Equation admitting linearization and describing waves in dissipative media with modular, quadratic, and quadratically cubic nonlinearities

Abstract

Abstract-A second-order partial differential equation admitting exact linearization is discussed. It contains terms with nonlinearities of three types-modular, quadratic, and quadratically cubic-which can be present jointly or separately. The model describes nonlinear phenomena, some of which have been studied, while others call for further consideration. As an example, individual manifestations of modular nonlinearity are discussed. They lead to the formation of singularities of two types, namely, discontinuities in a function and discontinuities in its derivative, which are eliminated by dissipative smoothing. The dynamics of shock fronts is studied. The collision of two single pulses of different polarity is described. The process reveals new properties other than those of elastic collisions of conservative solitons and inelastic collisions of dissipative shock waves.

Collisions, mutual losses and annihilation of pulses in a modular nonlinear medium

“…(8) and (9). The following is rather standard and is described, for example, in monographs [15,16] and in the textbook (collection of problems) [19].…”

“…The annihilation effect exists there too, and it is therefore interesting to consider this more general case. The existence of dissipation helps us to eliminate the two bothersome singularities, namely the breaks of the wave and the breaks of its derivative [8,10], which are eliminated by dissipative smoothing [8]. Consequently, in order to find a continuous solution for a wave in a modular medium one must solve Eq.…”

“…Equation (9) is a modified form of Burgers' equation. Other models of similar type have recently been investigated [8][9][10]17,18]. Here, we will now use Eq.…”

“…The term modular nonlinearity is used here as defined in the works [8][9][10]. From a mathematical point of view, it means that the modular term is present in the equation of motion.…”

“…The collision of two nonoverlapping pulses of different polarities was considered only for the case Γ = 0 [8]. First the distance between the pulses decreases as the pulses propagate linearly with different velocities.…”