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Generic Hamiltonian systems have a mixed phase space, where classically disjoint regions of regular and chaotic motion coexist. For many applications it is useful to approximate the regular dynamics of such a mixed system H by an integrable approximation Hreg. We present a new, iterative method to construct such integrable approximations. The method is based on the construction of an integrable approximation in action representation which is then improved in phase space by iterative applications of canonical transformations. In contrast to other known approaches, our method remains applicable to strongly non-integrable systems H. We present its application to 2D maps and 2D billiards. Based on the obtained integrable approximations we finally discuss the theoretical description of dynamical tunneling in mixed systems. / Typische Hamiltonsche Systeme haben einen gemischten Phasenraum, in dem disjunkte Bereiche klassisch regulärer und chaotischer Dynamik koexistieren. Für viele Anwendungen ist es zweckmäßig, die reguläre Dynamik eines solchen gemischten Systems H durch eine integrable Näherung Hreg zu beschreiben. Wir stellen eine neue, iterative Methode vor, um solche integrablen Näherungen zu konstruieren. Diese Methode basiert auf der Konstruktion einer integrablen Näherung in Winkel-Wirkungs-Variablen, die im Phasenraum durch iterative Anwendungen kanonischer Transformationen verbessert wird. Im Gegensatz zu bisher bekannten Verfahren bleibt unsere Methode auch auf stark nichtintegrable Systeme H anwendbar. Wir demonstrieren sie anhand von 2D-Abbildungen und 2D-Billards. Mit den gewonnenen integrablen Näherungen diskutieren wir schließlich die theoretische Beschreibung von dynamischem Tunneln in gemischten Systemen.

This dissertation is concerned with analysis of orbital stability of solitary waves and well-posedness of the Cauchy problem in the integrable evolution equations. The analysis is developed by using tools from integrable systems, such as higher-order conserved quantities, B\"{a}cklund transformation, and inverse scattering transform. The main results are obtained for the massive Thirring model, which is an integrable nonlinear Dirac equation, and for the derivative NLS equation. Both equations are related with the same Kaup-Newell spectral problem. Our studies rely on the spectral properties of the Kaup-Newell spectral problem, which convey key information about solution behavior of the nonlinear evolution equations. / Dissertation / Doctor of Philosophy (PhD)

We present a new spectral problem, a generalization of the D-Kaup-Newell spectral problem, associated with the Lie algebra sl(2,R). Zero curvature equations furnish the soliton hierarchy. The trace identity produces the Hamiltonian structure for the hierarchy. Lastly, a reduction of the spectral problem is shown to have a different soliton hierarchy with a bi-Hamiltonian structure. The first major motivation of this dissertation is to present spectral problems that generate two soliton hierarchies with infinitely many commuting conservation laws and high-order symmetries, i.e., they are Liouville integrable. We use the soliton hierarchies and a non-seimisimple matrix loop Lie algebra in order to construct integrable couplings. An enlarged spectral problem is presented starting from a generalization of the D-Kaup-Newell spectral problem. Then the enlarged zero curvature equations are solved from a series of Lax pairs producing the desired integrable couplings. A reduction is made of the original enlarged spectral problem generating a second integrable coupling system. Next, we discuss how to compute bilinear forms that are symmetric, ad-invariant, and non-degenerate on the given non-semisimple matrix Lie algebra to employ the variational identity. The variational identity is applied to the original integrable couplings of a generalized D-Kaup-Newell soliton hierarchy to furnish its Hamiltonian structures. Then we apply the variational identity to the reduced integrable couplings. The reduced coupling system has a bi-Hamiltonian structure. Both integrable coupling systems retain the properties of infinitely many commuting high-order symmetries and conserved densities of their original subsystems and, again, are Liouville integrable. In order to find solutions to a generalized D-Kaup-Newell integrable coupling system, a theory of Darboux transformations on integrable couplings is formulated. The theory pertains to a spectral problem where the spectral matrix is a polynomial in lambda of any order. An application to a generalized D-Kaup-Newell integrable couplings system is worked out, along with an explicit formula for the associated Bäcklund transformation. Precise one-soliton-like solutions are given for the m-th order generalized D-Kaup-Newell integrable coupling system.

