DOCSLIB.ORG

Random Matrix Theory, Quantum Chaos, and Eigenstate Thermalization

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Random Matrix Theory, Quantum Chaos, and Eigenstate Thermalization Random Matrix Theory, Quantum Chaos, and Eigenstate Thermalization Hiroki Nakayama Department of Applied Physics, Stanford University, Stanford, CA 94305 (Dated: July 2, 2020) Submitted as coursework for PH470, Stanford University, Spring 2020 In this paper, the main results from Random Matrix Theory (RMT), quantum chaos and Eigen- state Thermalization Hypothesis (ETH) will be surveyed. The theory behind quantum chaos and ETH is deeply rooted in RMT. It will be found that RMT can be used as an indicator for quantum chaos by analyzing the distribution of energy level spacing in quantum systems. ETH is a general- ization of RMT in that one can reproduce RMT from ETH in a narrow energy window. The paper explores the connections in RMT, quantum chaos, and ETH. c (Hiroki Nakayama). The author warrants that the work is the author's own and that Stanford University provided no input other than typesetting and referencing guidelines. The author grants permission to copy, distribute, and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All of the rights, including commercial rights, are reserved to the author. I. INTRODUCTION focusing on their wave functions, one should instead fo- cus on the observables. In his theorem, the quantum A classical system in which its motions are governed by ergodic theorem, he stated that observable will relax to 9 non-linear equations can exhibit chaotic motions. Chaos a microcanonical distribution at later times. This relax- in the classical sense means that the the system has an ation of observables to the microcanonical distribution exquisite sensitivity to small perturbations in its phase- is called thermalization, a concept deeply connected to space trajectories. A famous example is the ”Butterfly RMT. Since RMT was limited in describing physical ob- Effect”, illustrating the point that any small difference in servables in real systems, in the 1990's Srednicki gen- the initial condition can cause chaotic behavior. eralized its notion by making an ansatz, known as the 1,4{6 On the other hand, this notion of classical chaos does Eigenstate Thermalization Hypothesis (ETH). not apply directly to a quantum mechanical system, in Following the discussions given by D'Alessio et.al.1, which its dynamics is governed by the linear Schr¨odinger this paper will survey some of the key concepts in RMT, equation. As in classical chaotic systems, one can use quantum chaos and ETH. The goal of this paper is to phase-space methods to describe quantum mechanical illustrate how these concepts relate to one another. systems; however, the notion of trajectory is meaning- less since the uncertainty principle forbids simultaneous measurement of coordinates and momenta. In such case, one may ask what chaos is, in the quantum mechanical setting. II. CLASSICAL CHAOS Although, to this day, there is no precise definition of quantum chaos, it is referred to as chaos in the quan- A. Phase space and Liouville's Theorem tum mechanical setting. The basis for the understanding of quantum chaos has been formulated by Wigner and Dyson to understand the spectra of complex atomic nu- Suppose we have a system of N point particle systems clei, now called Random Matrix Theory (RMT).1 The in which the position qi and momenta pi of each par- idea of this theory is that rather than finding the exact ticle is known, where i indicates the i-th particle. For eigenspectra of chaotic Hamiltonians, one should instead instance, let there be N particles. In this case, if the probe their statistics. position and momentum for each particle have 3 compo- Following this idea, Bohigas, Giannoni, and Schmit nents, then the subscript i will run from 1 to 3N. A pair has formulated the BGS conjecture stating that quantum of position and momentum (qi; pi) is referred to as the systems with a classical chaotic analog are described by degree of freedom. In the example above, the degree of RMT.3 If the energy level spacing of arbitary quantum freedom is then, 3N. 3N of these pairs are needed to systems follow this he Wigner-Dyson distribution, then describe a dynamical system. it can be said that the system is a quantum chaotic sys- Consider a conserved system, or in other words, a sys- tem. RMT acts as a tool to indicate whether a quantum tem without any dissipation. A Hamiltonian of such sys- system is chaotic or not. tem is described as a function of qi and pi that completely RMT also has deep connections with thermalization