Exact accelerating solitons in nonholonomic deformation of the KdV equation with a two-fold integrable hierarchy

Abstract

Recently proposed nonholonomic deformation of the KdV equation is solved through inverse scattering method by constructing AKNS type Lax pair. Exact N-soliton solutions are found for the basic field and the deforming function showing unusual ac(de)celerated motion. Two-fold integrable hierarchy is revealed, one with usual higher order dispersion and the other with novel higher nonholonomic deformations.

Integrable twofold hierarchy of perturbed equations and application to optical soliton dynamics

“…On the other hand, in addition to the well-known Camassa-Holms and Degasperis-Procesi equations, some other unusual new integrable equations were recently discovered, namely, the KdV6 [4], LF DNLS [5], mKdV-SG [10], and higher mKdV [11] equations. We realized that these uncommon equations are merely particular cases of general integrable perturbations of the AKNS-Zakharov-Shabat and the KN systems [6]- [8].…”

“…One such area is the nonholonomic deformation of integrable systems [2], which leads to integrable perturbations of known integrable systems with a twofold hierarchy and rich soliton dynamics [3] and also to discovering new equations such as integrable perturbations of the KdV (mKdV), sine-Gordon (SG), nonlinear Schrödinger (NLS), and the derivative NLS (DNLS) equations, and to obtaining some recently proposed equations, for example, the KdV6 [4] and Lennels-Fokas (LF) [5] equations, and so on. Moreover, using nonholomorphic deformations allows explaining the integrability of these systems and obtaining exact soliton solutions [3], [6]- [8].…”

“…Further deforming the time Lax operator using negative powers of the spectral parameter, we could then generate new integrable perturbations with nonholonomic constraints for those well-known integrable equations. This novel class of deformed integrable equations exhibiting more interesting features such as a twofold hierarchy and an exact soliton with shape changing and accelerating motion [3], [6]- [8] is important not only from the theoretical standpoint but also from the application standpoint.…”

Integrability of Certain Deformed Nonlinear Partial Differential Equations

“…Kundu has shown that Eq. (1.4) admits a Lax pair [9]. Also Kupershmidt [8] has demonstrated that Eq.…”

“…Equations (1.3) known as KdV 6 in the literature and can also be written as u t − u xxx − 6uu x = g x (t, x), (1.4a) g xxx + 4ug x + 2u x g = 0, (1.4b) which can be viewed as a nonholonomic deformation of the Korteweg-deVries equation [9,11]. It is appropriate to mention that Eq.…”