<p>We derive new combinatorail formulas for vector-valued weight functions for the evolution modules over the Yangians Y (gln). We obtain them using the Nested Algebraic Bethe ansatz method.</p> <p>We also describe the asymptotic behavior of the radial solutions of the negative tt∗ equation via the Riemann-Hilbert problem and the Deift-Zhou nonlinear steepest descent method.</p>

Two Poisson brackets are called compatible if any linear combination of these brackets is a Poisson bracket again. The set of non-zero linear combinations of two compatible Poisson brackets is called a Poisson pencil. A system is called bihamiltonian (with respect to a given pencil) if it is hamiltonian with respect to any bracket of the pencil. The property of being bihamiltonian is closely related to integrability. On the one hand, many integrable systems known from physics and geometry possess a bihamiltonian structure. On the other hand, if we have a bihamiltonian system, then the Casimir functions of the brackets of the pencil are commuting integrals of the system. We consider the situation when these integrals are enough for complete integrability. As it was shown by Bolsinov and Oshemkov, many properties of the system in this case can be deduced from the properties of the Poisson pencil itself, without explicit analysis of the integrals. Developing these ideas, we introduce a notion of linearization of a Poisson pencil. In terms of linearization, we give a criterion for non-degeneracy of a singular point and describe its type. These results are applied to solve the stability problem for a free multidimensional rigid body.

Hamiltonian systems of hydrodynamic type occur in a wide range of applications including fluid dynamics, the Whitham averaging procedure and the theory of Frobenius manifolds. In 1 + 1 dimensions, the requirement of the integrability of such systems by the generalised hodograph transform implies that integrable Hamiltonians depend on a certain number of arbitrary functions of two variables. On the contrary, in 2 + 1 dimensions the requirement of the integrability by the method of hydrodynamic reductions, which is a natural analogue of the generalised hodograph transform in higher dimensions, leads to finite-dimensional moduli spaces of integrable Hamiltonians. We classify integrable two-component Hamiltonian systems of hydrodynamic type for all existing classes of differential-geometric Poisson brackets in 2D, establishing a parametrisation of integrable Hamiltonians via elliptic/hypergeometric functions. Our approach is based on the Godunov-type representation of Hamiltonian systems, and utilises a novel construction of Godunov's systems in terms of generalised hypergeometric functions. Furthermore, we develop a theory of integrable dispersive deformations of these Hamiltonian systems following a scheme similar to that proposed by Dubrovin and his collaborators in 1 + 1 dimensions. Our results show that the multi-dimensional situation is far more rigid, and generic Hamiltonians are not deformable. As an illustration we discuss a particular class of two-component Hamiltonian systems, establishing triviality of first order deformations and classifying Hamiltonians possessing nontrivial deformations of the second order.

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In order to perform a specific function, a protein needs to fold into the proper structure. Prediction the protein structure from its amino acid sequence has still been unsolved problem. The main focus of this thesis is to develop new approach on the protein structure modeling by means of differential geometry and integrable theory. The start point is to simplify a protein backbone as a piecewise linear polygonal chain, with vertices recognized as the central alpha carbons of the amino acids. Frenet frame and equations from differential geometry are used to describe the geometric shape of the protein linear chain. Within the framework of integrable theory, we also develop a general geometrical approach, to systematically derive Hamiltonian energy functions for piecewise linear polygonal chains. These theoretical studies is expected to provide a solid basis for the general description of curves in three space dimensions. An efficient algorithm of loop closure has been proposed.

This thesis describes a geometric approach to integrable systems. In the first part we describe the geometry of Drinfeld--Sokolov integrable hierarchies including the corresponding tau-functions. Motivated by a relation between Drinfeld--Sokolov hierarchies and certain physical partition functions, we define a dispersionless limit of Drinfeld--Sokolov systems. We introduce a class of solutions which we call string solutions and prove that the tau-functions of string solutions satisfy Virasoro constraints generalizing those familiar from two-dimensional quantum gravity. In the second part we explain how procedures of Hamiltonian and quasi-Hamiltonian reductions in symplectic geometry arise naturally in the context of shifted symplectic structures. All constructions that appear in quasi-Hamiltonian reduction have a natural interpretation in terms of the classical Chern-Simons theory that we explain. As an application, we construct a prequantization of character stacks purely locally. / text