describes the system. The dynamics of the system can be of quantum mechanical systems. von Neumann's insight expressed by considering the time evolution of the vari- for quantum mechanical systems was that rather than ables, qi and pi, namely, qi(t) and pi(t). In such case, the 2 following Hamilton's equations,2 conditions inside a volume in phase space will stay con- dρ 2 stant. This allows us to make the notion that dt = 0. @H(q; p) dq 2 = i (1) Applying this to (5), we get the following, @pi dt @ρ X @ρ @H @ρ @H + − = 0: (6) @t @q @p @p @q @H(q; p) dp i i i i i = − i (2) @qi dt This is called the Liouville's theorem. The terms inside the brackets describe how the density ρ varies with re- where i = 1;:::; 3N, if as in the above example, each spect to q and p. On the other hand, @ρ describes the dy- particle has 3 components for their momenta and posi- @t namics of ρ at a fixed point defined in phase space. What tion. It is worth noting that the Hamilton's equations this theorem implies is that ρ does not evolve chaotically are coupled equations for each degree of freedom. because it is linear in ρ. In other words, if the system The Hamiltonian for the conserved system is depen- starts in a different initial condition, then the probability dent on momenta p, and position q. Let the system distribution of the initial conditions will not evolve ex- not be subject to time dependent forces. The total time ponentially compared to the original distribution. This derivative of the system can be found by the following,2 notion can be extended not only for non-chaotic systems, but also for chaotic systems in which the trajectories are dH(q; p) X @H dpi @H dqi @H = + + : (3) 2 dt @p dt @q dt @t diverging exponentially. i i i Due to the assumption that the system is not subject to @H B. Integrable and nonintegrable systems time dependent forces, @t = 0. Now, using the coupled differential equations for the position and momenta,2 Consider a conserved system with its trajectory in @H @H @H @H phase space start at initial condition (q0; p0). Since the − = 0: (4) system is conserved, the energy of the system will have @qi @pi @pi @qi the same value at later times, which leads us to write If (4) holds true for each i in (3), it can be concluded that the Hamiltonian as H(q(t); p(t)) = H(q0; p0). Here, dH(q;p) (q(t); p(t)) is the trajectory in phase space. Recall Hamil- dt = 0 and H itself is a conserved quantity of the system. If H is the total energy, what ever the motion ton's equation considering momenta in (2). Consider a case where @H = − dpi = 0. From this, we can say of the trajectories are in phase space (discussed in this @qi dt section later), H is constant. that when H does not have dependence on qi, there is Next, the notion of Phase space of a conserved system no time dependence on pi. Therefore, in this system, will be mentioned. Phase space is a space created by the trajectories are constrained by the constant energy position p and momenta q. Let a system under consider- value H(q0; p0) and pi values. If the system has N de- ation have N degrees of freedom, then its dimensionality grees of freedom, and a total of M conserved quantities, is 2N since each degree of freedom is represented by the then the trajectories will be on a 2N − M dimensional pair q and p. In such conserved systems H(q; p) = E, surface. There are two types of conserved systems: inte- where E is a constant energy value defined by the initial grable and nonintegrable systems. A system is said to be conditions of q and p. The trajectory that the system integrable if the number of conserved quantities is equal will follow in phase space will be confined to this 2N − 1 or greater than the number of degrees of freedom. On the dimensional constant energy surface.2 other hand, a system is considered nonintegrable if the Notice that there are multiple possibilities of initial number of conserved quantities is fewer than the number conditions that the system can take. Therefore, we can of degrees of freedom. In an nonintegrable system there define a distribution of initial conditions in phase space. are less constraints on the trajectories, therefore mak- Let the probability density function of the distribution ing them able able to travel through phase space more be represented as ρ(q; p). In this case, this function is freely. In such system, chaotic motions are possible. In defined so that the probability P (q; p) of finding a tra- an integrable system, on the other hand, the motions of jectory in an infinitesimal volume dV in phase space is the system are constrained by the conserved quantities; defined by P (q; p) = ρ(q; p)dV .
