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

In this paper we construct a weakly-nonlinear d'Alembert-type solution of the Cauchy problem for a Boussinesq-Klein-Gordon 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 limiting case of an "infinite" interval with zero boundary conditions), and work with the equation describing a deviation from the mean value. Unlike our earlier paper, here we develop a novel multiple-scales procedure involving fast characteristic variables and two slow time scales, 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 follows from our earlier results, and is illustrated numerically for a number of instructive examples, both for periodic solutions on a finite interval, and the well-studied scenario for localised solutions on a large ("infinite") interval. Importantly, in all cases the initial conditions for the leading-order Ostrovsky equations by construction have zero mass. Thus, the so-called "zero-mass contradiction" has been completely by-passed.