New Localized Excitations in (2+1)-Dimensional Integrable Systems

Abstract

The new localized excitations of some (2+1)-dimensional integrable models obtained by the multi-linear variable separation approach (MLVSA) are reviewed. A universal formula with some arbitrary functions is obtained for some suitable physical quantities for various integrable models. By selecting the arbitrary functions appropriately, one may obtain abundant interesting localized excitations like the multi-dromions, lumps, ring solitons, breathers, instantons, peakons, compactons, foldons, chaotic and fractal patterns and so on. The localized excitations possess quite rich interaction behaviors.

Binary Bell Polynomials Approach to Generalized Nizhnik—Novikov—Veselov Equation

“…As another application of the binary Bell polynomials, Fan [14] derived the infinite conservation laws of soliton equations through decoupling binary Bell polynomials into a Riccati type equation and a divergence type equation. In fact, the conservation laws have been hinted in the two-field constraint system (22) and (23), which can be rewritten in the conserved form…”

Fractal structure of ferromagnets: The singularity structure analysis

“…As mentioned above, following the recent investigation of many types of solitons and other coherent structures with chaotic (sensitive dependence on the initial conditions) and fractal (self-similar structures) behaviors in integrable or nonintegrable (2 + 1)-dimensional systems, [51][52][53][54][55][56][57][58][59][60][61][62][63] additionally, as lower dimensional arbitrary functions are present in the exact or approximate solutions to these model systems, we can use lower dimensional fractal solutions in view of obtaining higher dimensional fractal pattern formations to such models. Throughout this paper, we classify our fractal patterns through three considerations based upon the expressions of a generic lower dimensional arbitrary function as follows.…”

“…In fact, for each type of the fractal patterns, the generic function is defined in such a way that with the details presented in the figure captions, one can straightforwardly generate the related waveguide channels. The different expressions of above, although appearing quite peculiar in their forms, have been chosen from the lower dimensional integrable models [51][52][53][54][55][56][57][58][59][60][61][62][63] in view of unearthing some hidden properties of the higher dimensional ferromagnetic systems that were not ascertained previously. We actually believe that these fractal pattern formations are particularly underlying both from the viewpoint of the investigation of the propagation of higher dimensional excitations and from the viewpoint of the existence of stable wave pattern formations.…”

Fission, fusion and annihilation in the interaction of localized structures for the (2+1)-dimensional generalized Broer–Kaup system

“…By solving the multi-linear form of these NLPDEs and introducing a prior ansatz, some special types of exact solutions can be obtained from two (1+1)-dimensional variable separated fields. This variable separation approach (VSA) is successfully applied to many (2+1)-dimensional integrable models such as the Davey Stewartson (DS) equation [10], the Nizhnik-Novikov-Veselov (NNV) equation [10], the dispersive long wave equation (DLWE) [11], the Broer-Kaup-Kupershmidt (BKK) system [12], the Burgers equation [13], the Maccari system [14], the general (N+M)components Ablowitz-Kaup-Newell-Segur (AKNS) system [15]. Because the formula of the field includes some arbitrary functions, abundant coherent structures such as, the solitoffs, dromions, lumps, ring-solitons, peakons, compactons, breathers and instantons are given.…”

“…Because the formula of the field includes some arbitrary functions, abundant coherent structures such as, the solitoffs, dromions, lumps, ring-solitons, peakons, compactons, breathers and instantons are given. More recently, Tang and Lou have proposed a more general variable separation approach (GVSA) for several (2+1)-dimensional integrable models [13,16,17].…”