Load more

Recommended publications
  • Aspects of Integrable Models
    Aspects of Integrable Models Undergraduate Thesis Submitted in partial fulfillment of the requirements of BITS F421T Thesis By Pranav Diwakar ID No. 2014B5TS0711P Under the supervision of: Dr. Sujay Ashok & Dr. Rishikesh Vaidya BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE PILANI, PILANI CAMPUS January 2018 Declaration of Authorship I, Pranav Diwakar, declare that this Undergraduate Thesis titled, `Aspects of Integrable Models' and the work presented in it are my own. I confirm that: This work was done wholly or mainly while in candidature for a research degree at this University. Where any part of this thesis has previously been submitted for a degree or any other qualification at this University or any other institution, this has been clearly stated. Where I have consulted the published work of others, this is always clearly attributed. Where I have quoted from the work of others, the source is always given. With the exception of such quotations, this thesis is entirely my own work. I have acknowledged all main sources of help. Signed: Date: i Certificate This is to certify that the thesis entitled \Aspects of Integrable Models" and submitted by Pranav Diwakar, ID No. 2014B5TS0711P, in partial fulfillment of the requirements of BITS F421T Thesis embodies the work done by him under my supervision. Supervisor Co-Supervisor Dr. Sujay Ashok Dr. Rishikesh Vaidya Associate Professor, Assistant Professor, The Institute of Mathematical Sciences BITS Pilani, Pilani Campus Date: Date: ii BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE PILANI, PILANI CAMPUS Abstract Master of Science (Hons.) Aspects of Integrable Models by Pranav Diwakar The objective of this thesis is to study the isotropic XXX-1/2 spin chain model using the Algebraic Bethe Ansatz.
    [Show full text]
  • A New Two-Component Integrable System with Peakon Solutions
    A new two-component integrable system with peakon solutions 1 2 Baoqiang Xia ∗, Zhijun Qiao † 1School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, Jiangsu 221116, P. R. China 2Department of Mathematics, University of Texas-Pan American, Edinburg, Texas 78541, USA Abstract A new two-component system with cubic nonlinearity and linear dispersion: 1 1 mt = bux + 2 [m(uv uxvx)]x 2 m(uvx uxv), 1 − −1 − nt = bvx + [n(uv uxvx)]x + n(uvx uxv), 2 − 2 − m = u uxx, n = v vxx, − − where b is an arbitrary real constant, is proposed in this paper. This system is shown integrable with its Lax pair, bi-Hamiltonian structure, and infinitely many conservation laws. Geometrically, this system describes a nontrivial one-parameter family of pseudo-spherical surfaces. In the case b = 0, the peaked soliton (peakon) and multi-peakon solutions to this two-component system are derived. In particular, the two-peakon dynamical system is explicitly solved and their interactions are investigated in details. Moreover, a new integrable cubic nonlinear equation with linear dispersion 1 2 2 1 ∗ ∗ arXiv:1211.5727v4 [nlin.SI] 10 May 2015 mt = bux + [m( u ux )]x m(uux uxu ), m = u uxx, 2 | | − | | − 2 − − is obtained by imposing the complex conjugate reduction v = u∗ to the two-component system. The complex valued N-peakon solution and kink wave solution to this complex equation are also derived. Keywords: Integrable system, Lax pair, Peakon. Mathematics Subject Classifications (2010): 37K10, 35Q51. ∗E-mail address: [email protected] †Corresponding author, E-mail address: [email protected] 1 2 1 Introduction In recent years, the Camassa-Holm (CH) equation [1] mt bux + 2mux + mxu = 0, m = u uxx, (1.1) − − where b is an arbitrary constant, derived by Camassa and Holm [1] as a shallow water wave model, has attracted much attention in the theory of soliton and integrable system.
