Computable Integrability. Chapter 1: General notions and ideas (original) (raw)
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Arxiv preprint arXiv:1103.2423, 2011
iv Being unaware of the work of FPU, Toda was looking for an exact model of heat conduction, and eventually arrived at his exponential lattice. His method was to go back the usual way from equations to solutions: starting from a nonlinear wave given by an elliptic function, he attempted to derive a nonlinear equation to be satisfied by the nonlinear wave, and eventually arrived to his exponential lattice. Within less than a decade, the method of [21] and was extended to many other soliton equations, some of the most results in this direction are briefly presented below.
Aspects of Integrability of Differential Systems and Fields
SpringerBriefs in Physics, 2019
This monograph, written at an intermediate level for educational purposes, serves as an introduction to the concept of integrability as it applies to systems of differential equations (both ordinary and partial) as well as to vector-valued fields. We stress from the outset that this is not a treatise on the theory or the methods of solution of differential equations! Instead, we have chosen to focus on specific aspects of integrability that are often encountered in a variety of problems in Applied Mathematics, Physics and Engineering. an important application in first-order PDEs, the solution process of which is briefly described. Finally, we study the case of a linear system of ODEs, the solution of which reduces to an eigenvalue problem. Chapter 5 examines systems of ODEs from the geometric point of view. Concepts of Differential Geometry such as the integral and phase curves of a differential system, the differential-operator representation of vector fields, the Lie derivative, etc., are introduced at a fundamental level. The geometric significance of first-order PDEs is also studied, revealing a close connection of these equations with systems of ODEs and vector fields. Two notions of importance in the theory of integrable nonlinear PDEs are Bäcklund transformations and Lax pairs. In both cases a PDE is expressed as an integrability condition for solution of an associated system of PDEs. These ideas are briefly discussed in Chapter 6. A familiar system of PDEs in four dimensions, namely, the Maxwell equations for the electromagnetic field, is shown to constitute a Bäcklund transformation connecting solutions of the wave equations satisfied by the electric and the magnetic field. The solution of the Maxwell system for the case of a monochromatic plane electromagnetic wave is derived in detail. Finally, the use of Bäcklund transformations as recursion operators for producing symmetries of PDEs is described. I would like to thank my colleague and friend Aristidis N. Magoulas for an excellent job in drawing a number of figures, as well as for several fruitful discussions on the issue of integrability in Electromagnetism!
An Algebraic Approach to Integrability
We present an algebraic definition of complete integrability of Hamiltonian dynamical systems in an algebraic framework constructed out of an arbitrary algebra F which plays the rôle of the algebra of functions on a manifold.
On the complete integrability of completely integrable systems
Communications in Mathematical Physics, 1991
The question of complete integrability of evolution equations associated to n × n first order isospectral operators is investigated using the inverse scattering method. It is shown that for n > 2, e.g. for the three-wave interaction, additional (nonlinear) pointwise flows are necessary for the assertion of complete integrability. Their existence is demonstrated by constructing action-angle variables. This construction depends on the analysis of a natural 2-form and symplectic foliation for the groups GL(n) and SU(n).
On the Integrability of a Class of Differential Equations
Математички билтен/BULLETIN MATHÉMATIQUE DE LA SOCIÉTÉ DES MATHÉMATICIENS DE LA RÉPUBLIQUE MACÉDOINE, 2021
In this paper, a class of second-order linear di erential equations is reviewed. For this class of B.S.Popov necessary and su cient condition for reductable according to Frobenius is obtained. By using another method, the same condition is obtained where the existence of the natural number n is replaced by the existence of an integer n. For the same class of second-order linear di erential equations, the case for reductable according to Frobenius which is independent from an exist of a number n is reviewed. In both cases, formulas of one particular solution and transformation to a system of rstorder di erential equations are obtained. In end, this theory is supported by examples.