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

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## Abstract

One of the most important sections of nonlinear wave theory is related to the collisions of single pulses. These often exhibit corpuscular properties. For example, it is well known that solitons described by the Korteweg–de Vries equation and a few other conservative model equations exhibit properties of elastic particles, while shock waves described by dissipative models like Burgers’ equation stick together as absolutely inelastic particles when colliding. The interactions of single pulses in media with modular nonlinearity considered here reveal new physical features that are still poorly understood. There is an analogy between the single pulses collision and the interaction of clots of chemical reactants, such as fuel and oxidant, where the smaller component disappears and the larger one decreases after a reaction. At equal “masses” both clots can be annihilated. In this work various interactions of two and three pulses are considered. The conditions for which a complete annihilation of the pulses occurs are indicated.

## Keywords

Nonlinear partial differential equation Mutual loss Annihilation Modular nonlinearity Bimodular media Pulse interaction## Mathematics Subject Classification

35C07 35G20 35K55 74J30 74J40 76E30## 1 Introduction

Nonlinear interactions of single pulse waves propagating in dispersive and dissipative media have been extensively studied. Historically, special attention was paid to solitary waves (solitons) behaving like elastic particles [1]. The analogy between nonlinear waves and quasi-stable particles is interesting both for the understanding of the fundamental properties of matter [2] and for analyzing processes of a different physical nature [3, 4, 5]. Collisions between solitons were studied in conservative media with diversified types of nonlinearity [6]. However, corpuscular properties are present also for nonlinear waves in media with energy dissipation where weak shock waves collide as inelastic particles. As a result of multiple mergers, a “large particle” is formed, the mass being equal to the sum of the masses of the colliding smaller particles [7]. As far as we know, collisions in dissipative media have been studied in depth only for the case of quadratic nonlinearity. The collisions of single pulses in media with modular nonlinearity considered in this work demonstrate new properties. Before the collision, the pulses preserve their shape. Once the collision begins, a mutual absorption starts which under certain conditions leads to total annihilation of the pulses.

*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 following anharmonic oscillator is a simple example of such a nonlinear system:

*g*|

*u*| replaces in Eq. (1) the usual quadratic nonlinear term \(g\, u^{2}\) . Equation (1) describes the vibration of a mass on a spring, which is piecewise linear and has higher elasticity with respect to the compression (\(k(1+g)\)) than when extended (\(k(1-g)\)). When the nonlinear parameter \(|g|<1\), Eq. (1) describes a periodic motion in a force field with the potential \(2W/k=u^{2}+g\, u|u|\).

Let the masses be located periodically along the *x*-direction, at equilibrium distance *a* from each other. The small black arrows mark their origins. The displacement of mass *n* is denoted by \(u_{n}\), where \(n=0, \pm 1, \pm 2, \ldots \) The nearest neighbors are connected by an ensemble of springs showing as a whole the properties of a modular nonlinearity (1). The first group of longer springs is rigidly connected to both masses, at the small solid dots in Fig. 2. The second group of shorter springs is each attached to only one mass. These springs begin to influence the system (by their compression) only after the distance between the masses is less than the spring equilibrium length.

*n*in this chain is described by the equation:

*g*. As if linear, oscillation (5) is isochronous and its period does not depend on the amplitude

*D*:

*g*. It increases with the increase in the nonlinear parameter—as \(g\rightarrow 1\). This tendency of nonlinear decrease of resonator natural frequency was observed experimentally [12, 13, 14].

We will now pass to the key issue of this paper, namely, to the collisions of single pulses. Equation (4) can govern the interaction of two types: collision of counter-propagating pulses; and collision of pulses running at different speeds in the same direction.

*x*-axis (3 in Fig. 4) for which Eq. (4) can now be simplified for weak nonlinearity, i.e., for \(g \ll 1\). The standard method of slowly varying profiles will be used, described in detail for example in the books [15, 16]. For a wave traveling in the positive direction of the

*x*-axis, a solution is sought in the form

*c*, and \(x_{1}\) is the slow variable (\(\mu \ll 1\), see “Appendix” section) related to the smallness of the nonlinear parameter \(g\sim \mu \). Substituting (7) in (4) and neglecting the second derivative with respect to the slow coordinate, we obtain:

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. (9) to study the interesting phenomenon of annihilation of pulses traveling in the same direction.

