Periodic solutions of coupled Boussinesq equations and Ostrovsky-type models free from zero-mass contradiction

Coupled Boussinesq equations describe long weakly-nonlinear longitudinal strain waves in a bi-layer with a soft bonding between the layers (e.g. a soft adhesive). From the mathematical viewpoint, a particularly difficult case appears when the linear long-wave speeds in the layers are significantly different (high-contrast case). The traditional derivation of the uni-directional models leads to four uncoupled Ostrovsky equations, for the right- and left-propagating waves in each layer. However, the models impose a ``zero-mass constraint'' i.e. the initial conditions should necessarily have zero mean, restricting the applicability of that description. Here, we bypass the contradiction in this high-contrast case by constructing the solution for the deviation from the evolving mean value, using asymptotic multiple-scale expansions involving two pairs of fast characteristic variables and two slow-time variables. By construction, the Ostrovsky equations emerging within the scope of this derivation are solved for initial conditions with zero mean while initial conditions for the original system may have non-zero mean values. Asymptotic validity of the solution is carefully examined numerically. We apply the models to the description of counter-propagating waves generated by solitary wave initial conditions, or co-propagating waves generated by cnoidal wave initial conditions, as well as the resulting wave interactions, and contrast with the behaviour of the waves in bi-layers when the linear long-wave speeds in the layers are close (low-contrast case). One local (classical) and two non-local (generalised) conservation laws of the coupled Boussinesq equations for strains are derived, and these are used to control the accuracy of the numerical simulations.Comment: 25 pages, 11 figures; previously this version appeared as arXiv:2210.14107 which was submitted as a new work by acciden

Fission of a longitudinal strain solitary wave in a delaminated bar

The aim of the paper is to show that splitting of a waveguide leads to fission of bulk solitons in solids. We study the dynamics of a longitudinal bulk solitary wave in a delaminated, symmetric layered elastic bar. First, we consider a two-layered bar and assume that there is a perfect interface when x 0 and complete debonding splitting when x 0, where the axis Ox is directed along the bar. We derive the so-called doubly dispersive equation DDE for a long nonlinear longitudinal bulk wave propagating in an elastic bar of rectangular cross section. We formulate the problem for a delaminated two-layered bar in terms of the DDE with piecewise constant coefficients, subject to continuity of longitudinal displacement and normal stress across the “jump” at x=0. We find the weakly nonlinear solution to the problem and consider the case of an incident solitary wave. The solution describes both the reflected and transmitted waves in the far field, as well as the diffraction in the near field in the vicinity of the jump . We generalize the solution to the case of a symmetric n-layered bar. We show that delamination can lead to the fission of an incident solitary wave, and obtain explicit formulas for the number, amplitudes, velocities, and positions of the secondary solitary waves propagating in each layer of the split waveguide. We establish that generally there is a higher-order reflected wave even when the leading order reflected wave is absent

Periodic solutions of coupled Boussinesq equations and Ostrovsky-type models free from zero-mass contradiction

Coupled Boussinesq equations are used to describe long weakly-nonlinear longitudinal strain waves in a bi-layer with a soft bonding between the layers (e.g. a soft adhesive). From the mathematical viewpoint, a particularly difficult case appears when the linear long-wave speeds in the layers are significantly different (high-contrast case). The traditional derivation of the uni-directional models leads to four uncoupled Ostrovsky equations, for the right-and left-propagating waves in each layer. However, the models impose a "zero-mass constraint" i.e. the initial conditions should necessarily have zero mean, restricting the applicability of that description. Here, we bypass the contradiction in this high-contrast case by constructing the solution for the deviation from the evolving mean value, using asymptotic multiple-scale expansions involving two pairs of fast characteristic variables and two slow-time variables. By construction, the Ostro-vsky equations emerging within the scope of this derivation are solved for initial conditions with zero mean while initial conditions for the original system may have non-zero mean values. Asymptotic validity of the solution is carefully examined numerically. We apply the models to the description of counter-propagating waves generated by solitary wave initial conditions, or co-propagating waves generated by cnoidal wave initial conditions, as well as the resulting wave interactions, and contrast with the behaviour of the waves in bi-layers when the linear long-wave speeds in the layers are close (low-contrast case). One local (classical) and two non-local (generalised) conservation laws of the coupled Boussinesq equations for strains are derived and used to control the accuracy of the numerical simulations. A weakly-nonlinear solution to the coupled Boussinesq equations on a finite interval with periodic boundary conditions is constructed, resolving the zero-mass contradiction. The solution is shown to be asymptotically valid by comparison to direct numerical simulations of the original coupled Boussinesq equations, with the additional control of derived generalised conservation laws. Examples include counter-propagating radiating solitary waves and Ostrovsky-type wave packets when the period of the solution is large compared to the size of a localised initial condition, while decreasing the period of the solution for the localised perturbations and using non-localised initial conditions leads to more complicated scenarios. We observe that, in many cases, the waves appear to interact in a nearly-elastic manner, similarly to that of solitary waves, with small phase shift and amplitude changes compared to the case with no interaction, while in other cases strong interactions lead to formation of new wave structures

