Mid Sweden University

An overview on the study of nonlinear evolution equations of soliton type is provided. In addition, 5th-order nonlinear evolution equations are shown to be connected to the Caudrey–Dodd–Gibbon–Sawada–Kotera (CDGSK) equation via Bäcklund transformations. The links are depicted in a wide net of links which we term a Bäcklund Chart. The links obtained previously by Rogers and Carillo and by Carillo and Fuchssteiner are revisited, and new results are obtained. A 5th-order nonlinear evolution equation, which does not seem to appear in any list of integrable equations, is provided. All the connected equations exhibit a very interesting symmetry structure enjoyed by the corresponding full hierarchies. Indeed, they all admit a hereditary recursion operator. Hence, each one of the mentioned equations represents the base member of a corresponding hierarchy of equations. These hierarchies are constructed via the recursive application of the respective recursion operators. The symmetry properties of such equations are recalled. Finally, we compare the net of links, derived via Bäcklund transformations, in the case of the fifth-order nonlinear evolution equations with an analog net of links connecting third-order Korteweg-de Vries (KdV) and modified Korteweg-de Vries (mKdV) equations. Analogies and discrepancies between the connections established in the case of fifth-order equations with respect to those established in the case of third-order equations are analyzed. This study aims to open the way for the construction of corresponding non-Abelian equations of the fifth order.

In this article, a general solution formula is derived for the d×d-matrix modified Korteweg–de Vries equation. Then, a solution class corresponding to special parameter choices is examined in detail. Roughly, this class can be described as N-solitons (in the sense of Goncharenko) with common phase matrix. It turns out that such a solution even takes values in a commutative subalgebra of the d×d-matrices. We arrive at a rich picture of possibilities for generalized 1-solitons and at visual patterns of N-solitons which combine nonlinear with linear features. The impact of the phase matrix is visualized in computer plots.

In this paper, which has a partially introductory character, the point of departure is very general solution formulas to the operator-valued Kadomtsev-Petviashvili equations (KPI and KPII). Their generality relies on the presence of parameters, which are allowed to be linear mappings between Banach spaces. While this freedom gives access to very complicated solutions, like countable nonlinear superpositions of solitons, the applications in the present article are restricted to matrix parameters. The necessary techniques to ‘project’ to scalar and matrix-valued solutions are explained in detail. The applications part is focused on the KPII. For scalar solutions, the testing ground is the celebrated classification of web structures of interacting line-solitons by Biondini, Chakravarty, Kodama, et al. We establish an explicit link between the different approaches and add some details about interactions of few line-solitons and solutions with singularities. Starting from more complicated algebraic data, we construct solutions beyond web structures, for which we prove asymptotic movement along logarithmic curves in time slices. The article is concluded by a section on solutions to the d × d matrix KPII. We realize N-solitons in the sense of Gilson, Nimmo, and Sooman (and originally Goncharenko) and discuss the question about nonscalar Miles structures. 

This article is a sequel to 

Carillo, S., Lo Schiavo, M., Schiebold, C.: Matrix soliton solutions of the modified Korteweg-de Vries equation. In: Lacarbonara, W., Balachandran, B., Ma, J., Tenreiro Machado, J.A., Stepan, G. (eds.) Nonlinear Dynamics of Structures, Systems and Devices, pp. 75–83. Springer, Cham (2020)

Carillo, S., Schiebold, C.: Construction of soliton solutions of the matrix modified Korteweg-de Vries equation. In: Lacarbonara, W., Balachandran, B., Leamy, M.J., Ma, J., Tenreiro Machado, J.A., Stepan, G. (eds.) Advances in Nonlinear Dynamics, pp. 481–491. Springer, Cham (2022)

A couple of applications of Bäcklund 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 Bäcklund transformations, a Bäcklund 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 Bäcklund chart which connects various different third order nonlinear evolution equations an invariance admitted by the Korteweg-de Vries interacting soliton (int.sol.KdV) equation is obtained and a related new explicit solution is constructed. Then, the corresponding non-Abelian Bäcklund chart, showshow to construct matrix solutions of the mKdV equations: some recently obtained solutions are reconsidered. 

