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.

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.

With a view to obtain exact analytic solutions to the time-dependent Schrödinger equation for a few potentials of physical interest in three dimensions, transformation-group method is used. Interestingly, the integrals of motion in the new coordinates turn out to be the desired invariants of the systems.

We deal with the difficulties claimed by the author of [Ann. Phys. 206, 90 (1991)] while solving the Schrödinger equation for the ground states of two-dimensional anharmonic potentials. It is shown that the ground state energy eigenvalues and eigen-functions for the coupled quadratic and quartic potentials can be obtained by making some simple assumptions. Expressions for the energy eigenvalues and the eigenfunctions for the first and second excited states of these systems are also obtained.

We investigate the quasi-exact solutions of an analogous Schrödinger wave equation for two-dimensional non-Hermitian complex Hamiltonian systems within the framework of an extended complex phase space characterized by $x = x_{1} + ip_{3}$, $y = x_{2} + ip_{4}$, $p_{x} = p_{1} + ix_{3}$, $p_{y} = p_{2} + ix_{4}$. Explicit expressions for the energy eigenvalues and eigenfunctions for ground and first excited states of a two-dimensional $\mathcal{PT}$-symmetric sextic potential and some of its variants are obtained. The eigenvalue spectra are found to be real within some parametric domains.

The Schrödinger equation was solved for a generalized $\mathcal{PT}$-symmetric quartic potential in two dimensions. It was found that, under a suitable ansatz for the wave function, the system possessed real and discrete energy eigenvalues. Analytic expressions for the energy eigenvalues and the eigenfunctions for the first four states were obtained. Some constraining relations among the wave function parameters rendered the problem quasi-solvable.

The ($G'/G$)-expansion method and its simplified version are used to obtain generalized travelling wave solutions of five nonlinear evolution equations (NLEEs) of physical importance, viz. the ($2+1$)-dimensional Maccari system, the Pochhammer–Chree equation, the Newell–Whitehead equation, the Fitzhugh–Nagumo equation and the Burger–Fisher equation. A variety of special solutions like periodic, kink–antikink solitons, bell-type solitons etc. can easily be derived from the general results. Three-dimensional profile plots of some of the solutions are also drawn.

We use rationalization method to study two-dimensional complex dynamical systems (shifted harmonic oscillator in complex plane) on the extended comples phase space (ECPS). The role and scope of the derived invatiants in the context of various physical problems are high-lighted.

Complex dynamical invariants are searched out for two-dimensional complex potentials using rationalization method within the framework of an extended complex phase space characterized by $x = x_{1} + ip_{3}. y = x_{2} + ip_{4}, p_{x} = p_{1} + ix_{3}, p_{y} = p_{2} + ix_{4}$. It is found that the cubic oscillator and shifted harmonic oscillator admit quadratic complex invariants. THe obtained invariants may be useful for studying non-Hermitian Hamiltonian systems.