The effect of the negative particle velocity in a soliton gas within Korteweg–de Vries-type equations

• Published in: Moscow University physics bulletin Vol. 72; no. 5; pp. 441 - 448
• Main Authors: Shurgalina, E. G., Pelinovsky, E. N., Gorshkov, K. A.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01.09.2017; Springer Nature B.V
• Subjects: Criteria; Defects; Kinetic theory; Korteweg-Devries equation; Lattices (mathematics); Mathematical and Computational Physics; Physics; Physics and Astronomy; Solitary waves; Theoretical; Theoretical and Mathematical Physics; soliton gas; negative soliton velocity; solitons; modified Korteweg–de Vries equation; Korteweg–de Vries equation
• ISSN: 0027-1349; 1934-8460
• DOI: 10.3103/S0027134917050101

The effect of changing the direction of motion of a defect (a soliton of small amplitude) in soliton lattices described by the Korteweg–de Vries and modified Korteweg–de Vries integrable equations (KdV and mKdV) was studied. Manifestation of this effect is possible as a result of the negative phase shift of a small soliton at the moment of nonlinear interaction with large solitons, as noted in [1], within the KdV equation. In the recent paper [2], an expression for the mean soliton velocity in a “cold” KdV-soliton gas has been found using kinetic theory, from which this effect also follows, but this fact has not been mentioned. In the present paper, we will show that the criterion of negative velocity is the same for both the KdV and mKdV equations and it can be obtained using simple kinematic considerations without applying kinetic theory. The averaged dynamics of the “smallest” soliton (defect) in a soliton gas consisting of solitons with random amplitudes has been investigated and the average criterion of changing the sign of the velocity has been derived and confirmed by numerical solutions of the KdV and mKdV equations.