A. Roy Chowdhury

On the complete integrability of the supersymmetric nonlinear Schrödinger equation

The Painleve criterion has been applied to the supersymmetric nonlinear Schrodinger equation. This particular system of fermionic and bosonic fields shows up a rich spectrum of resonances and it can be explicitly proved that the expansion coefficients at the resonance positions can remain arbitrary. At this point is is worth noting that even when the extra nonlinear field (which is fermionic in this case) is considered to be bosonic, the resulting system turns out to satisfy the Painleve test so that this second system may be thought of as a new completely integrable system whose Lax pair is still to be found.

On Some Cnoidal Wave Solutions of Some Non-Linear Equations in One and Two Dimension

We have obtained, following Hirotas direct approach, some cnoidal wave solutions of some new nonlinear equations in one and two dimension. In the course of our calculation we have also extended the method for application to nonlinear equations of higher order, where auxiliary independent variables are to be introduced for converting it into Hirotas framework. Lastly, solutions of some equations are obtained for which still now no inverse scattering is known to exist.

On the prolongation approach in three dimensions for the conservation laws and Lax pair of the Benjamin–Ono equation

The prolongation structure approach of Wahlquist and Estabrook [J. Math. Phys. 16, 1 (1975)] is used effectively in a new situation in relation to the integrodifferential type BO equation (the Benjamin–Ono equation). The main clue lies in the possible differential equation representation of such equations in three dimensions. Here it is shown how the usual analysis of prolongation structure can be utilized to deduce a Lax pair for a BO type equation in three dimensions. Effectiveness of the present approach is further demonstrated by an independent derivation of some conservation laws associated with the equation. Last, the whole formalism is reduced to two dimensions to make contact with known results.

Group-theoretic approach to the conservation laws of the KP equation in lagrangian and Hamiltonian formalism

The conservation laws—precisely speaking, the basis of the conservation laws—are obtained through the use of Noethers theorem, Lie symmetry, and a theorem due to Ibragimov. Though in principle for each generator of Lie symmetry there should be a different conserved vector, due to the closed Lie algebra generated by the generators, some of these vectors become no longer independent. The theorem of Ibragimov is used to construct a basis in the case of the KP equation in three dimensions. It is then shown how the same analysis can be performed through the Hamiltonian formalism.

A Memory Function Approach to Stochastic KdV Equation

We have developed a memory function approach to stochastic KdV equation. The behaviour of a KdV soliton and the corresponding many soliton solution are analyzed when under the influence of a random force obeying a Gaussian law of fluctuation. An interesting feature of our calculation is that in spite of the presence of the random perturbation, the solutions can be obtained and analyzed in an exact manner through the technique of memory function.

Prolongation structure of a new integrable system

We have obtained the inverse scattering equations associated with a new pair of coupled nonlinear evolution equations in two dimensions. The spectral parameter is introduced by invoking the invariance of the equation set, and imposing those on the Lax pair.

On a painleve test for the complete integrability of Bogomolny's monopole equation

Abstract We have made an analysis of the monopole equation of Bogomolny from the standpoint of the Painleve test. The idea that any non-linear partial differential equation admitting a Lax representation should conform to the criterion of the Painleve analysis seems to hold well in case of the Bogomolny equation. We have determined the position for resonances and have proved that at each of these the coefficients in the Frobenius type expansion of the gauge potentials do become arbitrary signalling the complete integrability of the system.