Noncommutative Korteweg deVries and modified Korteweg deVries Hierarchies via Recursion Methods

Here, noncommutative hierarchies of nonlinear equations are studied. They represent a generalization to the operator level of corresponding hierarchies of scalar equations, which can be obtained from the operator ones via a suitable projection. A key tool is the application of Bäcklund transformations to relate different operator-valued hierarchies. Indeed, in the case when hierarchies in 1+1-dimensions are considered, a “Bäcklund chart” depicts links relating, in particular, the Korteweg–de Vries (KdV) to the modified KdV (mKdV) hierarchy. Notably, analogous links connect the hierarchies of operator equations. The main result is the construction of an operator soliton solution depending on an infinite-dimensional parameter. To start with, the potential KdV hierarchy is considered. Then Bäcklund transformations are exploited to derive solution formulas in the case of KdV and mKdV hierarchies. It is remarked that hierarchies of matrix equations, of any dimension, are also incorporated in the present framework

A new extended matrix KP hierarchy and its solutions

With the square eigenfunctions symmetry constraint, we introduce a new extended matrix KP hierarchy and its Lax representation from the matrix KP hierarchy by adding a new $\tau_B$ flow. The extended KP hierarchy contains two time series ${t_A}$ and ${\tau_B}$ and eigenfunctions and adjoint eigenfunctions as components. The extended matrix KP hierarchy and its $t_A$-reduction and $\tau_B$ reduction include two types of matrix KP hierarchy with self-consistent sources and two types of (1+1)-dimensional reduced matrix KP hierarchy with self-consistent sources. In particular, the first type and second type of the 2+1 AKNS equation and the Davey-Stewartson equation with self-consistent sources are deduced from the extended matrix KP hierarchy. The generalized dressing approach for solving the extended matrix KP hierarchy is proposed and some solutions are presented. The soliton solutions of two types of 2+1-dimensional AKNS equation with self-consistent sources and two types of Davey-Stewartson equation with self-consistent sources are studied.Comment: 17 page

Solitons of the sine-gordon equation coming in clusters

In the present paper, we construct a particular class of solu- tions of the sine-Gordon equation, which is the exact analogue of the so-called negatons, a solution class of the Korteweg-de Vries equation discussed by Matveev [17] and Rasinariu et al. [21]. Their characteristic properties are: Each solution consists of a _nite number of clusters. Roughly speaking, in such a cluster solitons are grouped around a center, and the distance between two of them grows logarithmically. The clusters themselves rather behave like solitons. Moving with constant velocity, they collide elastically with the only e_ect of a phase- shift. The main contribution of this paper is the proof that all this { including an explicit calculation of the phase-shift { an be ex- pressed by concrete asymptotic formulas, which generalize very naturally the known expressions for solitons. Our results con_rm expectations formulated in the context of the Korteweg-de Vries equation by Matveev [17] and Rasinariu et al. [21].In the present paper, we construct a particular class of solu- tions of the sine-Gordon equation, which is the exact analogue of the so-called negatons, a solution class of the Korteweg-de Vries equation discussed by Matveev [17] and Rasinariu et al. [21]. Their characteristic properties are: Each solution consists of a _nite number of clusters. Roughly speaking, in such a cluster solitons are grouped around a center, and the distance between two of them grows logarithmically. The clusters themselves rather behave like solitons. Moving with constant velocity, they collide elastically with the only e_ect of a phase- shift. The main contribution of this paper is the proof that all this { including an explicit calculation of the phase-shift { an be ex- pressed by concrete asymptotic formulas, which generalize very naturally the known expressions for solitons. Our results con_rm expectations formulated in the context of the Korteweg-de Vries equation by Matveev [17] and Rasinariu et al. [21]

Solitons of the sine-Gordon equation coming in clusters

In the present paper, we construct a particular class of solutions of the sine-Gordon equation, which is the exact analogue of the so-called negatons, a solution class of the Korteweg-de Vries equation discussed by Matveev [17] and Rasinariu et al. [21]. Their characteristic properties are: Each solution consists of a finite number of clusters. Roughly speaking, in such a cluster solitons are grouped around a center, and the distance between two of them grows logarithmically. The clusters themselves rather behave like solitons. Moving with constant velocity, they collide elastically with the only effect of a phase-shift. The main contribution of this paper is the proof that all this -including an explicit calculation of the phase-shift - can be expressed by concrete asymptotic formulas, which generalize very naturally the known expressions for solitons. Our results confirm expectations formulated in the context of the Korteweg-de Vries equation by Matveev [17] and Rasinariu et al. [21]

Recursion Techniques and Explicit Solutions of Integrable Noncommutative Hierarchies

A recently developed approach is discussed which, via the examination of noncommutative integrable systems, leads to very general solution formulas for whole hierarchies of KdV type. Applications to matrix systems are indicated

Baecklund Charts: commutative versus non-commutative Equation Hierarchies

Here di↵erent B¨acklund Charts are considered both in the case of Commutative Equation Hierarchies as well as in the case of their Non-Commutative analogues. The aim is to point out di↵erences and analogies[1]. Specifically, the case of the Cole-Hopf link between Burgers and Heat Equations [2, 3] and its extension to the corresponding Hierarchies are considered [4]. Furthermore, links connecting third order nonlinear evolution equations, such as KdV, mKdV are analyzed, again, in both the commutative [5] and noncommutative case [6].si veda sopr