submitted to: J. Mathematical Physics

Nonlinear Dynamics of Structures, Systems and Devices, 2020

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 [10] 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 [11]. 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.

Mathematische Nachrichten, 2000

We derive an improved fully explicit expression for the right-hand sides of the matrix KdV hierarchy using the relation to the heat kernel of the one-dimensional Schrödinger operator. Our method of "matrix elements" produces, moreover, an explicit expression for the powers of a Schrödinger-like differential operator of any order.

This is a review of two of the fundamental tools for analysis of soliton equations: i) the algebraic ones based on Kac-Moody algebras, their central extensions and their dual algebras which underlie the Hamiltonian structures of the NLEE; ii) the construction of the fundamental analytic solutions (FAS) of the Lax operator and the Riemann-Hilbert problem (RHP) which they satisfy. The fact that the inverse scattering problem for the Lax operator can be viewed as a RHP gave rise to the dressing Zakharov-Shabat, one of the most effective ones for constructing soliton solutions. These two methods when combined may allow one to prove rigorously the results obtained by the abstract algebraic methods. They also allow to derive spectral decompositions for non-self-adjoint Lax operators. 2000 Mathematics Subject Classification. [. Key words and phrases. graded algebras, soliton equations. Financial support in part from Gruppo collegato di INFN at Salerno, Italy and project PRIN 2000 (contract 323/2002) is acknowledged.

Computer Physics Communications, 1987

The hereditariness of recursion operators is discussed for some 5th order nonlinear partial differential equations as well as for several coupled systems. Consequences of the hereditary property are surveyed. An outline of the corresponding computer algebra proofs (based on the formula manipulation systems MAPLE and MACSYMA) is given. Several new hierarchies of completely integrable systems are presented.

Il Nuovo Cimento A

2013

The Lax representations for the soliton equations with Z h and D h reductions are analyzed. Their recursion operators are shown to possess factorization properties due to the grading in the relevant Lie algebra. We show that with each simple Lie algebra one can relate r fundamental recursion operators Λ m k and a master recursion operator Λ generating NLEEs of MKdV type and their Hamiltonian hierarchies. The Wronskian relations are formulated and shown to provide the tools to understand the inverse scattering method as a generalized Fourier transform. They are also used to analyze the conservation laws of the above mentioned soliton equations.

Journal of Mathematical Sciences, 2018

UDC 517.9 We investigate the procedures of discretization of the integrable nonlinear Schrödinger dynamical system, well known as the Ablowitz-Ladik equation, the corresponding symplectic structures, and the finitedimensional invariant reductions. We develop an efficient scheme of invariant reduction of the corresponding infinite system of ordinary differential equations to an equivalent finite system of ordinary differential equations with respect to the evolution parameter. We construct a finite set of recurrence algebraic regular relations that allows one to generate solutions of the discrete nonlinear Schrödinger dynamical system and discuss the related functional spaces of solutions. Finally, we analyze the Fourier-transform approach to the study of the set of solutions of the discrete nonlinear Schrödinger dynamical system and its functional-analytic aspects.

Methods and Applications of Analysis, 2010

In this paper, we present the differential operators of the generalized fifth-order KdV equation. We give formal proofs on the Hamiltonian property including the skew-adjoint property and Jacobi identity by the use of prolongation method. Our results show that there are five 3-order Hamiltonian operators, which can be used to construct the Hamiltonians, and no 5-order operators are shown to pass the Hamiltonian test, although there are infinite number of them, and are skewadjoint.

Journal of Mathematical Analysis and Applications, 2018

The inverse scattering transform is developed for a combined modified Korteweg-de Vrie equation through the technique of Riemann-Hilbert problems. From special Riemann-Hilbert problems with an identity jump matrix, soliton solutions are generated, which corresponds to the inverse scattering problems with reflectionless coefficients. A specific example of two-soliton solutions is explicitly presented, together with its 3d plots, contour plots and x-curve plots.