Abelian versus non-Abelian Baecklund Charts: some remarks

Connections via Baecklund transformations among different non-linear evolution equations are investigated aiming to compare corresponding Abelian and non Abelian results. Specifically, links, via Baecklund transformations, connecting Burgers and KdV-type hierarchies of nonlinear evolution equations are studied. Crucial differences as well as notable similarities between Baecklund charts in the case of the Burgers - heat equation, on one side and KdV -type equations are considered. The Baecklund charts constructed in [16] and [17], respectively, to connect Burgers and KdV-type hierarchies of operator nonlinear evolution equations show that the structures, in the non-commutative cases, are richer than the corresponding commutative ones.Comment: 18 page

A novel noncommutative KdV-type equation, its recursion operator, and solitons

A noncommutative KdV-type equation is introduced extending the Baecklund chart in [S. Carillo, M. Lo Schiavo, and C. Schiebold, SIGMA 12 (2016)]. This equation, called meta-mKdV here, is linked by Cole-Hopf transformations to the two noncommutative versions of the mKdV equations listed in [P.J. Olver and V.V. Sokolov Commun. Math. Phys. 193 (1998), Theorem 3.6]. For this meta-mKdV, and its mirror counterpart, recursion operators, hierarchies and an explicit solution class are derived

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

Construction of soliton solutions of the matrix modified Korteweg-de Vries equation

An explicit solution formula for the matrix modified KdV equation is presented, which comprises the solutions given in Ref. 7 (S. Carillo, M. Lo Schiavo, and C. Schiebold. Matrix solitons solutions of the modified Korteweg-de Vries equation. In: Nonlinear Dynamics of Structures, Systems and Devices, edited by W. Lacarbonara, B. Balachandran, J. Ma, J. Tenreiro Machado, G. Stepan (Springer, Cham, 2020), pp. 75-83). In fact, the solutions in Ref.7 are part of a subclass studied in detail by the authors in a forthcoming publication. Here several solutions beyond this subclass are constructed and discussed with respect to qualitative properties.Comment: 10 pages, 6 figures, Proceedings of the Second International Nonlinear Dynamics Conference (NODYCON 2021), W. Lacarbonara et al, Ed.

Soliton equations: admitted solutions and invariances via Bäcklund transformations

A couple of applications of Baecklund transformations in the study of nonlinear evolution equations is here given. Specifically, we are concerned about third order nonlinear evolution equations. Our attention is focussed on one side, on proving a new invariance admitted by a third order nonlinear evolution equation and, on the other one, on the construction of solutions. Indeed, via B ̈acklund transformations, a Baecklund chart, connecting Abelian as well as non Abelian equations can be constructed. The importance of such a net of links is twofold since it indicates invariances as well as allows to construct solutions admitted by the nonlinear evolution equations it relates. The present study refers to third-order nonlinear evolution equations of KdV type. On the basis of the Abelian wide Baecklund chart which connects various different third order nonlinear evolution equations an invariance admitted by the Korteweg-deVries interacting soliton (int.sol.KdV) equation is obtained and a related new explicit solution is constructed. Then, the corresponding non-Abelian Baecklund chart, shows how to construct matrix solutions of the mKdV equations: some recently obtained solutions are reconsidered

Soliton equations: admitted solutions and invariances via B\"acklund transformations

A couple of applications of B\"acklund transformations in the study of nonlinear evolution equations is here given. Specifically, we are concerned about third order nonlinear evolution equations. Our attention is focussed on one side, on an invariance admitted by the interacting soliton equation and, on the other one, on the construction of solutions. Indeed, via B\"acklund transformations, a B\"acklund chart, connecting Abelian as well as non Abelian equations can be constructed. The importance of such a net of links is twofold since it indicates invariances as well as allows to construct solutions admitted by the nonlinear evolution equations it relates. The present study refers to third order nonlinear evolution equations of KdV type. On the basis of the Abelian wide B\"acklund chart which connects various different third order nonlinear evolution equations an invariance admitted by the int sol. KdV equation is obtained and an explicit solution is constructed. Then, the corresponding non-Abelian B\"acklund chart, shows how to construct matrix solutions of the mKdV equations: some recently obtained solutions are reconsidered.Comment: 11 pages, 6 figures. arXiv admin note: text overlap with arXiv:2101.0924

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

Construction of soliton solutions of the matrix modified Korteweg-de Vries equation

An explicit solution formula for the matrix modified KdV equation is presented, which comprises the solutions given in [ S. Carillo, M. Lo Schiavo, and C. Schiebold. Matrix solitons solutions of the modified Korteweg-de Vries equation. In: Nonlinear Dynamics of Structures, Systems and Devices, edited by W. Lacarbonara, B. Balachandran, J. Ma, J. Tenreiro Machado, G. Stepan. (Springer, Cham, 2020), pp. 75–83]. In fact, the solutions therein are part of a subclass studied in detail by the authorsin a forthcoming publication. Here several solutions beyond this subclass are constructed and discussed with respect to qualitative properties