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Soliton molecules in Sharma-Tasso-Olver-Burgers equation

Zhaowen Yan, Senyue Lou

arXiv:1912.13324v2nlin.PS

TL;DR

Soliton molecules have been observed in optics and discussed for coupled systems, but their formation in integrable systems is less understood. This paper applies velocity resonance to multiple solitary-wave solutions of the noncoupled STOB equation, deriving molecule types and analyzing fission and fusion among them.

  • Problem

    Soliton molecules have been observed in optics and studied in coupled systems, but less is known about them in integrable systems, motivating their construction in noncoupled STOB.

    Source: §1 · p. 2

  • Method

    The paper introduces velocity resonance into multiple solitary-wave solutions of STOB to investigate kink, HPK, and breathing soliton molecules and their fissions and fusions.

    Source: §4 · p. 13

  • Results

    The analysis identifies soliton molecules, HPK molecules, and central periodic kinks, and reports both fission and fusion in STOB, whereas Burgers admits fusion only.

    Source: §5 · p. 13

  • Takeaways & Limitations

    The paper establishes molecule and solitary-wave structures in STOB that the authors report are absent from the Burgers and STO equations.

    Source: §4 · p. 8, §5 · p. 13

    In the solution derivation, c is a function of t and is taken as c = 0 without loss of generality.

    Source: §2 · p. 3

Abstract

from arXiv · show

Soliton molecules have been experimentally discovered in optics and theoretically investigated for coupled systems. This paper is concerned with the formation of soliton molecules by the resonant mechanism for a noncoupled system, the Sharma-Tasso-Olver-Burgers (STOB) equation. In terms of introducing velocity resonance conditions, we derive the soliton (kink) molecules, half periodic kink (HPK) molecules and breathing soliton molecule of STOB equation. Meanwhile, the fission and fusion phenomena among kinks, kink molecules, HPKs and HPK molecules have been revealed. Moreover, we also discuss the central periodic kink solutions from the multiple solitary wave solutions.

1 Introduction

Soliton molecules have been observed in optical systems and investigated in coupled systems, while less is known about molecules in integrable systems. This work targets their formation in the noncoupled STOB equation using velocity resonance conditions.

1 Introduction

  • The STO equation has been studied using methods including first integrals, bi-Hamiltonian formulation, fractional sub-equations, sine-cosine methods, q-functions, and transformations.The cited transformations include Cole-Hopf, fractional complex, Darboux, and Bäcklund transformations.Source: §1 · p. 1
  • Soliton molecules, or soliton bound states, have been observed in optical systems and analyzed in Bose-Einstein condensates, with theoretical proposals for their formation.The introduction also notes that molecules in coupled systems have been well discussed.Source: §1 · p. 2
  • The paper identifies a gap in understanding soliton molecules in integrable systems and sets out to construct them for the noncoupled STOB equation using velocity resonance conditions.The introduction places this objective alongside prior resonance-based molecule studies in other systems.Source: §1 · p. 2

2 The fission and fusion in Sharma-Tasso-Olver-Burgers equation

The section constructs multiple solitary-wave solutions for STOB and examines two-wave fission and fusion. These collisions can change wave characteristics, and STOB permits both processes, unlike the Burgers equation, which permits fusion only.

2 The fission and fusion in Sharma-Tasso-Olver-Burgers equation

  • The section introduces a truncated Painlevé expansion and derives a Cole-Hopf transformation for constructing STOB solutions.The derivation substitutes a zero choice for u1 into the equation and considers one case for u0.Source: §2 · p. 3
  • The section treats c as a function of t and takes c = 0 without loss of generality.This choice is stated while solving for the STOB solution.Source: §2 · p. 3
  • The multiple solitary-wave ansatz uses wave numbers, frequencies, and original positions, with dispersion relations making it a STOB solution.For two waves, the authors derive a solution and its potential field before analyzing fission and fusion.Source: §2 · pp. 3–4
  • The two-wave analysis separates cases according to whether k1k2 is positive or negative, and identifies the k1k2 < 0 single waves as exact STOB solutions that fuse resonantly.Before fusion, the two single waves have their own locations and amplitudes; the resonant wave after fusion has a changed location and amplitude.Source: §2 · pp. 4–5
  • Two solitary waves can fuse into one resonant solitary wave, while one resonant solitary wave can split into two; the plotted examples show these fusion and fission events.The section reports that collision changes velocity, amplitude, width, and wave shape.Source: Figure 1 · p. 5, §2 · pp. 5–6, Figure 2 · p. 6
  • Unlike the Burgers equation, which has fusion but no fission, both fission and fusion occur for the STO and STOB equations.The authors report changes in velocity, amplitude, width, and wave shape after collision.Source: §2 · p. 6

