A R Chowdhury

On the quasi-periodic solutions to the discrete nonlinear Schrodinger equation

The authors have obtained the quasi-periodic solutions to the discrete non-linear Schrodinger equation by a variant of a method due to Date and Tanaka (1976). It is shown explicitly that the non-linear field variables at the different lattice points can be determined in a recursive fashion in terms of combinations of Riemanns theta functions depending on lattice position and time.

Towards the conservation laws and Lie symmetries for the Khokhlov-Zabolotskaya equation in three dimensions

The authors have obtained the explicit structure for the generating function of Lie symmetries for the Khokhlov-Zabolotskaya equation describing the propagation of a sound beam in a non-linear medium. In the absence of a canonical framework they have derived the set of conservation laws through the technique of differential forms and prolongation. Such conservation laws are of utmost importance for the analysis of a sound beam in the medium.

Heisenberg subalgebra, dressing approach and super bi-Hamiltonian integrable system

Starting with the affine super-algebra Osp(2,2), we have shown how different choices of Heisenberg subalgebras in conjunction with the dressing operator approach of Wilson (1958) leads to different super-symmetric bi-Hamiltonian systems. Subsequently, the explicit form of the symplectic operators are deduced with the help of the trace identity. Lastly a possible relation with the tau -function is indicated.

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

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.

On a different approach to the bi-Hamiltonian structure of higher-order water-wave equations

Conservation laws and the bi-Hamiltonian structure of higher-order water-wave equations are obtained by a method different from the usual approach of symmetry analysis. The method is based on the technique of Fourier analysis and small amplitude expansion. Some comments are made about the symmetries of the equation. The recursion operator Lambda is then constructed from the two symplectic operators and it is explicitly verified that it is both a strong operator and a hereditary one.

On the complete integrability of the Hirota-Satsuma system

The authors have applied the Painleve Test to the coupled nonlinear system advocated by Hirota and Satsuma (1981). They have also made a Lie point symmetry analysis of these equations and have shown that the reduced ordinary nonlinear equations are not members of the Painleve class. Also the Painleve test itself is seen to fail, so that on both counts the equations are not completely integrable in the usual sense.

Current algebra, AKS theorem and new super evolution equations

The authors have deduced a new class of integrable super evolution equations by using a supersymmetric version of the AKS (Adler-Kostant-Sym) theorem in conjunction with the homogeneous space reduction technique of Marshall (1988) and current algebra of Samenov-Tian-Shansky (1983).

Stokes phenomena and the monodromy deformation problem for the non-linear Schrodinger equation

Following Flaschka and Newell (1979) the authors have formulated the inverse problem for Painleve IV, with the help of similarity variables. The Painleve IV arises as the eliminant of the two second-order differential equations originating from the non-linear Schrodinger equation. They have obtained the asymptotic expansions near the singularities at 0 and infinity of the complex eigenvalue plane. The corresponding analysis then displays Stokes phenomena. The monodromy matrices connecting the solution Yj in the sector Sj to that in Sj+1 are fixed in structure by the imposition of certain conditions. They then show that a deformation keeping the monodromy data fixed leads to the non-linear Schrodinger equation. At this point they may mention that, while Flaschka and Newell did not make any absolute determination of the Stokes parameter, the approach yields the values of the Stokes parameter in an explicit way, which in turn can determine the matrix connecting the solutions near 0 and infinity . Such a realisation was not possible in the approach of Flashchka and Newell. Lastly they show that the integral equation originating from the analyticity and asymptotic nature leads to the similarity solution previously determined by Boiti and Pempinelli (1979).

Bi-Hamiltonian structure and Lie-Backlund symmetries for a modified Harry-Dym system

The authors have obtained the complete Lie-Backlund symmetry for a modified Harry-Dym system and hence deduced the bi-Hamiltonian structure associated with it. It is shown that these Lie-Backlund symmetries generate the recursion operator inherent in the theory and the conserved quantities can also be computed according to the Dorfman prescription.

Algebraic Bethe ansatz with boundary condition for SUp,q(2) invariant spin chain

We have analysed a generalized Heisenberg spin chain invariant under the quantum group SUp,q(2). Instead of the usual periodic boundary condition, non-trivial boundary conditions are imposed a la Sklyanin(1988). R-matrix and Bethe eigenstates are constructed explicitly. The R-matrix describes a vertex type model which can be shown to be connected to the xxz type six vertex model under the Akutsu-Deguchi-Wadati-type symmetry-breaking transformation.