S. C. Mishra

Exact travelling wave solutions of some nonlinear equations by G′G-expansion method

Abstract Here we find exact solutions of some nonlinear evolution equations within the framework of the G ′ G -expansion method. Exact solutions of five nonlinear equations of physical importance viz. the coupled Schrodinger–KdV equation, the coupled nonlinear Reaction–Diffusion equation, the Foam Drainage equation, the Phi-Four equation and the Dodd–Bullough–Mikhailov equation are obtained. These general solutions can be reduced in some standard results derived by some other methods. Two and three dimensional plots of some of the results are also presented.

Solution of Schrödinger Equation for Two-Dimensional Complex Quartic Potentials

We investigate the quasi-exact solutions of the Schr?dinger wave equation for two-dimensional non-hermitian complex Hamiltonian systems within the frame work of an extended complex phase space characterized by x = x1 + ip3, y = x2 + ip4, px = p1 + ix3, py = p2 + ix4. Explicit expressions of the energy eigenvalues and the eigenfunctions for ground and first excited states for a complex quartic potential are obtained. Eigenvalue spectra of some variants of the complex quartic potential, including PT-symmetric one, are also worked out.

The solution of the Schrödinger equation for complex Hamiltonian systems in two dimensions

We investigate the ground state solutions of the Schrodinger equation for complex (non-Hermitian) Hamiltonian systems in two dimensions within the framework of an extended complex phase-space approach. The eigenvalues and eigenfunctions of some two-dimensional complex potentials are found.

Construction of exact dynamical invariants of two-dimensional classical system

A general method is used for the construction of second constant of motion of fourth order in momenta using the complex coordinates (z,z-). A fourth-order potential equation is obtained whose solutions directly provide a large class of integrable systems. The potential equation is tested with an interesting example which admits second constants of motion.

Eigenspectra of a complex coupled harmonic potential in three dimensions

Within the framework of extended complex phase space approach characterized by position and momentum coordinates, we investigate the quasi-exact solutions of the Schrodinger equation for a coupled harmonic potential and its variants in three dimensions. For this purpose ansatz method is employed and nature of the eigenvalues and eigenfunctions is determined by the analyticity property of the eigenfunctions alone. The energy eigenvalue is real for the real coupling parameters and becomes complex if the coupling parameters are complex. However, in case of complex coupling parameters, the imaginary component of energy eigenvalue reduces to zero if the P T -symmetric condition is satisfied. Thus a non-hermitian Hamiltonian possesses real eigenvalue if it is P T -symmetric.

Closed-form solutions of the Schrödinger equation for a coupled harmonic potential in three dimensions

Using the ansatz method, we obtain the exact closed-form solutions of a time independent Schrodinger equation for a coupled harmonic potential and its variant in three dimensions. Some authors [R.S. Kaushal, Quantum mechanics of noncentral harmonic and anharmonic potentials in two dimensions, Ann. Phys. (N.Y.) 206 (1991) 90-105] have raised the difficulty to solve the Schrodinger equation for coupled potentials but we found that by imposing some restrictions on ansatz parameters and at the cost of certain constraints on the potential parameters, ground state as well as excited state solutions of the Schrodinger equation are obtained. The number of constraints increases with increase in anharmonicity of the potential.

Second invariant for two-dimensional classical super systems

Construction of superpotentials for two-dimensional classical super systems (forN > 2) is carried out. Some interesting potentials have been studied in their super form and also their integrability.