A curious simplicity in the equations with modular nonlinearity is that they are linear all the while the function preserves its sign, that is, when it is purely \(u>0\) or purely \(u<0\). Nonlinear effects occur only in alternating solutions. For example, a positive pulse of arbitrary shape \(u=\varPhi _{+}(\theta )\) at \(\varGamma =0\) propagates as the stationary wave \(u=\varPhi _{+}(\theta + z)\) . A purely negative pulse also propagates without changing its shape, but at a different speed, \(u=\varPhi _{-}(\theta - z)\). Because of this difference in velocities, pulses can approach each other and start to interact.

The collision of two nonoverlapping pulses of different polarities was considered only for the case \(\varGamma = 0\) [8]. First the distance between the pulses decreases as the pulses propagate linearly with different velocities. When they reach each other, a nonlinear interaction begins. After the disappearance of one of the two pulses, the interaction stops, and once again the surviving pulse propagates linearly. Consequently, the interaction of two solitary waves in a modular medium reveals properties that are different from the effects observed at elastic collisions of solitons and inelastic mergers of shock waves. There exists an analogy between the single wave collision in modular media and the interaction of clots of chemical reactants, such as a fuel and an oxidant. A chemical reaction (for example, burning) results in that the smaller component disappears, and the larger second mass decreases. At equal masses of fuel and oxidant, these clots can completely disappear or annihilate.

*c*. Substituting (11) into (9), we arrive at an ordinary differential equation, which is then integrated once to

Thus, we obtain a system of three nonlinear equations for the three unknown functions \(\theta _{FR}(z)\), \(|u_{1}(z)|\) and \(|u_{2}(z)|\). A similar approach was used to calculate the parameters of shock described by the usual Burgers equation (see [19, Problems 2.18–2.23]). Of course, this system cannot be solved for all possible forms of pulse signals.

*L*is conserved during propagation and interactions follows from Eq. (9):

The process of collision of two pulses of equal durations but different amplitudes and polarities (curve 1) is shown in Fig. 5. Before the collision, the negative and positive pulses approach each other while propagating at constant speeds and stable shapes. A nonlinear interaction begins once the pulses start overlapping (curve 2). As a result, the negative pulse with a smaller amplitude is absorbed by the positive pulse (i.e., at a certain distance negative pulse disappears, curve 3). In turn, the amplitude of the positive pulse decreases. During this interaction, the speed of the positive pulse is reduced. After the disappearance of the negative pulse, the nonlinear interaction is terminated and the speed of the positive pulse returns to its original value. At this stage, again only linear dissipation distorts the shape, resulting in a broadening of the positive pulse and reduction in its peak amplitude.

In Fig. 6 dependencies of energy on distance *E*(*z*) are constructed for different values of \(\varGamma = 0.002,\) 0.04, 0.2, 0.6 (curves 1–4). Let us first discuss how the energy decreases in a weakly dissipative medium (curve 1). During the first stage, the pulses are spatially separated and approach each other. Their total energy slowly decreases due to the usual linear absorption in the medium. During the second stage, the collision of pulses begins (curve 2 in Fig. 5). As a result of their intersection, a nonlinear absorption is activated, and the energy decreases noticeably faster. During the third stage, when the negative pulse has been completely absorbed (curves 3 and 4 in Fig. 5), the nonlinearity is de-activated and the energy loss of the surviving positive pulse again goes slowly, within the framework of linear theory.

As the dissipative parameter increases, the stage of strong nonlinear damping becomes less pronounced and is for \(\varGamma = 0.6\) in curve 4 in Fig. 6 not noticeable.

As final remark should be pointed out that we do not know of other works on nonlinear annihilation and, apparently, research in this direction will be continued. At the same time, it is of interest to study processes that are opposite to annihilation, namely the creation of new waves from a wave vacuum. The birth of counter-propagating waves as a result of self-reflection has long been studied both theoretically and experimentally (see [15, Chapter 7]). However, the corpuscular analogy was not analyzed in these prior works. Evidently, studies based on these new models with new types of nonlinearity represent mathematically and physically intriguing issues.

## Notes

### Acknowledgements

This work is supported by the Russian Scientific Foundation Grant No. 14-22-00042.

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