    [Show full text]
  • Remarks on the Complete Integrability of Quantum and Classical Dynamical
    Remarks on the complete integrability of quantum and classical dynamical systems Igor V. Volovich Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia [email protected] Abstract It is noted that the Schr¨odinger equation with any self-adjoint Hamiltonian is unitary equivalent to a set of non-interacting classical harmonic oscillators and in this sense any quantum dynamics is completely integrable. Higher order integrals of motion are presented. That does not mean that we can explicitly compute the time dependence for expectation value of any quantum observable. A similar re- sult is indicated for classical dynamical systems in terms of Koopman’s approach. Explicit transformations of quantum and classical dynamics to the free evolution by using direct methods of scattering theory and wave operators are considered. Examples from classical and quantum mechanics, and also from nonlinear partial differential equations and quantum field theory are discussed. Higher order inte- grals of motion for the multi-dimensional nonlinear Klein-Gordon and Schr¨odinger equations are mentioned. arXiv:1911.01335v1 [math-ph] 4 Nov 2019 1 1 Introduction The problem of integration of Hamiltonian systems and complete integrability has already been discussed in works of Euler, Bernoulli, Lagrange, and Kovalevskaya on the motion of a rigid body in mechanics. Liouville’s theorem states that if a Hamiltonian system with n degrees of freedom has n independent integrals in involution, then it can be integrated in quadratures, see [1]. Such a system is called completely Liouville integrable; for it there is a canonical change of variables that reduces the Hamilton equations to the action-angle form.
    [Show full text]
  • "Integrable Systems, Random Matrices and Random Processes"
    Contents Integrable Systems, Random Matrices and Random Processes Mark Adler ..................................................... 1 1 Matrix Integrals and Solitons . 4 1.1 Random matrix ensembles . 4 1.2 Large n–limits . 7 1.3 KP hierarchy . 9 1.4 Vertex operators, soliton formulas and Fredholm determinants . 11 1.5 Virasoro relations satisfied by the Fredholm determinant . 14 1.6 Differential equations for the probability in scaling limits . 16 2 Recursion Relations for Unitary Integrals . 21 2.1 Results concerning unitary integrals . 21 2.2 Examples from combinatorics . 25 2.3 Bi-orthogonal polynomials on the circle and the Toeplitz lattice 28 2.4 Virasoro constraints and difference relations . 30 2.5 Singularity confinement of recursion relations . 33 3 Coupled Random Matrices and the 2–Toda Lattice . 37 3.1 Main results for coupled random matrices . 37 3.2 Link with the 2–Toda hierarchy . 39 3.3 L U decomposition of the moment matrix, bi-orthogonal polynomials and 2–Toda wave operators . 41 3.4 Bilinear identities and τ-function PDE’s . 44 3.5 Virasoro constraints for the τ-functions . 47 3.6 Consequences of the Virasoro relations . 49 3.7 Final equations . 51 4 Dyson Brownian Motion and the Airy Process . 53 4.1 Processes . 53 4.2 PDE’s and asymptotics for the processes . 59 4.3 Proof of the results . 62 5 The Pearcey Distribution . 70 5.1 GUE with an external source and Brownian motion . 70 5.2 MOPS and a Riemann–Hilbert problem . 73 VI Contents 5.3 Results concerning universal behavior . 75 5.4 3-KP deformation of the random matrix problem .
