On the bifurcation of energy in media governed by (2 + 1)-dimensional modified Klein–Gordon equations

Abstract

In the present work, we develop a numerical technique to approximate solutions of a general class of two-dimensional modified Klein–Gordon equations that appears (amongst other applications) in the study of lattices consisting of Josephson junctions coupled through superconducting wires when dissipative and dispersive effects are taken into account. The basic motivation for our study is the need to posses a numerically accurate finite-difference scheme to approximate the occurrence of the process of nonlinear supratransmission when the mentioned systems are submitted to Dirichlet, Neumann or, more generally, boundary conditions of mixed type. The computational technique that we present in this paper is nonstandard, implicit, nonlinear, and not only consistent order O ( Δ t ) 2 and conditionally stable on the approximation of solutions but it also yields consistent approximations of the local energies of the individual junctions, of the total energy of the system and, moreover, of the rate of change of energy of the system with respect to time, making it an ideal method in the analysis of the process of supratransmission. Generalizations to higher-dimensional cases may be induced from the one- and two-dimensional versions of the method, and applications of the technique are given to the determination of the occurrence of nonlinear supratransmission under the presence of several parameters. Our results show that the transmission of energy in the system starts at a well-defined nonnegative lower threshold, after which the creation of localized intrinsic modes that move away from the driving boundaries is imminent. The study of the effects of nonzero mass, and internal and external damping is carried out from a computational perspective.