Model of modular nonlinearity in the problem of nonlinear wave propagation

Gusev V.A.

Lomonosov's Moscow State University, Physical Faculty, Dep. of Acoustics, Russia, 199991, Moscow, Leninskie gory, Tel.: 8(495)939-29-43, E-mail:

The study of high-intensity wave fields is usually associated with the solution of nonlinear equations. However, these equations usually do not have a common exact solution. Therefore, it is necessary to develop new approaches to the construction of analytical solutions and to identify the qualitative dependences of the solution on the parameters of the problem in order to optimize and create fields of a given structure. One of such approaches is the method of modular nonlinearity, which consists in the qualitative replacement of nonlinear terms of a power type with terms of a lower degree or linear ones, but containing the modulus function. For example, a quadratic term can be replaced by a term with a modulus. At the same time, the qualitative behavior is generally preserved. Instead of a non-linear equation, a system of linear equations is obtained for intervals of different polarity, which can be solved exactly. Next, it is necessary to carry out the matching of solutions, during which nonlinear effects appear in the form of the formation of a shock front.

The model of modular nonlinearity is effective in such specific problems as the three-dimensional problem of the propagation of intense acoustic beams, as well as the calculation of an intense surface wave at the interface between media. In the case of beams, within the framework of this model, it is possible to obtain an exact discontinuous solution on the beam axis [1]. The time profile consists of a sequence of narrow pulses of large positive amplitude and long intervals of small negative amplitude. The decrease in the negative amplitude is due to two factors: nonlinear damping at the shock front and diffraction spreading. In addition, the narrower the characteristic beam width, the stronger these effects manifest themselves. In the case of a surface wave, it is possible to write down the dispersion equation and construct the depth distribution of the field [2]. At the same time, in the case of quadratic nonlinearity, only approximate methods for taking into account nonlinear effects are possible. It can be expected that such a model will also be useful in other problems of nonlinear dynamics, from the classical predator-prey model to problems of dynamical chaos.


1. Gusev V.A. Calculation of the field of a high-intensity focused ultrasonic beam using the modular nonlinearity model // 2022 Days on Diffraction (DD), 2022.

2. Gusev V.A. Transformation of acoustic waves in layered media with modular nonlinearity // Proceedings of the All-Russian Acoustic Conference. St. Petersburg: POLYTECH-PRESS, 2020. S. 101-105.


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