    [Show full text]
  • On the Non-Integrability of the Popowicz Peakon System
    DISCRETE AND CONTINUOUS Website: www.aimSciences.org DYNAMICAL SYSTEMS Supplement 2009 pp. 359–366 ON THE NON-INTEGRABILITY OF THE POPOWICZ PEAKON SYSTEM Andrew N.W. Hone Institute of Mathematics, Statistics & Actuarial Science University of Kent, Canterbury CT2 7NF, UK Michael V. Irle Institute of Mathematics, Statistics & Actuarial Science University of Kent, Canterbury CT2 7NF, UK Abstract. We consider a coupled system of Hamiltonian partial differential equations introduced by Popowicz, which has the appearance of a two-field coupling between the Camassa-Holm and Degasperis-Procesi equations. The latter equations are both known to be integrable, and admit peaked soliton (peakon) solutions with discontinuous derivatives at the peaks. A combination of a reciprocal transformation with Painlev´eanalysis provides strong evidence that the Popowicz system is non-integrable. Nevertheless, we are able to con- struct exact travelling wave solutions in terms of an elliptic integral, together with a degenerate travelling wave corresponding to a single peakon. We also describe the dynamics of N-peakon solutions, which is given in terms of an Hamiltonian system on a phase space of dimension 3N. 1. Introduction. The members of a one-parameter family of partial differential equations, namely mt + umx + buxm =0, m = u − uxx (1) with parameter b, have been studied recently. The case b = 2 is the Camassa-Holm equation [1], while b = 3 is the Degasperis-Procesi equation [3], and it is known that (with the possible exception of b = 0) these are the only integrable cases [14], while all of these equations (apart from b = −1) arise as a shallow water approximation to the Euler equations [6].
    [Show full text]
  • What Are Completely Integrable Hamilton Systems
    THE TEACHING OF MATHEMATICS 2011, Vol. XIII, 1, pp. 1–14 WHAT ARE COMPLETELY INTEGRABLE HAMILTON SYSTEMS BoˇzidarJovanovi´c Abstract. This paper is aimed to undergraduate students of mathematics and mechanics to draw their attention to a modern and exciting field of mathematics with applications to mechanics and astronomy. We cared to keep our exposition not to go beyond the supposed knowledge of these students. ZDM Subject Classification: M55; AMS Subject Classification: 97M50. Key words and phrases: Hamilton system; completely integrable system. 1. Introduction Solving concrete mechanical and astronomical problems was one of the main mathematical tasks until the beginning of XX century (among others, see the work of Euler, Lagrange, Hamilton, Abel, Jacobi, Kovalevskaya, Chaplygin, Poincare). The majority of problems are unsolvable. Therefore, finding of solvable systems and their analysis is of a great importance. In the second half of XX century, there was a breakthrough in the research that gave a basis of a modern theory of integrable systems. Many great mathemati- cians such as P. Laks, S. Novikov, V. Arnold, J. Mozer, B. Dubrovin, V. Kozlov contributed to the development of the theory, which connects the beauty of classi- cal mechanics and differential equations with algebraic, symplectic and differential geometry, theory of Lie groups and algebras (e.g., see [1–4] and references therein). The aim of this article is to present the basic concepts of the theory of integrable systems to readers with a minimal prior knowledge in the graduate mathematics, so we shall not use a notion of a manifold, symplectic structure, etc. The most of mechanical and physical systems are modelled by Hamiltonian equations.
