Showing papers on "Integrable system published in 2010"
Book•
19 Nov 2010
TL;DR: This book presents cutting-edge developments in the theory and experiments of nonlinear waves, and contains a large number of simple and efficient MATLAB codes for various nonlinear wave computations, which readers can easily adapt to solve their own problems.
Abstract: This book presents cutting-edge developments in the theory and experiments of nonlinear waves. Its comprehensive coverage of analytical methods for nonintegrable systems is the first of its kind. It also covers in great depth analytical methods for integrable equations, and comprehensively describes efficient numerical methods for all major aspects of nonlinear wave computations. In addition, the book presents the latest experiments on nonlinear waves in optical systems and Bose- Einstein condensates, especially in periodic media. The book contains a large number of simple and efficient MATLAB codes for various nonlinear wave computations, which readers can easily adapt to solve their own problems. The codes can also be found on an associated Web page. Audience: This book is intended for researchers and graduate students working in applied mathematics and various physical subjects, such as nonlinear optics, Bose Einstein condensates, and fluid dynamics, where nonlinear wave phenomena arise. Contents: List of Figures; Preface; Chapter 1. Derivation of nonlinear wave equations; Chapter 2. Integrable theory for the nonlinear Schrdinger equation; Chapter 3. Theories for integrable equations with higher-order scattering operators; Chapter 4. Soliton perturbation theories and applications; Chapter 5. Theories for nonintegrable equations; Chapter 6. Nonlinear wave phenomena in periodic media; Chapter 7. Numerical methods for nonlinear wave equations; Bibliography; Index.
881 citations
TL;DR: In this article, the authors reformulate the Ω-deformation of four-dimensional gauge theory in a way that is valid away from fixed points of the associated group action.
Abstract: We reformulate the Ω-deformation of four-dimensional gauge theory in a way that is valid away from fixed points of the associated group action. We use this reformulation together with the theory of coisotropic A-branes to explain recent results linking the Ω-deformation to integrable Hamiltonian systems in one direction and Liouville theory of two-dimensional conformal field theory in another direction.
317 citations
TL;DR: In this article, a general integrable coupled nonlinear Schrodinger system is investigated, where the coefficients of the self-phase modulation, cross-phase and four-wave mixing terms are more general while still maintaining integrability.
Abstract: In this paper, a general integrable coupled nonlinear Schrodinger system is investigated. In this system, the coefficients of the self-phase modulation, cross-phase modulation, and four-wave mixing terms are more general while still maintaining integrability. The N-soliton solutions in this system are obtained by the Riemann–Hilbert method. The collision dynamics between two solitons is also analyzed. It is shown that this collision exhibits some new phenomena (such as soliton reflection) which have not been seen before in integrable systems. In addition, the recursion operator and conservation laws for this system are also derived.
309 citations
01 Jan 2010
TL;DR: In this paper, the integrability of Hamiltonian systems has been discussed in very different connections and connections have been found between the geodesics on an ellipsoid and Hill's equation with finite gap potential.
Abstract: (a) In these informal lecture notes we discuss a number of integrable Hamiltonian systems which have surfaced recently in very different connections. It is our goal to discuss various aspects underlying the integrability of a system like that of group representation, isospectral deformation and geometrical considerations. Since this subject is still far from being understood or being systematic we discuss a number of examples which are seemingly disconnected. In fact, there are some rather unexpected connections like between the inverse square potential of Calogero (Section 4) and the Korteweg de Vries equation. Here we show a surprising new connection between the geodesics on an ellipsoid and Hill's equation with finite gap potential.
256 citations
TL;DR: In this paper, the authors studied the non-equilibrium time evolution of an integrable field theory in 1+1 dimensions after a sudden variation of a global parameter of the Hamiltonian.
Abstract: We study the non equilibrium time evolution of an integrable field theory in 1+1 dimensions after a sudden variation of a global parameter of the Hamiltonian. For a class of quenches defined in the text, we compute the long times limit of the one point function of a local operator as a series of form factors. Even if some subtleties force us to handle this result with care, there is a strong evidence that for long times the expectation value of any local operator can be described by a generalized Gibbs ensemble with a different effective temperature for each eigenmode.
254 citations
TL;DR: The T and Y-systems are ubiquitous structures in classical and quantum integrable systems as discussed by the authors, having a variety of aspects related to commuting transfer matrices in solvable lattice models, q-characters of Kirillov-Reshetikhin modules of quantum affine algebras.
