Soliton (topological)

In mathematics and physics, a topological soliton or a topological defect is a solution of a system of partial differential equations or of a quantum field theory homotopically distinct from the vacuum solution.

A topological defect can be proven to exist because the boundary conditions entail the existence of homotopically distinct solutions. Typically, this occurs because the boundary on which the conditions are specified has a non-trivial homotopy group which is preserved in differential equations; the solutions to the differential equations are then topologically distinct, and are classified by their homotopy class. Topological defects are not only stable against small , but cannot decay or be undone or be de-tangled, precisely because there is no continuous transformation that will map them (homotopically) to a uniform or "trivial" solution.

Topological defects occur in partial differential equations and are believed to drivephase transitions in condensed matter physics.

The authenticity of a topological defect depends on the nature of the vacuum in which the system will tend towards if infinite time elapses; false and true topological defects can be distinguished if the defect is in a false vacuum and a true vacuum, respectively.

Examples include the soliton or solitary wave which occurs in exactly solvable models, such as

Topological defects in lambda transition universality class systems including:

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