On radiating solitary waves in bi-layers with delamination and coupled Ostrovsky equations

2019

Stud Appl Math

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In this paper, we construct a weakly‐nonlinear d'Alembert‐type solution of the Cauchy problem for the Boussinesq‐Klein‐Gordon (BKG) equation. Similarly to our earlier work based on the use of spatial Fourier series, we consider the problem in the class of periodic functions on an interval of finite length (including the case of localized solutions on a large interval), and work with the nonlinear partial differential equation with variable coefficients describing the deviation from the oscillating mean value. Unlike our earlier paper, here we develop a novel multiple‐scales procedure involving fast characteristic variables and two slow time scales and averaging with respect to the spatial variable at a constant value of one or another characteristic variable, which allows us to construct an explicit and compact d'Alembert‐type solution of the nonlinear problem in terms of solutions of two Ostrovsky equations emerging at the leading order and describing the right‐ and left‐propagating waves. Validity of the constructed solution in the case when only the first initial condition for the BKG equation may have nonzero mean value follows from our earlier results, and is illustrated numerically for a number of instructive examples, both for periodic solutions on a finite interval, and localized solutions on a large interval. We also outline an extension of the procedure to the general case, when both initial conditions may have nonzero mean values. Importantly, in all cases, the initial conditions for the leading‐order Ostrovsky equations by construction have zero mean, while initial conditions for the BKG equation may have nonzero mean values.

“…and the functions ± are to be found at the next order. Substituting (24) and (25) into (21) and integrating with respect to the characteristic variables, we obtain ( − , + , ,…”

Section: Weakly-nonlinear D'alembert-t Ype Solutionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

D'Alembert‐type solution of the Cauchy problem for the Boussinesq‐Klein‐Gordon equation

2019

Stud Appl Math

Self Cite

“…Φ n = Φ n,n+1 +Φ n,n+1 +Φ n,n+1 +Φ n,n+1 +Φ n−1,n +Φ n−1,n +Φ n−1,n +Φ n−1,n +Φ ⊥ , (5) where overlines denote particles in the second ("bottom") row. Let the potential energy of interaction between any two neighbouring particles have the form…”

Section: Then (3) Gives Usmentioning

confidence: 99%

Nonlinear Longitudinal Bulk Strain Waves in Layered Elastic Waveguides

Applied Wave Mathematics II

2019

We consider long longitudinal bulk strain waves in layered waveguides using Boussinesq-type equations. The equations are developed using lattice models, and this is viewed as an extension of the Fermi-Pasta-Ulam problem. We describe semi-analytical approaches to the solution of scattering problems in delaminated waveguides, and to the construction of the solution of an initial-value problem in the class of periodic functions, motivated by the scattering problems.

Weakly-Nonlinear Solution of Coupled Boussinesq Equations and Radiating Solitary Waves

Advanced Structured Materials

2019

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Weakly-nonlinear waves in a layered waveguide with an imperfect interface (soft bonding between the layers) can be modelled using coupled Boussinesq equations. We assume that the materials of the layers have close mechanical properties, in which case the system can support radiating solitary waves. We construct a weakly-nonlinear d'Alembert-type solution of this system, considering the problem in the class of periodic functions on an interval of finite length. The solution is constructed using a novel multiple-scales procedure involving fast characteristic variables and two slow time variables. Asymptotic validity of the solution is carefully examined numerically. We also discuss the limiting case of an infinite interval for localised initial conditions. The solution is applied to study interactions of radiating solitary waves.