Swapna Roy

On the Bäcklund transformation and Hamiltonian properties of superevaluation equations

Backlund and Darboux–Backlund transformations are deduced for the superevaluation equations recently deduced by Kupershmidt [B. Kuperschmidt, ‘‘A super‐K–dV equation—An integrable system,’’ preprint UTSI‐Tullahoma, 1984; Phys. Lett. A 102, 213 (1983); J. Phys. A 17, L 863 (1984)] and Gurses [H. Gurses and O. Oguz, Phys. Lett. A 108, 437 (1985)]. By a extension of the technique of BPT [M. Boiti, F. Pempinelli, and G. Z. Tu, Nuovo Cimento B 79, 231 (1984)] to anticommuting variables the bi‐Hamiltonian structure and hence the form of the recursion operator for the Lie–Backlund symmetry for such equations are deduced. Incidentally some explicit forms of the Lie–Backlund symmetry are also deduced.

Variational Approach to the Hamiltonian Structure of a New Hierarchy of Nonlinear Equations

We determine the symplectic Hamiltonian structure associated with the nonlinear evolution equations obtained from two new isospectral problems. We follow the method of variation with respect to the field variables. An explicit example is given to demonstrate the new class of equations that are generated.

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.

On the Hamiltonian structure and generalised Lie-Backlund symmetries of Langmuir solitons

The structure of generalised Lie-Backlund symmetries for the coupled equations of Langmuir solitons are analysed in detail. The form of these symmetries, when compared with those of the conservation laws, yields the first symplectic form. The spectra gradient method is then seen to generate the recursion operator for these symmetries which, on factorisation, leads immediately to the second Hamiltonian structure. The same recursion operator, when used along with the (x,t)-dependent symmetries, yields a new class of generalised symmetries for the equations under consideration. Lastly it is observed that these symmetries are in involution with respect to a Jacobi bracket.

Loop Algebra of Lie symmetries for a short-wave equation

It is demonstrated that Lie point symmetries associated with a nonlinear equation for short waves in three dimensions generate an infinite-dimensional Lie algebra—a loop Algebra. Classification of the independent sets of the subalgebra is done through the adjoint action of the corresponding generators. Different forms of similarity solutions are discussed.

Prolongation theory. A new nonlinear Schrödinger equation

We discuss a new kind of nonlinear Schrödinger equation from the viewpoint of prolongation theory. It is shown that the equation possess a Lax pair with a 3 × 3 matrix structure. It is further demonstrated that by a multiple scale perturbation of Zakharovet al. it can be reduced to the usual KdV equation.

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.

ON THE GLOBAL SOLUTION TO THE DIFFERENTIAL EQUATION FOR THE N-POINT CONFORMAL CORRELATOR

We have obtained compact expressions for the global solutions of the second order differential equations for the n-point conformal correlation functions. These equations were initially deduced by Belavin, Polyakov and Zamolodchikov. The monodromy property of such solutions can be ascertained from these expressions very easily.

Kac-Moody Algebra, Nonlocal Symmetries, and Backlund Transformation for KdV Equation

Gauge transformation of the Lax eigenfunction through the explicit use of Lie group generators is seen to generate a two-parameter Backlund transformation. Explicit integration of this in two particular cases leads to sech2θ type and rational solutions starting from the trivial one. A method is indicated to generate infinitesimal transformations aroundu in the sense of Steudel, which in this case leads to a nonlocal structure of transformations.