Algorithms for Mappings and Symmetries of Differential Equations

Differential Equations are used to mathematically express the laws of physics and models in biology, finance, and many other fields. Examining the solutions of related differential equation systems helps to gain insights into the phenomena described by the differential equations. However, finding exact solutions of differential equations can be extremely difficult and is often impossible. A common approach to addressing this problem is to analyze solutions of differential equations by using their symmetries. In this thesis, we develop algorithms based on analyzing infinitesimal symmetry features of differential equations to determine the existence of invertible mappings of less tractable systems of differential equations (e.g., nonlinear) into more tractable systems of differential equations (e.g., linear). We also characterize features of the map if it exists. An algorithm is provided to determine if there exists a mapping of a non-constant coefficient linear differential equation to one with constant coefficients. These algorithms are implemented in the computer algebra language Maple, in the form of the MapDETools package. Our methods work directly at the level of systems of equations for infinitesimal symmetries. The key idea is to apply a finite number of differentiations and eliminations to the infinitesimal symmetry systems to yield them in the involutive form, where the properties of Lie symmetry algebra can be explored readily without solving the systems. We also generalize such differential-elimination algorithms to a more frequently applicable case involving approximate real coefficients. This contribution builds on a proposal by Reid et al. of applying Numerical Algebraic Geometry tools to find a general method for characterizing solution components of a system of differential equations containing approximate coefficients in the framework of the Jet geometry. Our numeric-symbolic algorithm exploits the fundamental features of the Jet geometry of differential equations such as differential Hilbert functions. Our novel approach establishes that the components of a differential equation can be represented by certain points called critical points

On the Topic of Motion Integrals

An integral of motion is a function of the states of a dynamical system that is constant along the system’s trajectories. Integrals are known for their utility as a means of reducing the dimension of a system, effectively leaving only one differential – or in some cases algebraic – equation to be solved. Invariants of dynamical systems have also proven useful in other contexts, such as in estimation, numerical integration and optimal control. Regardless of the manner in which an integral is employed, finding an analytic form for the integrals of a system generally requires solution of a system of non-linear partial differential equations, with the exception of cases in which certain symmetries of the system are apparent. The objective of this work is to investigate a generalized method for determining motion integrals for non-linear dynamical systems. This method will not work for all nonlinear systems. Indeed, it is expected that the results will test the limitations of this method. In this we consider a method for determining integrals of motion for a small class of dynamical systems akin to the traditional series expansion method for solving partial differential equations. This method involves posing a candidate integral of motion as a series expansion in terms of some set of polynomials. The coefficients of the candidate polynomial are treated as the unknowns in a system of equations. The system of equations is constructed by sampling simulated trajectories of the dynamical system in question. Then the coefficients are solved for using singular value decomposition. There are a number of parameters that can potentially affect this method’s ability to generate an integral of motion that effectively approximates the phase space of the full nonlinear dynamical system. Part of this thesis proposes what some of these parameters might be and investigates how they affect the outcome. A couple of well-known systems are used to conduct these tests: the simple pendulum and the rotating rigid body. The simple pendulum is one of the simplest examples of a non-linear system, and examples of the rotating rigid body in aerospace engineering are ubiquitous

A Characterization of Reduced Forms of Linear Differential Systems

A differential system $[A] : \; Y'=AY$, with $A\in \mathrm{Mat}(n, \bar{k})$ is said to be in reduced form if $A\in \mathfrak{g}(\bar{k})$ where $\mathfrak{g}$ is the Lie algebra of the differential Galois group $G$ of $[A]$. In this article, we give a constructive criterion for a system to be in reduced form. When $G$ is reductive and unimodular, the system $[A]$ is in reduced form if and only if all of its invariants (rational solutions of appropriate symmetric powers) have constant coefficients (instead of rational functions). When $G$ is non-reductive, we give a similar characterization via the semi-invariants of $G$. In the reductive case, we propose a decision procedure for putting the system into reduced form which, in turn, gives a constructive proof of the classical Kolchin-Kovacic reduction theorem.Comment: To appear in : Journal of Pure and Applied Algebr

Tyurin parameters and elliptic analogue of nonlinear Schr\"odinger hierarchy

Two "elliptic analogues'' of the nonlinear Schr\"odinger hiererchy are constructed, and their status in the Grassmannian perspective of soliton equations is elucidated. In addition to the usual fields $u,v$, these elliptic analogues have new dynamical variables called ``Tyurin parameters,'' which are connected with a family of vector bundles over the elliptic curve in consideration. The zero-curvature equations of these systems are formulated by a sequence of $2 \times 2$ matrices $A_n(z)$, $n = 1,2,...$, of elliptic functions. In addition to a fixed pole at $z = 0$, these matrices have several extra poles. Tyurin parameters consist of the coordinates of those poles and some additional parameters that describe the structure of $A_n(z)$'s. Two distinct solutions of the auxiliary linear equations are constructed, and shown to form a Riemann-Hilbert pair with degeneration points. The Riemann-Hilbert pair is used to define a mapping to an infinite dimensional Grassmann variety. The elliptic analogues of the nonlinear Schr\"odinger hierarchy are thereby mapped to a simple dynamical system on a special subset of the Grassmann variety.Comment: latex2e, 36 pp, no figure; (v2) minor changes, mostly typos; (v3) Title changed, text fully revised with new results; (v4) serious errors in section 5 corrected; (v5) proof of main results is improved; (v6) minor change in proof of Lemma 10 etc; (v7) final version for publication; (v8) typos corrected. Journal of Mathematical Sciences, University of Tokyo (to appear

A density theorem for parameterized differential Galois theory

We study parameterized linear differential equations with coefficients depending meromorphically upon the parameters. As a main result, analogously to the unparameterized density theorem of Ramis, we show that the parameterized monodromy, the parameterized exponential torus and the parameterized Stokes operators are topological generators in Kolchin topology, for the parameterized differential Galois group introduced by Cassidy and Singer. We prove an analogous result for the global parameterized differential Galois group, which generalizes a result by Mitschi and Singer. These authors give also a necessary condition on a group for being a global parameterized differential Galois group; as a corollary of the density theorem, we prove that their condition is also sufficient. As an application, we give a characterization of completely integrable equations, and we give a partial answer to a question of Sibuya about the transcendence properties of a given Stokes matrix. Moreover, using a parameterized Hukuhara-Turrittin theorem, we show that the Galois group descends to a smaller field, whose field of constants is not differentially closed.Comment: To appear in Pacific Journal of Mathematic

A Reduced Form for Linear Differential Systems and its Application to Integrability of Hamiltonian Systems

Let $[A]: Y'=AY$ with $A\in \mathrm{M}_n (k)$ be a differential linear system. We say that a matrix $R\in {\cal M}_{n}(\bar{k})$ is a {\em reduced form} of $[A]$ if $R\in \mathfrak{g}(\bar{k})$ and there exists $P\in GL_n (\bar{k})$ such that $R=P^{-1}(AP-P')\in \mathfrak{g}(\bar{k})$. Such a form is often the sparsest possible attainable through gauge transformations without introducing new transcendants. In this article, we discuss how to compute reduced forms of some symplectic differential systems, arising as variational equations of hamiltonian systems. We use this to give an effective form of the Morales-Ramis theorem on (non)-integrability of Hamiltonian systems.Comment: 28 page