Foldons in 2+1 Dimensions

The (2+1)-dimensional “universal” variable separation formula (Tang, Lou and Zhang, Physical Review E 66, 046601 (2002)) is extended to a more general form for some (2+1)-dimensional integrable systems. Various independent arbitrary functions have been included in the extended formula. Starting from the universal formula and its extended form, a new type of solitons, foldons, is studied both analytically and graphically. The foldons may be “folded” in quite complicated ways and then possess quite rich structures and abundant interaction behaviors. The completely elastic interaction property for a quite general class of foldons is proven and the corresponding phase shifts are explicitly given. PACS.02.30.Jr, 02.30.Ik, 05.45.Yv. In the study of nonlinear physical systems, which are diverged at many physical fields such as the fluid physics, condense matter physics, plasma physics, nonlinear optics, quantum field theory and particle physics, various mathematical models have been established. Usually these models have to be expressed by complicated nonlinear partial differential equations (PDEs) or nonlinear differential-difference equations (DDEs). To solve these complicated models, the best and natural starting point is to study some first order approximate models that are usually the integrable systems and possess intrinsic and essential nonlinear behaviors. To solve the integrable models especially in high dimensions is still very difficult. Though some types of powerful methods, say, the inverse scattering transformation (IST), Lie group approaches, bilinear method, Bäclund and Darboux transformations etc.[1] have been established to find some exact solutions, we still need more approaches to find more general exact solutions. Recently, it is found that a quite “universal” formula [2] U ≡ −2∆qypx (a0 + a1p+ a2q + a3pq) , ∆ ≡ a0a3 − a1a2, (1) is valid for suitable fields or potentials of various (2+1)-dimensional physically interesting integrable models including the Davey-Stewartson (DS) equation, the Nizhnik-Novikov-Veselov 1 (NNV) equation, the asymmetric NNV (ANNV) equation, the asymmetric DS (ADS) equation, the dispersive long wave equation (DLWE)[3], the Broer-Kaup-Kupershmidt (BKK), the Maccari system, the general (N+M)-component Ablowitz-Kaup-Newell-Segur (AKNS) system, the long wave-short wave interaction model, and the (2+1)-dimensional sine-Gordon (2DsG) model. A differential-difference form of the universal formula (1) has also been given for a special differential-difference Toda system and the differential-difference ANNV system[4]. In Eq. (1), p ≡ p(x, t) is an arbitrary function of {x, t}, q ≡ q(y, t) may be either an arbitrary function for some kinds of models such as the DS, NNV and 2DsG systems or an arbitrary solution of a Riccati equation for some other types of systems, say, the ANNV, ADS and AKNS systems while a0, a1, a2 and a3 are taken as constants. One of the most important results obtained from (1) is that for all the models mentioned above there are quite rich localized excitations, for example, the solitoffs, dromions, lumps, breathers, instantons, ring solitons, peakons, compactons, localized chaotic and fractal patterns and so on[2]. In this letter, we mainly focus our attentions on the following two important problems: (i). Can the “universal” formula be extended to some more general forms? (ii). What kinds of new localized excitations especially with completely elastic interaction property can we find from the universal formula or its extended forms (if there exist)? For the first question, the formula (1) will be extended to a much more general form with any number of arbitrary functions for the dispersive long wave equation and the result may be valid for some other models. For the second problem, a new type of localized excitations, foldons, is defined and the possible completely elastic interaction behaviors of foldons are investigated. For the (2+1)-dimensional DLWE uyt + ηxx + uxuy + uuxy = 0, ηt + ux + ηux + uηx + uxxy = 0, (2) the universal formula (1) is valid for the field v ≡ −η − 1 and the related known exact solution reads v ≡ −η − 1 = U, u = ± 2px(a1 + a3q) a0 + a1p+ a2q + a3pq + u0, (3) with p being an arbitrary functions of {x, t}, q = q(y, t) being an arbitrary solution of the Riccati equation, qt − a0c0 − (a1c1 + a2c0 − a0c2)q − (a3c1 − a2c2)q 2 = 0, and u0 = −p x (pt ± pxx − a0c1 − (a1c1 + a2c0 + a0c2)p − (a1c2 + a3c0)p ). The (1+1)dimensional DLWE (y = x of (2)) is also called the classical Boussinesq equation. There exist a large 2 number of papers to discuss the possible applications and exact solutions of the (1+1)dimensional DLWE[5]. The (2+1)-dimensional DLWE was firstly obtained by Boiti et al [6] as a compatibility condition for a “weak” Lax pair. In [7], Paquin and Winternitz showed that the symmetry algebra of (2) is infinite-dimensional and has a Kac-Moody-Virasoro structure. Some special similarity solutions are also given in [7] by using symmetry algebra and the classical theoretical analysis. The more general symmetry algebra, W∞ symmetry algebra, is given in [8]. In [9], nine types of two-dimensional similarity reductions and thirteen types of ordinary differential equation reductions are given. Though the model is Lax or IST integrable, it does not pass the Painlevé test[10]. To extend the universal formula to a more general form, we use the multi-linear variable separation approach (MLVSA) again. For the DLWE (2), the Bäcklund transformation u = ±2 fx f + u0 and η = η0 + 2 fxy f − 2 fxfy f with {u0, η0} being a known solution of (2) degenerates two equations of (2) to a single trilinear form, [fxxxy ± (fxyt + u0xfxy + u0fxxy)]f 2 − [(fxxx ± u0fxx ± fxt)fy ± fxfyu0x +(fxxy ± 2u0fxy ± fyt)fx + fxyfxx ± ftfxy]f + 2fxfy(fxx ± u0fx ± ft) = 0, (4) where the seed solution has been fixed as {u0 ≡ u0(x, t), η0 = −1} for simplicity. To solve the trilinear equation (4), one has to use some prior ansatz. If one use the variable separation ansatz f = a0 +a1p+a2q+a3pq, then the variable separation solution (3) follows immediately. In [8], it has been pointed out that for the DLWE, there are two sets of infinitely many generalized symmetries and every symmetry possesses an arbitrary function of t or y. That means infinitely many arbitrary functions of y and t can be entered into the general solutions of (2). So we hope to extend the solution (3) to a more general one such that more arbitrary functions can be included. After finishing the detailed analysis and calculations, we find that the following variable separation ansatz,