    [Show full text]
  • Introduction to Integrable Systems 33Rd SERB Theoretical High Energy Physics Main School, Khalsa College, Delhi Univ
    Introduction to Integrable Systems 33rd SERB Theoretical High Energy Physics Main School, Khalsa College, Delhi Univ. December 17-26, 2019 (updated Feb 26, 2020) Govind S. Krishnaswami, Chennai Mathematical Institute Comments and corrections may be sent to [email protected] https://www.cmi.ac.in/˜govind/teaching/integ-serb-thep-del-19/ Contents 1 Some reference books/review articles 2 2 Introduction 3 2.1 Integrability in Hamiltonian mechanics.............................................3 2.2 Liouville-Arnold Theorem...................................................6 2.3 Remarks on commuting flows.................................................9 2.4 Integrability in field theory...................................................9 2.5 Some features of integrable classical field theories.......................................9 2.6 Integrability of quantum systems................................................ 10 2.7 Examples of integrable systems................................................ 11 2.8 Integrable systems in the wider physical context........................................ 12 3 Lax pairs and conserved quantities 12 3.1 Lax pair for the harmonic oscillator.............................................. 12 3.2 Isospectral evolution of the Lax matrix............................................. 13 4 KdV equation and its integrability 15 4.1 Physical background and discovery of inverse scattereing................................... 15 4.2 Similarity and Traveling wave solutions............................................ 17 4.3
    [Show full text]
  • Integrable Systems, Symmetries and Quantization
    Integrable systems, symmetries, and quantization Daniele Sepe∗ Vu˜ Ngo. c Sany October 18, 2017 Keywords : classical and quantum integrable systems, moment(um) maps, spectral theory, quantization. MS Classification: 37J35, 53D05, 70H06, 81Q20. Abstract These notes are an expanded version of a mini-course given at the Poisson 2016 conference in Geneva. Starting from classical integrable systems in the sense of Liouville, we explore the notion of non-degenerate singularities and expose recent research in connection with semi-toric systems. The quantum and semiclassical counterpart are also pre- sented, in the viewpoint of the inverse question: from the quantum mechanical spectrum, can one recover the classical system? 1 Foreword These notes, after a general introduction, are split into four parts: • Integrable systems and action-angle coordinates (Section3), where the basic notions about Liouville integrable systems are recalled. • Almost-toric singular fibers (Section4), where emphasis is laid on a Morse- arXiv:1704.06686v2 [math.SG] 16 Oct 2017 like theory of integrable systems. • Semi-toric systems (Section5), where we introduce the recent semi-toric systems and their classification. ∗Universidade Federal Fluminense, Instituto de Matemática, Campus do Valonguinho CEP 24020-140, Niterói (Brazil), email: [email protected] yIRMAR (UMR 6625), Université de Rennes 1, Campus de Beaulieu, 35042 Rennes cedex (France), email: [email protected] 1 • Quantum systems and the inverse problem (Section6), where the geomet- ric study is applied to the world of quantum mechanics. The main objective is to serve as an introduction to recent research in the field of classical and quantum integrable systems, in particular in the relatively new and expanding theory of semi-toric systems, in which the au- thors have taken an active part in the last ten years (cf.[78, 62, 63, 35] and references therein).
    [Show full text]
  • INTEGRABLE SYSTEMS and GROUP ACTIONS Contents 1
    INTEGRABLE SYSTEMS AND GROUP ACTIONS EVA MIRANDA Abstract. The main purpose of this paper is to present in a uni¯ed approach to see di®erent results concerning group actions and integrable systems in symplectic, Poisson and contact manifolds. Rigidity problems for integrable systems in these manifolds will be explored from this perspective. Contents 1. Introduction 2 2. The Symplectic case 4 2.1. Liouville-Mineur-Arnold Theorem: Torus actions meet integrable systems 4 2.2. Rigidity 7 2.3. Singular counterparts to Arnold-Liouville 8 2.4. Additional symmetries 14 3. The Contact case 16 3.1. Toric and non-toric contact manifolds and integrability 18 3.2. Additional symmetries 25 4. The Poisson case 27 4.1. Examples 27 4.2. The local structure of a Poisson manifold 29 4.3. Integrable systems, normal forms and action-angle coordinates for Poisson manifolds 30 4.4. Equivariant theorems for Poisson manifolds 32 References 33 Date: November 15, 2011. 1991 Mathematics Subject Classi¯cation. 53D17. Key words and phrases. integrable system, momentum map, Poisson manifold, Contact man- ifold, symplectic manifold, group action. Partially supported by the DGICYT/FEDER project MTM2009-07594: Estructuras Geomet- ricas: Deformaciones, Singularidades y Geometria Integral. 1 2 EVA MIRANDA 1. Introduction From the very beginning: symmetries, group actions and integrable systems have been close allies. The study of symmetries of the di®erential systems given by Hamilton's equation associated to an energy function H, led naturally to the existence of Poisson commuting functions and the method of integration by quadra- tures. When the number of commuting functions is maximal, the resolution by quadratures is possible and the set of commuting functions de¯nes an integrable system on the phase space T ¤(Rn).
