Journal of Nonlinear Mathematical Physics

We study representations of the generalized Poincaré group and its extensions in the class of Lie vector fields acting in a space of n + m independent and one dependent variables. We prove that an arbitrary representation of the group P(n, m) with max {n, m} ≥ 3 is equivalent to the standard one, while the conformal group C (n, m) has non-trivial nonlinear representations. Besides that, we investigate in detail representations of the Poincaré group P (2, 2), extended Poincaré groups ̃P (1, 2), ̃P (2, 2), and conformal groups C (1, 2), C (2, 2) and obtain their linear and nonlinear representations.

Formalism of differential forms is developed for a variety of Quantum and noncommutative situations.

We discuss integrable extensions of real nonlinear wave equations with multi-soliton solutions, to their bicomplex, quaternionic, coquaternionic and octonionic versions. In particular, we investigate these variants for the local and nonlocal Korteweg-de Vries equation and elaborate on how multi-soliton solutions with various types of novel qualitative behaviour can be constructed. Corresponding to the different multicomplex units in these extensions, real, hyperbolic or imaginary, the wave equations and their solutions exhibit multiple versions of antilinear or PT-symmetries. Utilizing these symmetries forces certain components of the conserved quantities to vanish, so that one may enforce them to be real. We find that symmetrizing the noncommutative equations is equivalent to imposing a PT-symmetry for a newly defined imaginary unit from combinations of imaginary and hyperbolic units in the canonical representation.

The subject of this paper is to propose a numerical algorithm for solving 2D diffusion and diffusion-wave equations of distributed order fractional derivatives. Such equations arise in modelling complex systems and have many important applications. Existence of integral term over the order of fractional derivative causes the high complexity of these equations and so their numerical solutions needs special cares. Using Gauss quadrature approach for discretizing the integral term of fractional derivative converts the distributed equation into a multi-term fractional differential equation. Then, the time variable is discretized with a suitable finite difference approach. The resultant semi-discretized equations are fully discretized by a radial basis function-generated finite difference based method. Convergence of the method are studied numerically. Various kind of test problems are considered for a comprehensive numerical study and the results confirm the efficiency of the method.

In this paper, we bring out the Lie symmetries and associated similarity reductions of the recently proposed (2+1) dimensional long dispersive wave equation. We point out that the integrable system admits an infinite-dimensional symmetry algebra along with Kac-Moody-Virasoro-type subalgebras. We also bring out certain physically interesting solutions.