours) of bosons or fermions. The matrix periodic boundary conditions for bosons and fermions are obtained and the relationship between them discussed. The eigenstates for the infinite conserved quantities of the system are constructed. Lastly, the author generalises these results to the non-linear Schrodinger model with the mirror-image delta potentials proposed by Schulz (1987).