Abstract: The T and Y-systems are ubiquitous structures in classical and quantum integrable systems. They are difference equations having a variety of aspects related to commuting transfer matrices in solvable lattice models, q-characters of Kirillov-Reshetikhin modules of quantum affine algebras, cluster algebras with coefficients, periodicity conjectures of Zamolodchikov and others, dilogarithm identities in conformal field theory, difference analogue of L-operators in KP hierarchy, Stokes phenomena in 1d Schrodinger problem, AdS/CFT correspondence, Toda field equations on discrete space-time, Laplace sequence in discrete geometry, Fermionic character formulas and combinatorial completeness of Bethe ansatz, Q-system and ideal gas with exclusion statistics, analytic and thermodynamic Bethe ansatze, quantum transfer matrix method and so forth. This review article is a collection of short reviews on these topics which can be read more or less independently.
179 citations
TL;DR: In this article, the authors consider a completely integrable Hamiltonian system on the Hardy space on the circle and show that it admits an infinite sequence of conservation laws in involution, and that it can be approximated by a sequence of finite dimensional completely integral Hamiltonian systems.
Abstract: We consider the following Hamiltonian equation on the $L^2$ Hardy space on the circle, $$i\partial _tu=\Pi(|u|^2u)\ ,$$ where $\Pi $ is the Szego projector. This equation can be seen as a toy model for totally non dispersive evolution equations. We display a Lax pair structure for this equation. We prove that it admits an infinite sequence of conservation laws in involution, and that it can be approximated by a sequence of finite dimensional completely integrable Hamiltonian systems. We establish several instability phenomena illustrating the degeneracy of this completely integrable structure. We also classify the traveling waves for this system.
164 citations
01 Jan 2010
TL;DR: In this paper, the authors give a complete treatment not only of the basic facts about QRT maps, but also the background theory on which these maps are based, assuming Theorem 3.7.
Abstract: 10.1007/978-1-4419-9126-3 Copyright owner: Springer Science+Buisness Media, LLC, 2010 Data set: Springer Source Springer Monographs in Mathematics The rich subject matter in this book brings in mathematics from different domains, especially from the theory of elliptic surfaces and dynamics.The material comes from the authorâ€TMs insights and understanding of a birational transformation of the plane derived from a discrete sine-Gordon equation, posing the question of determining the behavior of the discrete dynamical system defined by the iterates of the map. The aim of this book is to give a complete treatment not only of the basic facts about QRT maps, but also the background theory on which these maps are based. Readers with a good knowledge of algebraic geometry will be interested in Kodairaâ€TMs theory of elliptic surfaces and the collection of nontrivial applications presented here. While prerequisites for some readers will demand their knowledge of quite a bit of algebraicand complex analytic geometry, different categories of readers... more Identifiers series ISSN : 1439-7382 ISBN 978-1-4419-7116-6 e-ISBN 978-1-4419-9126-3 DOI Authors Additional information Publisher Springer New York book Read online Download Add to read later Add to collection Add to followed Share Export to bibliography J.J. Duistermaat Utrecht University, Department of Mathematics, Utrecht, Netherlands Terms of service Accessibility options Report an error / abuse © 2015 Interdisciplinary Centre for Mathematical and Computational Modelling Discrete integrable systems independent: it neither relies on nor used in the proof of integrability. Section 6 is not used in the proof of integrability. It discusses more specic discrete cluster integrable systems, assuming Theorem 3.7. Proof of part i) Take a pair of matchings (M1, M2) on Γ. Let us assign to them another pair of matchings (M1, M2) on Γ. Observe that [M1] − [M2] is a 1-cycle.
151 citations
Book•
08 Feb 2010
TL;DR: In this article, the integrability in classical mechanics has been studied in the context of conformal structures and asymmetric reductions in the Lagrangian formalism and field theory, as well as the integration of ASDYM and twistor theory.
Abstract: Preface 1. Integrability in classical mechanics 2. Soliton equations and the Inverse Scattering Transform 3. The hamiltonian formalism and the zero-curvature representation 4. Lie symmetries and reductions 5. The Lagrangian formalism and field theory 6. Gauge field theory 7. Integrability of ASDYM and twistor theory 8. Symmetry reductions and the integrable chiral model 9. Gravitational instantons 10. Anti-self-dual conformal structures Appendix A: Manifolds and Topology Appendix B: Complex analysis Appendix C: Overdetermined PDEs Index
138 citations
TL;DR: In this article, the existence of semilinear differential equations with nonlocal conditions was studied using the techniques of approximate solutions and fixed point, where the nonlocal item is Lipschitz in the space of piecewise continuous functions.
133 citations
TL;DR: In this paper, the integrable nonlinear motion in accelerators has the potential to introduce a large betatron tune spread to suppress instabilities and to mitigate the effects of space charge and magnetic field errors.
