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A two-dimensional self-similarity transformation theory is established, and the focusing (parabolic) (2 + 1)-dimensional NLS equation is taken as the model.  The two-dimensional self-similarity transformation is proposed for converting the focusing (2 + 1)-dimensional NLS equation into the focusing (1 + 1) dimensional NLS equations, and the excitation of its novel line-rogue waves is further investigated. It is found that the spatial coherent structures induced by the Akhmediev breathers (AB) and Kuznetsov-Ma solitons (KMS) also have the short-lived characteristics which are possessed by the line-rogue waves induced by the Peregrine solitons, and the other higher-order rogue waves and the multi-rogue waves of the (1 + 1) dimensional NLS equations. This is completely different from the evolution characteristics of spatially coherent structures induced by bright solitons (including multi-solitons and lump solutions), with their shapes and amplitudes kept unchanged. The diagram shows the evolution characteristics of  all kinds of resulting line rogue waves. The new excitation mechanism of line rogue waves revealed contributes to the new understanding of the coherent structure of high-dimensional nonlinear wave models.

Two-dimensional self-similarity transformation theory and line rogue waves excitation