МОДИФИЦИРОВАННАЯ ТЕОРИЯ МОДУЛЯЦИИ УИЗЕМА ДЛЯ НЕЛИНЕЙНОГО УРАВНЕНИЯ КЛЕЙНА-ГОРДОНА-ФОКА С ПРОСТЕЙШИМ ДРОБНО-РАЦИОНАЛЬНЫМ ПОТЕНЦИАЛОМ

Abstract

It is well-known that one can construct approximate solution of the nonlinear KleinGordon-Fock equation (NKGF) by means of the Whitham modulation theory (Whitham, 1977). In this work in the framework of the modified Whitham modulation theory presented at (Alekseeva, Rassadin, 2018) and (Kostromina et al., 2017) for NKGF with potential U(x) = (x–1/x)2 its asymptotic solution v(x, t) has been found. Due to isochronism of onedimensional movement of classical particle with unit mass in this potential amplitude a(x, t) of asymptotical solution obeys the linear transfer equation ∂a/∂t + V∂a/∂x = 0 with velocity V belonging to the interval –1<V<1. Peculiarity of the constructed solution is absence of gradient catastrophe therefore it is convenient for investigation of the next terms of asymptotic expansion of the NKGF solution with considered potential (Maslov, Omel’yanov, 1981). Product of Whitham-analogs of group and phase velocities of wave from constructed asymptotic solution of NKGF is equal to unit. This is the same value as for the linear Klein-Gordon-Fock equation. The mean square <v2(x, t)> of asymptotic solution of NKGF has been calculated under assumption that its initial phase shift is random value with stable distribution of probabilities. The obtained asymptotic solution due to its simplicity and informativity can be used by lecturers to illustrate abilities of the Whitham modulation theory.