Abstract: Integrable systems appeared in physics long ago at the onset of classical dynamics with examples being Kepler's and other famous problems. Unfortunately, the majority of nonlinear problems turned out to be nonintegrable. In accelerator terms, any 2D nonlinear nonintegrable mapping produces chaotic motion and a complex network of stable and unstable resonances. Nevertheless, in the proximity of an integrable system the full volume of such a chaotic network is small. Thus, the integrable nonlinear motion in accelerators has the potential to introduce a large betatron tune spread to suppress instabilities and to mitigate the effects of space charge and magnetic field errors. To create such an accelerator lattice one has to find magnetic and electric field combinations leading to a stable integrable motion. This paper presents families of lattices with one invariant where bounded motion can be easily created in large volumes of the phase space. In addition, it presents 3 families of integrable nonlinear accelerator lattices, realizable with longitudinal-coordinate-dependent magnetic or electric fields with the stable nonlinear motion, which can be solved in terms of separable variables.
TL;DR: In this article, the repulsive Lieb-Liniger model can be obtained as the nonrelativistic limit of the sinh-Gordon model: all physical quantities of the latter model ($S$-matrix, Lagrangian, and operators) can be put in correspondence with those of the former.
Abstract: The repulsive Lieb-Liniger model can be obtained as the nonrelativistic limit of the sinh-Gordon model: all physical quantities of the latter model ($S$-matrix, Lagrangian, and operators) can be put in correspondence with those of the former. We use this mapping, together with the thermodynamical Bethe ansatz equations and the exact form factors of the sinh-Gordon model, to set up a compact and general formalism for computing the expectation values of the Lieb-Liniger model both at zero and finite temperatures. The computation of one-point correlators is thoroughly detailed and when possible compared with known results in the literature.
TL;DR: The transmission probabilities on the outgoing bonds are obtained, which are inversely proportional to the bond-dependent strength of nonlinearity, with use of reflectionless propagation of Zakharov-Shabat's soliton through networks.
Abstract: We study the case in which the nonlinear Schrodinger equation (NLSE) on simple networks consisting of vertices and bonds has an infinite number of constants of motion and becomes completely integrable just as in the case of a simple one-dimensional (1D) chain. Here the strength of cubic nonlinearity is different from bond to bond, and networks are assumed to have at least two semi-infinite bonds with one of them working as an incoming bond. The connection formula at vertices obtained from norm and energy conservation rules shows (1) the solution on each bond is a part of the universal (bond-independent) soliton solution of the completely integrable NLSE on the 1D chain, but is multiplied by the inverse of square root of bond-dependent nonlinearity; (2) nonlinearities at individual bonds around each vertex must satisfy a sum rule. Under these conditions, we also showed an infinite number of constants of motion. The argument on a branched chain or a primary star graph is generalized to other graphs, i.e., general star graphs, tree graphs, loop graphs and their combinations. As a relevant issue, with use of reflectionless propagation of Zakharov-Shabat's soliton through networks we have obtained the transmission probabilities on the outgoing bonds, which are inversely proportional to the bond-dependent strength of nonlinearity. Numerical evidence is also given to verify the prediction.
TL;DR: In this paper, a general topological approach is proposed for the study of stability of periodic solutions of integrable dynamical systems with two degrees of freedom, which can be used to find non-degenerate periodic solutions, which is especially topical in those cases where no general solution (for example, by separation of variables) is known.
Abstract: In this paper a general topological approach is proposed for the study of stability of periodic solutions of integrable dynamical systems with two degrees of freedom. The methods developed are illustrated by examples of several integrable problems related to the classical Euler-Poisson equa- tions, the motion of a rigid body in a fluid, and the dynamics of gaseous expanding ellipsoids. These topological methods also enable one to find non-degenerate periodic solutions of integrable systems, which is especially topical in those cases where no general solution (for example, by separation of variables) is known. Bibliography: 82 titles.
TL;DR: In this article, the authors proposed integrable semi-discrete and fulldiscrete analogues of the short-pulse (SP) equation, where the key construction is the bilinear form and determinant structure of solutions of the SP equation, from which the multi-loop and multi-breather solutions can be generated.
Abstract: In this paper, we propose integrable semi-discrete and full-discrete analogues of the short pulse (SP) equation. The key construction is the bilinear form and determinant structure of solutions of the SP equation. We also give the determinant formulas of N-soliton solutions of the semi-discrete and full-discrete analogues of the SP equations, from which the multi-loop and multi-breather solutions can be generated. In the continuous limit, the full-discrete SP equation converges to the semi-discrete SP equation, and then to the continuous SP equation. Based on the semi-discrete SP equation, an integrable numerical scheme, i.e. a self-adaptive moving mesh scheme, is proposed and used for the numerical computation of the short pulse equation.
