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Ring solitons, dromions, breathers and instantons of the nonlinear Schrödinger equation. (Chinese. English summary) Zbl 1202.35306

Summary: We study the abundant localized coherent structures of the \((2+1)\)-dimensional nonlinear Schrödinger (NLS) equation which have been derived from fluid dynamics and plasma physics. Using a Bäcklund transformation and the separation of variables approach, we find that there exist many other abundant localized structures for the \((2+1)\)-dimensional NLS equation. The abundance of the localized structures of the model is introduced by an arbitrary function of the seed solution. Some special types of dromion solutions, breathers, instantons and ring-type solitons are discussed by selecting arbitrary functions appropriately. The dromion solutions can be driven by some sets of straight-line and curved-line ghost solitons. The dromion solutions may be located not only at the cross points of the lines, but also at the closed points of the curves. The breathers may breath both in amplitudes and in shapes.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35Q51 Soliton equations
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems