Novel variable separation solutions and exotic localized excitations via the ETM in nonlinear soliton systems

Abstract

In this paper, first, the ETM is applied to obtain variable separation solutions of (2+1)-dimensional systems. A common formula with some arbitrary functions is derived to describe suitable physical quantities for some (2+1)-dimensional models such as the generalized Nizhnik-Novikov-Veselov, Davey-Stewartson, Broer-Kaup-Kupershmidt, Boiti-Leon-Pempinelli, integrable Kortweg-de Vries (KdV), breaking soliton and Burgers models. The universal formula in Tang, Lou, and Zhang [Phys. Rev. E 66, 046601 (2002)] can be simplified to the common formula in the present paper, which indicates that redundant process is included there since the easier variable separation form can be employed without loss of generality. Second, this method is successfully generalized to (1+1)-dimensional systems, such as coupled integrable dispersionless, long-wave–short-wave resonance interaction and negative KdV models, and obtain another common formula to describe suitable physical fields or potentials of these (1+1)-dimensional models, which is similar to the one in (2+1)-dimensional systems. Moreover, it also is extended to (3+1)-dimensional Burgers system, and find that the common formula in (2+1)-dimensional systems is also appropriate to describe the (3+1)-dimensional Burgers system. The only differences are that the function q is a solution of a constraint equation and p is an arbitrary function of three independent variables. Finally, based on the common formula for (2+1)-dimensional systems and by selecting appropriate multivalued functions, interactions among special dromion, special peakon and foldon are investigated. The interactions between two special dromions, and between two special peakons, both possess novel properties, that is, there exist a multivalued foldon in the process of their collision, which is different from the reported cases in previous literature. Furthermore, the explicit phase shifts for all the local excitations offered by the common formula have been given, and are applied to these novel interactions in detail.

Exact Solutions and Localized Excitations of Burgers System in (3+1) Dimensions

“…The answer is positive as we choose the parameters of function q(y,t) shown by (26) appropriately. The similar cases have be reported in [34,35].…”

“…Therefore, similar to the (2+1)-dimensional cases, some special localized excitations based on the common formula may be re-derived in the (3+1)-dimensional Burgers system. Since these localized structures have been reported in the previous literature [34,35], we neglect the related discussions in this section. However, as far as we know, the physical field u of the Burgers system is little discussed in previous literature.…”

“…Similar to the case of Figure 4, one can find the interaction between the solitonic bubble and the taper-like soliton is also completely elastic since their amplitudes, velocities, and wave shapes are completely preserved after their collision. It should be mentioned that the completely elastic behaviours occurred in Figures 4 and 5 are rather determined by the selections of the function p: (35) and (36).…”

“…It is obvious that the prevenient mentioned ansatz [34,35,40] is a special case of the general ansatz (6).…”

“…The intrusion of the (3+1)-dimensional arbitrary function p(x, z,t) and the function q(y,t) (a solution of the linear heat equation) in the above solutions implies that the physical fields u, v, and w or their potentials may possess rich localized structures. In [34,35], the authors discussed some localized excitations of a potential R ≡ 2u x = 2v y , i.e.,…”