TL;DR: In this article, the authors study the non-equilibrium dynamics of one-dimensional Bose gas from the general perspective of the dynamics of integrable systems and develop a numerical procedure which allows explicit summation over intermediate states and analysis of the time evolution of non-local density correlation functions.
Abstract: In this paper we study the non-equilibrium dynamics of one-dimensional Bose gas from the general perspective of the dynamics of integrable systems. After outlining and critically reviewing methods based on the inverse scattering transform, intertwining operators, q-deformed objects, and extended dynamical conformal symmetry, we focus on the form-factor based approach. Motivated by possible applications in nonlinear quantum optics and experiments with ultracold atoms, we concentrate on the regime of strong repulsive interactions. We consider dynamical evolution starting from two initial states: a condensate of particles in a state with zero momentum and a condensate of particles in a Gaussian wavepacket in real space. Combining the form-factor approach with the method of intertwining operators we develop a numerical procedure which allows explicit summation over intermediate states and analysis of the time evolution of non-local density–density correlation functions. In both cases we observe a tendency toward the formation of crystal-like correlations at intermediate timescales.
TL;DR: In this paper, the main theorem of Morales, Ramis and Simo (2007) about Galoisian obstructions to meromorphic integrability of Hamiltonian systems can be naturally extended to the non-Hamiltonian case.
TL;DR: In this paper, the dressing method was used for a novel integrable generalization of the nonlinear Schrodinger equation, and explicit formulas for the N-soliton solutions were derived.
Abstract: We implement the dressing method for a novel integrable generalization of the nonlinear Schrodinger equation. As an application, explicit formulas for the N-soliton solutions are derived. As a by-product of the analysis, we find a simplification of the formulas for the N-solitons of the derivative nonlinear Schrodinger equation given by Huang and Chen.
TL;DR: In this article, a complete classification of quantum and classical superintegrable systems in E2 is presented that allow the separation of variables in polar coordinates and admit an additional integral of motion of order 3 in the momentum.
Abstract: A complete classification of quantum and classical superintegrable systems in E2 is presented that allow the separation of variables in polar coordinates and admit an additional integral of motion of order 3 in the momentum. New quantum superintegrable systems are discovered for which the potential is expressed in terms of the sixth Painleve transcendent or in terms of the Weierstrass elliptic function.
TL;DR: In this article, the modified form of Hirota's bilinear method, established by Hereman, is applied to derive multiple soliton solutions and multiple singular solutions for each model.
Abstract: In this work, integrable (2+1)-dimensional and integrable (3+1)-dimensional breaking soliton equations are examined. The modified form of Hirota's bilinear method, established by Hereman, is applied to derive multiple soliton solutions and multiple singular soliton solutions for each model. The resonance phenomenon does not exist for the two models.
TL;DR: In this article, a generalization of the coordinate Bethe ansatz that allows us to solve integrable open XXZ and ASEP models with non-diagonal boundary matrices, provided their parameters obey some relations.
Abstract: We present a generalization of the coordinate Bethe ansatz that allows us to solve integrable open XXZ and ASEP models with non-diagonal boundary matrices, provided their parameters obey some relations. These relations extend the ones already known in the literature in the context of the algebraic or functional Bethe ansatz. The eigenvectors are represented as sums over cosets of the BCn Weyl group.
TL;DR: This paper reformulated the Omega-deformation of four-dimensional gauge theory in a way that is valid away from fixed points of the associated group action and used this reformulation together with the theory of coisotropic A-branes to explain recent results linking the Omega deformation to integrable Hamiltonian systems in one direction and Liouville theory of two-dimensional conformal field theory in another direction.
Abstract: We reformulate the Omega-deformation of four-dimensional gauge theory in a way that is valid away from fixed points of the associated group action. We use this reformulation together with the theory of coisotropic A-branes to explain recent results linking the Omega-deformation to integrable Hamiltonian systems in one direction and Liouville theory of two-dimensional conformal field theory in another direction.
TL;DR: In this paper, the action-angle theorem in the context of integrable systems on Poisson manifolds was shown to be equivalent to the Caratheodory-Jacobi-Lie theorem.
