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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 sig...
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2022
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| Online Access: | https://hdl.handle.net/2134/21438078.v1 |
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| _version_ | 1818165679897968640 |
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| author | Karima Khusnutdinova Matthew Tranter |
| author_facet | Karima Khusnutdinova Matthew Tranter |
| author_sort | Karima Khusnutdinova (1247967) |
| collection | Figshare |
| description | 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 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 used to control the accuracy of the numerical simulations. |
| format | Default Article |
| id | rr-article-21438078 |
| institution | Loughborough University |
| publishDate | 2022 |
| record_format | Figshare |
| spelling | rr-article-214380782022-11-14T00:00:00Z Periodic solutions of coupled Boussinesq equations and Ostrovsky-type models free from zero-mass contradiction Karima Khusnutdinova (1247967) Matthew Tranter (7438670) Applied mathematics Numerical and computational mathematics Other physical sciences Fluids & Plasmas Coupled Boussinesq equations Ostrovsky-type models <p>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 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 used to control the accuracy of the numerical simulations.</p> 2022-11-14T00:00:00Z Text Journal contribution 2134/21438078.v1 https://figshare.com/articles/journal_contribution/Periodic_solutions_of_coupled_Boussinesq_equations_and_Ostrovsky-type_models_free_from_zero-mass_contradiction/21438078 All Rights Reserved |
| spellingShingle | Applied mathematics Numerical and computational mathematics Other physical sciences Fluids & Plasmas Coupled Boussinesq equations Ostrovsky-type models Karima Khusnutdinova Matthew Tranter Periodic solutions of coupled Boussinesq equations and Ostrovsky-type models free from zero-mass contradiction |
| title | Periodic solutions of coupled Boussinesq equations and Ostrovsky-type models free from zero-mass contradiction |
| title_full | Periodic solutions of coupled Boussinesq equations and Ostrovsky-type models free from zero-mass contradiction |
| title_fullStr | Periodic solutions of coupled Boussinesq equations and Ostrovsky-type models free from zero-mass contradiction |
| title_full_unstemmed | Periodic solutions of coupled Boussinesq equations and Ostrovsky-type models free from zero-mass contradiction |
| title_short | Periodic solutions of coupled Boussinesq equations and Ostrovsky-type models free from zero-mass contradiction |
| title_sort | periodic solutions of coupled boussinesq equations and ostrovsky-type models free from zero-mass contradiction |
| topic | Applied mathematics Numerical and computational mathematics Other physical sciences Fluids & Plasmas Coupled Boussinesq equations Ostrovsky-type models |
| url | https://hdl.handle.net/2134/21438078.v1 |