THE MODIFIED WHITHAM MODULATION THEORY FOR THE NONLINEAR KLEIN-GORDON-FOCK EQUATION WITH THE SIMPLEST RATIONAL FUNCTION AS ITS POTENTIAL
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.
References
- Alekseeva E.S. and Rassadin A.E. Asimptoticheskii analiz uravnenii Kleina-Gordona-Foka s chisto kubicheskoi nelineinost’yu (Asimptotic analysis of Klein-Gordon-Fock equations with purely cubic nonlinearity), Vos’maya Mezhdunarodnaya nauchnaya molodezhnaya shkola-seminar Matematicheskoe modelirovanie, chislennye metody i kompleksy programm imeni E.V. Voskresenskogo (The 8th International Scientific Yourth WorkShop “Mathematical Modelling, Numerical Methods and Program Complexes, named after E.V. Voskresenskii”), Saransk, 16–20 June, 2018, Book of Abstracts.
- Kostromina O.S., Potapov A.A., Rakut’ I.V., and Rassadin A.E. Kolebaniya v odnorodnoi tsepochke s segnetoelektricheskimi kondensatorami s otritsatel’noi emkost’yu (Oscillations in homogeneous circuit with ferroelectric capacitors with a negative capacitance). Uchenye zapiski fizicheskogo fakul’teta Moskovskogo universiteta, 2017, No. 6, p. 1760703.
- Maslov V.P. and Omel’yanov G.A. Asimptoticheskie solitonoobraznye resheniya uravnenii s maloi dispersiei (Asymptotic soliton-form solutions of equations with small dispersion). Uspekhi matematicheskikh nauk, 1981, Vol. 36, No. 3(219), pp. 63–126.
- Whitham G.B.Linear and nonlinear waves. Wiley, New York, 1974, 636 p.
Transfer of copyrights occurs on the basis of a license agreement between the Author and Shirshov Institute of Oceanology, RAS
