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The localized excitations propagating in a system with constant velocity and colliding with each other without change in their shapes are called solitons. During the collision of solitons the solution cannot be represented as a linear combination of two soliton solutions but after the collision solitons recover their shapes and the only result of collision is a phase shift.

Sine-Gordon Solitons and Soliton Collisions

In these pages we present the sine-Gordon solitons and their collisions.

The sine-Gordon equation

plays an important role in many branches of physics. It provides one of the simplest models of the unified field theory, can be found in the theory of dislocations in metals, in the theory of Josephson junctions and so on. It can be used also in interpreting certain biological processes like DNA dynamics.

Multi-Soliton Solutions: In the present pages, we demonstrate the derivation of the multi-soliton solutions to the sine-Gordon equation with the use of Backlund transformations. Animation and 3D graphics help to visualize one (Kink, Antikink), two (K-K, K-Ak, Breather), three soliton solutions and the basic properties of soliton collisions.

Pendulum Model: The soliton solutions to the sine-Gordon equation and soliton collisions are visualized for the system of coupled pendulums, having the sine-Gordon equation as the continuous analog.

Elastic Ribbon Model: The multi-soliton solutions to the sine-Gordon equation are discussed in the frame of the elastic ribbon model, which allows the consideration of solitons as the charged particles. For the presented animations we tried to choose the parameters in a way to demonstrate, in the clearest form, the charge conservation law.

Maple worksheets (code):

• Graphics Representation ( SGe_solitons.mws 0.95 Mb )
• Pendulum Model ( SGe_pend.mws 3.2 Mb )
• Elastic Ribbon Model ( SGe_ribbon.mws 3.9 Mb )

Test examples: A.E. Miroshnichenko, A.A. Vasiliev, S.V. Dmitriev A Maple package for the derivation of multi-soliton solutions to the sine-Gordon equation using the Backlund transformations.

• Link: Danko Georgiev Solitonic effects of the local electromagnetic field on neuronal microtubules – tubulin tail sine-Gordon solitons could control MAP attachment sites and microtubule motor protein function. Cogprints 2004, 3894; D.D. Georgiev, S.N. Papaioanou, and J.F. Glazebrook Biomedical Reviews 2004.

Discreteness Effects: Effects of inelastic collisions in a weakly discrete sine-Gordon system are presented. Existence and significance of the separatrix solutions and stochastic instability layers are underlined.

• Solitons Home Page (Heriot-Watt University, Edinburgh)
• Soliton-Lab Art Gallery.  Many Faces of Solitons (Kyoto University)
• Nonlinear Physics Group (Australian National University)
• Klaus Brauer's SOLITON Page (University of Osnabrück)
• Light Bullet Home Page (Simon Fraser University)
• Harmonic maps and integrable systems (ed. A.P. Fordy and J.C. Wood)

Multi-field solitons: In the present pages we introduce a model which supports stable periodic structures and multi-field solitons. Usually solitons are described by a single continuous function. The N-field soliton is described by N functions. We construct multi-field generalized continuum models to obtain them. You can see links for more information about generalized continuum models (Cosserat, micropolar, gradient-type, nonlocal, etc.).

Rotobreathers in underdamped Josephson junction ladders, representation in the framework of pendulum mechanical analogy are given on this page.

• Sergej Flach's Discrete Breather Homepage (Max-Planck-Institut für Physik komplexer Systeme, Dresden)
• Intrinsic Localized Modes (Cornell University)
• Breather solutions of the topological discrete sine-Gordon model (University of Texas)
• LOCNET: Localisation by Nonlinearity and Spatial Discreteness, and Energy Transfer, in Crystals, Biomolecules and Josephson Arrays

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