Abstract: We prove the action-angle theorem in the general, and most natural, context of integrable systems on Poisson manifolds, thereby generalizing the classical proof, which is given in the context of symplectic manifolds. The topological part of the proof parallels the proof of the symplectic case, but the rest of the proof is quite different, since we are naturally led to using the calculus of polyvector fields, rather than differential forms; in particular, we use in the end a Poisson version of the classical Caratheodory-Jacobi-Lie theorem, which we also prove. At the end of the article, we generalize the action-angle theorem to the setting of non-commutative integrable systems on Poisson manifolds.
TL;DR: In this paper, the spectrum of the integrable open XXX Heisenberg spin chain subject to non-diagonal boundary magnetic fields is studied and the spectral problem for this model can be formulated in terms of functional equations obtained by separation of variables or from the fusion of transfer matrices.
Abstract: We study the spectrum of the integrable open XXX Heisenberg spin chain subject to non-diagonal boundary magnetic fields. The spectral problem for this model can be formulated in terms of functional equations obtained by separation of variables or, equivalently, from the fusion of transfer matrices. For generic boundary conditions the eigenvalues cannot be obtained from the solution of finitely many algebraic Bethe equations. Based on careful finite size studies of the analytic properties of the underlying hierarchy of transfer matrices we devise two approaches to analyze the functional equations. First we introduce a truncation method leading to Bethe type equations determining the energy spectrum of the spin chain. In a second approach the hierarchy of functional equations is mapped to an infinite system of non-linear integral equations of TBA type. The two schemes have complementary ranges of applicability and facilitate an efficient numerical analysis for a wide range of boundary parameters. Some data are presented on the finite size corrections to the energy of the state which evolves into the antiferromagnetic ground state in the limit of parallel boundary fields.
TL;DR: For the infinite family of Hamiltonians Hk with odd k ≥ 3, the first member corresponds to the three-body Calogero-Marchioro-Wolfes model after elimination of the centre-of-mass motion.
Abstract: In a recent communication paper by Tremblay et al (2009 J. Phys. A: Math. Theor. 42 205206), it has been conjectured that for any integer value of k, some novel exactly solvable and integrable quantum Hamiltonian Hk on a plane is superintegrable and that the additional integral of motion is a 2kth-order differential operator Y2k. Here we demonstrate the conjecture for the infinite family of Hamiltonians Hk with odd k ≥ 3, whose first member corresponds to the three-body Calogero–Marchioro–Wolfes model after elimination of the centre-of-mass motion. Our approach is based on the construction of some D2k-extended and invariant Hamiltonian , which can be interpreted as a modified boson oscillator Hamiltonian. The latter is then shown to possess a D2k-invariant integral of motion , from which Y2k can be obtained by projection in the D2k identity representation space.
TL;DR: In this article, the integrable lattice regularizations of the sine-Gordon model with the help of the separation of variables method of Sklyanin and the Baxter operators were studied.
Abstract: We study integrable lattice regularizations of the sine–Gordon model with the help of the separation of variables method of Sklyanin and the Baxter -operators. This leads us to the complete characterization of the spectrum (eigenvalues and eigenstates), in terms of the solutions to the Bethe ansatz equations. The completeness of the set of states that can be constructed from the solutions to the Bethe ansatz equations is proven by our approach.
TL;DR: In this article, the authors give an overview of the integrability of the Hirota-Kimura discretization method applied to algebraically completely integrable (a.c.i.) systems with quadratic vector fields.
Abstract: We give an overview of the integrability of the Hirota-Kimura discretization method applied to algebraically completely integrable (a.c.i.) systems with quadratic vector fields. Along with the description of the basic mechanism of integrability (Hirota-Kimura bases), we provide the reader with a fairly complete list of the currently available results for concrete a.c.i. systems.
TL;DR: In this article, a general procedure to represent the integral Bethe Ansatz equations in the form of the Reimann-Hilbert problem is given, which allows us to study in simple way integrable spin chains in the thermodynamic limit.
Abstract: In this thesis a general procedure to represent the integral Bethe Ansatz equations in the form of the Reimann-Hilbert problem is given. This allows us to study in simple way integrable spin chains in the thermodynamic limit. Based on the functional equations we give the procedure that allows finding the subleading orders in the solution of various integral equations solved to the leading order by the Wiener-Hopf technics. The integral equations are studied in the context of the AdS/CFT correspondence, where their solution allows verification of the integrability conjecture up to two loops of the strong coupling expansion. In the context of the two-dimensional sigma models we analyze the large-order behavior of the asymptotic perturbative expansion. Obtained experience with the functional representation of the integral equations allowed us also to solve explicitly the crossing equations that appear in the AdS/CFT spectral problem.
TL;DR: In this article, a new integrable model which describes the motion of fluid is discussed, and the precise blow-up rate is investigated by analyzing the existence of global solutions.