3 The half periodic kink solutions and their fissions and fusions

The section identifies half periodic kink (HPK) behavior and shows HPKs fissioning and fusing with ordinary kinks in the STOB equation. It reports that these HPK–kink interactions are absent in the Burgers and STO equations.

3 The half periodic kink solutions and their fissions and fusions

  • A complex-conjugate two-soliton choice yields a singular kink, while adding a soliton under resonance is used to obtain a nonsingular kink with periodic behavior.The two-soliton expression is singular where its denominator vanishes.Source: §3 · p. 6
  • Solution (23) is analytic for b>|a|; the text also reports a periodic-wave limit when 27α^2−η>0 and a constant-zero limit in a half-plane.Source: §3 · p. 7
  • Solution (23) is called a half periodic kink when 27α^2−η<0.The section also identifies the half periodic kink (HPK) as a structure specific to the STOB equation.Source: §3 · pp. 6–7
  • An HPK fissions into an HPK and an ordinary kink, while two HPKs fuse into one ordinary kink.The figures illustrate the fission and fusion cases for the STOB equation.Source: §3 · pp. 7–8, Figure 4 · p. 8
  • The paper reports no corresponding HPK–kink fission or fusion solutions for the Burgers and STO equations.Source: §3 · p. 8

4 Kink molecules and Soliton molecules for the STOB equation

Velocity resonance produces bound soliton molecules in the STOB equation, with molecule structures and fission–fusion interactions among solitons, kinks, and HPKs. The section also derives breathing HPK molecules and central periodic kink solutions.

Soliton molecules

  • Velocity resonance binds two solitary waves into a soliton molecule, a structure reported here for STOB but not Burgers or STO.The molecule is identified as a bound state of solitons.Source: §4 · p. 8

Kink molecules and soliton molecules

  • A four-soliton interaction proceeds through fission of one soliton, followed by fusion that forms a soliton molecule.One split soliton fuses with another before combining with the remaining soliton.Source: §4 · p. 9, Figure 6 · p. 10

HPK molecules

  • For paired complex-conjugate parameters, the solutions include HPK molecules and a breathing dissipative kink–antikink molecule.The breathing solution is given by the field z = ux associated with the HPK molecule solution.Source: §4 · p. 10, Figure 7 · p. 11

Central periodic kinks

  • The parameter a1 controls the separation of parallel HPKs; when a1 = 1 they overlap and reduce to central periodic kinks.A central periodic kink comprises a periodic center and a kink, and the first type has higher amplitude than the second.Source: §4 · pp. 10–12, Figure 8 · p. 11

HPK molecule interactions

  • An HPK molecule can fuse with a usual kink to form an ordinary kink, or escape from a usual kink through splitting.The two processes are shown as absorption and escape interactions.Source: §4 · pp. 12–13

5 Summary

Velocity resonance in the STOB equation yields multiple solitary-wave and molecule types, with both fission and fusion; Burgers permits only fusion. The authors suggest possible connections between kink-kink molecules and layer dislocations or domain walls, and between breathing molecules and experimental observations.

5 Summary

  • STOB supports both fission and fusion under resonant conditions, whereas the Burgers equation admits only fusion.The interactions can change amplitude, width, velocity, and wave shape.Source: §5 · p. 13
  • Velocity resonance yields soliton molecules, kink molecules, HPK molecules, breathing soliton molecules, HPKs, and central periodic kink solutions for STOB.The authors state that some new solutions and molecules occur in STOB but not in the Burgers or STO equations.Source: §5 · p. 13
  • Kink-kink molecules may describe layer dislocations and domain walls, while breathing dissipative solitons may be candidates for describing known experimental observations.The authors say the structures and properties of molecules in other integrable systems deserve further study.Source: §5 · p. 13
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