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

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Khusnutdinova, K. and Tranter, M. ORCID: 0000-0002-6019-8819, 2022. Periodic solutions of coupled Boussinesq equations and Ostrovsky-type models free from zero-mass contradiction. Chaos, 32 (11): 113132. ISSN 1054-1500

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Official URL: https://doi.org/10.1063/5.0112982

Abstract

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.

Item Type:

Journal article

Publication Title:

Chaos

Creators:

Khusnutdinova, K. and Tranter, M.

Publisher:

AIP Publishing

Date:

14 November 2022

Volume:

Number: