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Physical Review A

Quantum inverse problem for an unstable nonlinear Schrödinger equation: A functional Bethe ansatz

N. Das Gupta, Asesh Roy Chowdhury

Published Jun 1, 1995• 1 citation
Publisher

Abstract

We have formulated the quantum inverse problem for the unstable nonlinear Schr\"odinger equation that forms the basis of nonlinear optics, following the modified approach of Sklyanin. Due to the nonexistence of the vacuum for the corresponding Lax L operator, the algebraic Bethe ansatz cannot be formulated. This difficulty is removed by recourse to the functional Bethe ansatz. The whole analysis is performed by considering different boundary conditions at the two ends of the interval on the x axis. Both signs of the nonlinearity are treated, corresponding to ``bright'' and ``dark'' solitons.

DOI
10.1103/physreva.51.4396

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We have formulated the quantum inverse problem for the unstable nonlinear Schr"odinger equation that forms the basis of nonlinear optics, following the modified approach of Sklyanin. Due to the nonexistence of the vacuum for the corresponding Lax L operator, the algebraic Bethe ansatz cannot be formulated. This difficulty is removed by recourse to the functional Bethe ansatz. The whole analysis is performed by considering different boundary conditions at the two ends of the interval on the x axis. Both signs of the nonlinearity are treated, corresponding to bright'' and dark'' solitons.
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.
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 extended the previous analysis of the quantum inverse problem for the Alfv\'en wave propogation by incorporating boundary conditions other than the periodic ones. Our approach is that of Sklyanin [Func. Ana. Appl. 21, 164 (1986)], which allows one to introduce different boundary conditions at the two ends. A generalized Bethe ansatz is used to deduce the eigenvalues of the Hamiltonian. An immediate outcome of our analysis is the effect of finite-size corrections, which is essential to study scale invariance and conformal properties of the model.
We consider the integrability problem for a quantum version of the perturbed nonlinear Schr\"odinger (NS) equation, including a higher spatial dispersion and nonlinear dispersion of the group velocity (the corresponding classical equations are well known in the nonlinear fiber optics and in other applications). Employing the Bethe ansatz (BA) technique, which is known to yield a complete spectrum including the so-called quantum solitons (multiparticle bound states) of the unperturbed NS system, we find that the particular cases of the model which correspond to integrable classical equations, viz., the derivative NS and Hirota equations, are also fully integrable at the quantum level. In the generic (nonintegrable) case, the model remains integrable in the two-particle sector. In the three-particle sector, the BA produces unphysical states with complex energy. It turns out that the Hamiltonian in this case becomes non-Hermitian. We propose a procedure for finding the physical eigenstates of the system. We build an example of such a state and we show that it describes inelastic scattering.