Complex Kraenkel-Manna-Merle system in a ferrite: N-fold Darboux transformation, generalized Darboux transformations and solitons

Abstract

Ferromagnetic materials such as the ferrites are used in the electronic and energy industries. Here, we concentrate on a complex Kraenkel-Manna-Merle system for the ultra-short waves in a ferrite. An N-fold Darboux transformation of that system is presented via an existing Lax pair, where N is a positive integer. A generalized (1, N − 1)- fold Darboux transformation, which admits one spectral parameter, and a generalized (m, N − m)-fold Darboux transformation, which admits m spectral parameters, are proposed based on that N-fold Darboux transformation, where m is a positive integer. One-, two- and three-soliton solutions of that system are determined via that N-fold Darboux transformation. Double-pole and triple-pole soliton solutions of that system are derived via that (1, N − 1)-fold Darboux transformation. Those solitons are graph- ically represented for the magnetization and external magnetic field related to a ferrite.

Generalized (m, N − m)-fold Darboux transformation and localized waves of the complex Kraenkel-Manna-Merle system in a ferrite

“…With the generalized (1, 0)-fold DT, we have obtained the first-order rogue-wave and breather solutions, i.e., Solutions (14) and (18). The second-order rogue-wave and breather solutions, i.e., Solutions (15) and (19), have been derived via the generalized (2, 0)-fold DT. By virtue of the generalized (1, 2)-fold DT, we have determined the third-order rogue-wave solutions, i.e., Solutions (16).…”

“…7(b) has one hump and one valley. On the basis of Solutions (19), the second-order breather is displayed in Fig. 8, which features the interaction between the two first-order breathers.…”

“…For certain short waves in a ferrite, Refs. [15][16][17][18][19] have studied the following complex Kraenkel-Manna-Merle system:…”

“…Painlevé integrability, Bäcklund transformation, bilinear form and one-soliton solutions of System (1) have been obtained [17,18]. N -fold DT, generalized (1, N − 1)-fold DT and soliton solutions of System (1) have been presented, with N being the positive integer [19].…”