Head-collision of nonlinear waves in a shallow basin: wave field and bottom pressure

<p> The collision of solitary waves has been studied analytically and numerically in numerated papers for last 50 years. In the weakly nonlinear theory, the soliton interaction is inelastic. Here we study more general class of the head-collision of nonlinear waves of various shape (Riemann waves of both polarities, shock waves and solitons) in the shallow water within nonlinear shallow-water theory, Serre-Green-Naghdi and Euler equations. The structure of wave field and induced bottom pressure at the moment of wave interaction is analysed analytically and numerically. It is shown that such an interaction leads to a phase shift and shape deformation in the moment of interaction. Estimates of the height of the Riemann waves as well solitons of moderate amplitudes at the moment of interaction are in agreement with theoretical predictions. The phase shift in the interaction of non-breaking waves is small enough, but becomes noticeable in the case of the shock waves motion. The approximated analytical solution for the wave field and bottom pressure distribution is obtained analytically within Serre-Green-Naghdi system. Computed bottom pressure in dispersive theories has two-bell shape for large amplitude solitary waves in quality agreement with theoretical analysis.</p>

Lie symmetry reductions and dynamics of solitary wave solutions of breaking soliton equation

In this paper, some new invariant solutions of breaking soliton (BS) equation have been derived by using similarity transformations method. The system represents the interaction of Riemann waves propagating along [Formula: see text]-axis and long waves along [Formula: see text]-axis. The commutative relation and symmetry analysis of BS equation are derived using Lie group theory. Meanwhile, the method reduces the number of independent variables by one in each step. A repeated application of similarity transformations method reduces the BS equation into overdetermined equations, which provide invariant solutions. The derived results are more general than previous findings. The obtained solutions are supplemented by numerical simulation, which makes this research physically meaningful. Eventually, doubly soliton, multisoliton and asymptotic profiles of solutions are analyzed in the analysis and discussion section.

Mathematical Modeling the Nonlinear 1D Dynamics of Elastic Heteromodular and Porous Materials

Approaches to mathematical modeling of nonlinear strain dynamics in heteromodular and porous materials are discussed; the mechanical properties of media are described in terms of the simple piecewise linear elastic models. Several nonstationary 1D boundary value problems show that the singularity of model relationships gives rise to shock waves and centered Riemann waves in generalized solutions. Nonstationary load modes leading to the listed nonlinear effects are indicated separately for heteromodular and porous media.

Generalized Riemann waves and their adjoinment through a shock wave

Generalized simple waves of the gas dynamics equations in Lagrangian and Eulerian descriptions are studied in the paper. As in the collision of a shock wave and a rarefaction wave, a flow becomes nonisentropic. Generalized simple waves are applied to describe such flows. The first part of the paper deals with constructing a solution describing their adjoinment through a shock wave in Eulerian coordinates. Even though the Eulerian form of the gas dynamics equations is most frequently used in applications, there are advantages for some problems concerning the gas dynamics equations in Lagrangian coordinates, for example, of being able to be reduced to an Euler–Lagrange equation. Through the technique of differential constraints, necessary and sufficient conditions for the existence of generalized simple waves in the Lagrangian description are provided in the second part of the paper.