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

Abstract

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.

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

“…Following our earlier work [21,22], we consider the equation system (1.1) -(1.2) on the periodic domain x ∈ [−L, L] and adjust the asymptotic expansions to the coupled system of Boussinesqtype equations. Firstly, we integrate (1.1) -(1.2) in x over the period 2L to obtain an evolution equation of the form…”

“…where we introduce fast characteristic variables and two slow time variables [21] ξ ± = x ± t, τ = √ t, T = t.…”

“…The acoustic mode has the dispersion relation ω = ω a (k) satisfying the two asymptotic approximations In this paper we revisit the weakly-nonlinear solution of the Cauchy problem for the coupled Boussinesq-type equations constructed in [20] with the view of extending the applicability of the solution to initial conditions with non-zero mass. We use the novel multiple-scales procedure recently devloped in [21].…”

“…The paper is organised as follows. In Section 2 we construct a weakly-nonlinear solution of the problem using asymptotic multiple-scales expansions for the deviations from the oscillating mean values (similarly to [21,22]), using fast characteristic variables and two slow time variables [21]. The validity of the constructed solution is examined in Section 3, where we compare it with direct numerical simulations of the Cauchy problem.…”

Nonlinear Longitudinal Bulk Strain Waves in Layered Elastic Waveguides

“…To simplify the problem, let us assume that the material of the lower layer has much greater density (greater inertia), and consider the reduction w = 0. Then, the dynamics of the top layer is described (after appropriate scalings) by the Boussinesq-Klein-Gordon (BKG) equation [8]…”