Bäcklund transformations, well known to represent a key tool in the study of nonlinear evolution equations, are shown to allow the construction of a net of nonlinear links, termed Bäcklund chart, connecting Abelian as well as non-Abelian equations. In particular, Bäcklund transformations are applied to reveal algebraic properties enjoyed by nonlinear evolution equations they connect. The present study concerns third-order nonlinear evolution equations, termed KdV-type, which are all connected to the KdV equation. The Abelian wide Bäcklund chart connecting these nonlinear evolution equations is recalled. Then, the links, originally established in the case of Abelian equations, are shown to conserve their validity when non-Abelian counterparts are considered. In addition, the non-commutative case reveals a richer structure since there may be more than a single non-Abelian counterpart of the same Abelian equation. Reduction from the non-commutative to the commutative case allows to show the connection of the KdV equation with KdV eigenfunction equation, in the scalar case. The main result presented refers to the KdV eigenfunction equation: some explicit solutions it admits are constructed on application of an invariance property proved via Bäcklund transformations. These, to the best of the authors? knowledge, new solutions represent an example of the powerfulness of the method devised. Matrix solutions of the mKdV equations, recently obtained, are mentioned in the closing remarks to stress the powerfulness of the method.

An explicit solution formula for the matrix modified KdV equation is presented, which comprises the solutions given in a previous article. In fact, the solutions in this study 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.

The present article is concerned with the interaction of solitary wave solutions of the matrix Korteweg-de Vries equation. The picture is essentially richer than in the classical scalar case since collisions may be less elasticin the sense that they do not only cause a position shift but also a change of shape. Our construction of solutions is based on a general solution formula with matrix parameters. After a discussion which parameters yield solutions con-sisting of one localized wave, an asymptotic description is obtained for the interaction of two such waves, the crucial point being explicit formulas for the change of shape. The main result extends previous work by Goncharenko, who studied waves coming from matrices of rank 1. As usual, it is to be expected that nonlinear superpositions of finitely many waves behave like combinations of 2-soliton interactions.

Nonlinear non-abelian Korteweg–de Vries (KdV) and modified Korteweg–de Vries (mKdV) equations and their links via Bäcklund transformations are considered. The focus is on the construction of soliton solutions admitted by matrix modified Korteweg–de Vries equation. Matrix equations can be viewed as a specialisation of operator equations in the finite dimensional case when operators admit a matrix representation. Bäcklund transformations allow to reveal structural properties Carillo and Schiebold (J Math Phys 50:073510, 2009) enjoyed by non-commutative KdV-type equations, such as the existence of a recursion operator. Operator methods combined with Bäcklund transformations allow to construct explicit solution formulae Carillo and Schiebold (J Math Phys 52:053507, 2011). The latter are adapted to obtain solutions admitted by the 2 × 2 and 3 × 3 matrix mKdV equation. Some of these matrix solutions are visualised to show the solitonic behaviour they exhibit. A further key tool used to obtain the presented results is an ad hoc construction of computer algebra routines to implement non-commutative computations.

The first main aim of this article is to derive an explicit solution formula for the scalar two-dimensional Toda lattice depending on three independent operator parameters, ameliorating work in [31]. This is achieved by studying a noncommutative version of the 2d-Toda lattice, generalizing its soliton solution to the noncommutative setting. The purpose of the applications part is to show that the family of solutions obtained from matrix data exhibits a rich variety of asymptotic behaviour. The first indicator is that web structures, studied extensively in the literature, see [4] and references therein, are a subfamily. Then three further classes of solutions (with increasingly unusual behaviour) are constructed, and their asymptotics are derived. © 2019, © 2019 the authors.