nLab soliton

Idea

The term soliton comes from an abbreviation of “solitary wave”.

A soliton solution of a nonlinear wave equation is a solution whose large amplitude part is localized in space and is asymptotically stable in time. This asymptotic stability (more precisely non-damping and asymptotic preservation of shape, up to translation) is typically a feature of an infinite number of conservation laws, and many models of equations allowing soliton solutions are in fact integrable systems (with infinitely many degrees of freedom). Soliton solution often combine to multisoliton solutions in a nonlinear way, with a period of interaction when they “meet”, but after a passage of some time, the waves gradually uncouple and regain their original shape when outgoing to infinity. A typical example of a nonlinear wave equation exhibiting soliton solutions is the exactly solvable “nonlinear Schroedinger equation” appearing in optics.

Solitons appear in description of many natural phenomena. For example, Davydov soliton (wikipedia) has a role in stabilizing dynamics of proteins.

Examples

• solitons in photonic crystals: see there

Related concepts

• vortex, instanton, caloron, skyrmion

• integrable model, Korteweg de Vries equation

• black hole, black brane

Refereneces

• M. Jimbo, T. Miwa, E. Date, Solitons: differential equations, symmetries and infinite dimensional algebras, Camb. tracts is math. 135 (transl. from Japanese by M. Reid)

• A. Newell, Solitons in mathematics and physics

• Ludwig D. Faddeev, Leon Takhtajan, Hamiltonian methods in the theory of solitons, Springer, (transl. from Russian Гамильтонов подход в теории солитонов. — М.: Наука. 1986.)

• L. D. Faddeev, V. E. Korepin, Quantum theory of solitons, 1987

• David Tong, TASI Lectures on Solitons (web),

• Anastasia Doikou, Iain Findlay, Solitons: conservation laws & dressing methods (arXiv:1812.11914)

• Fritz Gesztesy, Helge Holden, Soliton equations and their algebro-geometric solutions (vol. I), Cambridge Univ. Press 2003; vol. II, with Johanna Michor, Gerald Teschl, 2008.

• Emma Previato, Hyperelliptic quasi-periodic and soliton solutions of the nonlinear Schrödinger equation, Duke Math. J. 52(2): 329–377 (1985) doi