M. V. Flamarion¹, E. N. Pelinovsky^{2, 3}, T. G. Talipova²

¹Pontificia Universidad Cat´olica del Peru´ (Lima, Peru)

²Gaponov-Grekhov Institute of Applied Physics (Nizhny Novgorod, Russian Federation)

³Higher School of Economics (Nizhny Novgorod, Russian Federation)

Abstract. Analytical and numerical solutions of the damped Schamel equation, describing the dynamics of ion-acoustic waves in magnetized plasma, are presented. A small parameter is introduced in the equation before the dissipative term, ensuring that in its absence the solution reduces to a solitary wave (soliton). The asymptotic method employed for solving the equation is a variant of the Krylov-Bogolyubov-Mitropolsky multiple-scale technique. In the first-order approximation, the solution is described by a traveling solitary wave with slowly varying parameters. The second-order approximation yields the evolution laws for the soliton’s amplitude and phase as functions of «slow» time. Additionally, exact integral conservation laws (mass and energy of the wave field), derived directly from the original damped Schamel equation, are utilized. These integrals allow estimating the soliton’s radiative losses, particularly the mass of the so-called tail formed behind the soliton due to dissipation. Direct numerical solutions of the original equation, obtained via a pseudospectral method, confirm the asymptotic laws governing the soliton’s amplitude decay caused by dissipation. Another limiting case – strong dissipation (dominant over nonlinearity and dispersion), is also investigated, demonstrating that the soliton decays as a linear impulse, which is validated numerically.

Key Words: ion-acoustic waves, Shamel equation, solitary wave, method of multiple scales, pseudospectral method

For citation: M. V. Flamarion, E. N. Pelinovsky, T. G. Talipova. Asymptotic and numerical study to the damped Schamel equation. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 27:2(2025), 229–242. DOI: https://doi.org/10.15507/2079-6900.27.202502.229-242

Information about the authors:

Marcelo V. Flamarion, Ph.D. (Mathematics), Professor, Departamento Ciencias– Seccio´n Matema´ticas, Pontificia Universidad Cat´olica del Peru, (Av. Universitaria 1801, San Miguel 15088, Lima, Peru), ORCID: http://orcid.org/0000-0001-5637-7454, mvellosoflamarionvasconcellos@pucp.edu.pe

Efim N. Pelinovsky, D. Sc. (Phys. and Math.), Chief Researcher, Gaponov-Grekhov Institute of Applied Physics, (46 Uljanov Street, Nizhny Novgorod, 603120 Russian Federation); Professor, High School of Economics University (25 Bolshaya Pechorskaya Str., Nizhny Novgorod, 603120 Russian Federation), ORCID: http://orcid.org/0000-0002-5092- 0302, pelinovsky@ipfran.ru

Tatiana G. Talipova, D. Sc. (Phys. and Math.), Leading Researcher, Gaponov-Grekhov Institute of Applied Physics, (46 Uljanov Street, Nizhny Novgorod, 603120 Russian Federation); ORCID: http://orcid.org/0000-0002-1967-4174, tgtalipova@mail.ru