Oscillatory solitons of U(1)-invariant mKdV equations I: Envelope speed and temporal frequency

TL;DR

This paper investigates oscillatory solitons in U(1)-invariant mKdV equations, deriving explicit solutions and analyzing their properties, including speed, frequency, and interactions, revealing new behaviors like zero-speed standing waves.

Contribution

It introduces explicit oscillatory 1- and 2-soliton solutions for the Hirota and Sasa-Satsuma equations, highlighting novel features such as variable wave speeds and breather reductions.

Findings

Oscillatory solitons can have positive, negative, or zero speed.

Zero-speed oscillatory waves are time-periodic standing waves.

Oscillatory 2-solitons illustrate collision dynamics and breather formation.

Abstract

Harmonically modulated complex solitary waves which are a generalized type of envelope soliton (herein coined oscillatory solitons) are studied for the two U(1)-invariant integrable generalizations of the modified Korteweg-de Vries equation, given by the Hirota equation and the Sasa-Satsuma equation. A bilinear formulation of these two equations is used to derive the oscillatory 1-soliton and 2-soliton solutions, which are then written out in a physical form parameterized in terms of their speed, modulation frequency, and phase. Depending on the modulation frequency, the speeds of oscillatory waves (1-solitons) can be positive, negative, or zero, in contrast to the strictly positive speed of ordinary solitons. When the speed is zero, an oscillatory wave is a time-periodic standing wave. Properties of the amplitude and phase of oscillatory 1-solitons are derived. Oscillatory 2-solitons are graphically illustrated to describe collisions between two oscillatory 1-solitons in the case when the speeds are distinct. In the special case of equal speeds, oscillatory 2-solitons are shown to reduce to harmonically modulated breather waves.