Nonlinear layered lattice model and generalized solitary waves in imperfectly bonded structures

We study nonlinear waves in a two-layered imperfectly bonded structure using a nonlinear lattice model. The key element of the model is an anharmonic chain of oscillating dipoles, which can be viewed as a basic lattice analog of a one-dimensional macroscopic waveguide. Long nonlinear longitudinal waves in a layered lattice with a soft middle or bonding layer are governed by a system of coupled Boussinesq-type equations. For this system we find conservation laws and show that pure solitary waves, which exist in a single equation and can exist in the coupled system in the symmetric case, are structurally unstable and are replaced with generalized solitary waves

S-functions, reductions and hodograph solutions of the r-th dispersionless modified KP and Dym hierarchies

We introduce an S-function formulation for the recently found r-th dispersionless modified KP and r-th dispersionless Dym hierarchies, giving also a connection of these $S$-functions with the Orlov functions of the hierarchies. Then, we discuss a reduction scheme for the hierarchies that together with the $S$-function formulation leads to hodograph systems for the associated solutions. We consider also the connection of these reductions with those of the dispersionless KP hierarchy and with hydrodynamic type systems. In particular, for the 1-component and 2-component reduction we derive, for both hierarchies, ample sets of examples of explicit solutions.Comment: 35 pages, uses AMS-Latex, Hyperref, Geometry, Array and Babel package

On a class of second-order PDEs admitting partner symmetries

Recently we have demonstrated how to use partner symmetries for obtaining noninvariant solutions of heavenly equations of Plebanski that govern heavenly gravitational metrics. In this paper, we present a class of scalar second-order PDEs with four variables, that possess partner symmetries and contain only second derivatives of the unknown. We present a general form of such a PDE together with recursion relations between partner symmetries. This general PDE is transformed to several simplest canonical forms containing the two heavenly equations of Plebanski among them and two other nonlinear equations which we call mixed heavenly equation and asymmetric heavenly equation. On an example of the mixed heavenly equation, we show how to use partner symmetries for obtaining noninvariant solutions of PDEs by a lift from invariant solutions. Finally, we present Ricci-flat self-dual metrics governed by solutions of the mixed heavenly equation and its Legendre transform.Comment: LaTeX2e, 26 pages. The contents change: Exact noninvariant solutions of the Legendre transformed mixed heavenly equation and Ricci-flat metrics governed by solutions of this equation are added. Eq. (6.10) on p. 14 is correcte

An experimental and theoretical study of longitudinal bulk strain waves generated by fracture

From a dipole lattice model, we derive a Boussinesq-type equation that governs the propagation of a longitudinal strain wave in a uniform pre-strained bar with constant cross section. We then describe an experiment based on high-speed, single-point photoelasticity to measure strain waves in a Polymethylmethacrylate (PMMA) bar generated when it breaks whilst undergoing a tensile test. The experimental results are finally compared to the model predictions, showing a good correspondence in terms of wave propagation, strain rate at the leading edge and amplitude and frequency of the following ripple

MEDICO-SOCIAL AND INDIVIDUAL PSYCHOLOGICAL CHARACTERISTICS OF DRUG ADDICTED TEENAGERS AND THEIR HEALTHY RELATIVES

We conducted the research for detection of premorbid medico-social and personal features of related teenagers, using and not using psychoactive agents. Comparative analysis of 90 teenagers with drug addiction and their 90 healthy relatives (control group) was realized. Research was conducted by the method of anonymous questioning with use of specially developed questionnaire consisting of 88 questions, and also psychological methods -16 personal factors Kettell test, SMIL, and Zung depression test. Dispersive analysis was used to determine factors of medico-social character which can be considered informative for diagnostics of addiction deviations. Method of experimental psychology allowed to reveal individual and psychological features of teenagers leading to formation of addictive behavior. Teenagers suffering from drug addiction are much less socially successful than their healthy relatives. As compared to their healthy relatives, drug addicts' anamneses had more medico-social and premorbid factors of addictive risk: various neurotic episodes during childhood, craniocereberal traumas. They were exposed to actions of sexual violence more often. Their typical characteristics include tendency to internal aggression, parasuicide thoughts, suicide attempts and a paracriminal circle of contacts. Results of research at this stage gave the grounds to assume that under identical family conditions of education some teenagers are getting drug addiction, while others don't. This can be explained by the existence of individual and personal features which can already be induced by factors outside of the family. Experimental and psychological method revealed the individual and psychological features of teenagers promoting formation of dependent behavior. Unlike their healthy relatives teenage addicts have peculiar features such as aggressive tendencies, explosive nature of reaction, uneasiness and nervousness reflecting a condition of the extreme stress, hyper compensatory involvement of various protective mechanisms, hyperactivity in their search of an exit to their difficult situation, lower background of mood and narrow zone of contacts. Thus, in comparison with healthy relatives, anamnesis of teenagers addicts more often contain medico-social, behavioral and premorbid predictors of addictive behavior, and a number of specific features. All this needs to be considered while setting up individual programs of rehabilitation, and during development of primary prevention programs of psychoactive substances use in the youth environment