    [Show full text]
  • Integrable Systems (1834-1984)
    Integrable Systems (1834-1984) Da-jun Zhang, Shanghai University Da-jun Zhang • Integrable Systems • Discrete Integrable Systems (DIS) • Collaborators: Frank Nijhoff (Leeds), Jarmo Hietarinta (Turku) R. Quispel, P. van der Kamp (Melbourne) 6 lectures • A brief review of history of integrable systems and soliton theory (2hrs) • Lax pairs of Integrable systems (2hrs) • Integrability: Bilinear Approach (6hrs=2hr × 3) 2hrs for 3-soliton condition bilinear integralility 2hrs for Bäcklund transformations and vertex operators 2hrs for Wronskian technique • Discrete Integrable Systems: Cauchy matrix approach (2hrs) Integrable Systems • How does one determine if a system is integrable and how do you integrate it? …. categorically, that I believe there is no systematic answer to this question. Showing a system is integrable is always a matter of luck and intuition. • Viewpoint of Percy Deift on Integrable Systems Linearisable-resolve-clear dependence of parameters trigonometric, special, Pinleve functions • “Fifty Years of KdV: An Integrable System” Interaction of Integrable Methods • dynamical systems • probability theory and statistics • geometry • combinatorics • statistical mechanics • classical analysis • numerical analysis • representation theory • algebraic geometry • …… Solitons John Scott Russell (9 May 1808-8 June 1882) Education: Edinburgh, St. Andrews, Glasgow • August, 1834 • z Russell’s observation • A large solitary elevation, a rounded, smooth and well defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed … Its height gradually diminished, and after a chase of one or two miles I lost it in the windings of the channel. Such, in the month of August 1834, was my first chance interview with that singular and beautiful phenomenon.
    [Show full text]
  • Primitive Solutions of the Korteweg–De Vries Equation
    Theoretical and Mathematical Physics, 202(3): 334–343 (2020) PRIMITIVE SOLUTIONS OF THE KORTEWEG–DE VRIES EQUATION S. A. Dyachenko,∗ P. Nabelek, † D. V. Zakharov,‡ andV.E.Zakharov§ We survey recent results connected with constructing a new family of solutions of the Korteweg–de Vries equation, which we call primitive solutions. These solutions are constructed as limits of rapidly vanishing solutions of the Korteweg–de Vries equation as the number of solitons tends to infinity. A primitive solution is determined nonuniquely by a pair of positive functions on an interval on the imaginary axis and a function on the real axis determining the reflection coefficient. We show that elliptic one-gap solutions and, more generally, periodic finite-gap solutions are special cases of reflectionless primitive solutions. Keywords: integrable system, Korteweg–de Vries equation, primitive solution DOI: 10.1134/S0040577920030058 1. Introduction The Korteweg–de Vries (KdV) equation ut(x, t)=6u(x, t)ux(x, t) uxxx(x, t)(1) − plays a fundamental role in the modern theory of integrable systems and is the prototypical example of an infinite-dimensional integrable system. The KdV equation is the first equation of an infinite sequence of commuting equations called the KdV hierarchy. The auxiliary linear operator for the KdV hierarchy is the one-dimensional Schr¨odinger operator on the real axis ψ′′ + u(x)ψ = Eψ, <x< . (2) − −∞ ∞ ∗Department of Mathematics, University of Washington, Seattle, Washington, USA, e-mail: urrfi[email protected]. †Department of Mathematics, Oregon State University, Corvallis, Oregon, USA, e-mail: [email protected]. ‡Department of Mathematics, Central Michigan University, Mount Pleasant, Michigan, USA, e-mail: [email protected].
    [Show full text]
  • Integrable Couplings of the Kaup-Newell Soliton Hierarchy Mengshu Zhang University of South Florida, [email protected]
    University of South Florida Scholar Commons Graduate Theses and Dissertations Graduate School January 2012 Integrable Couplings of the Kaup-Newell Soliton Hierarchy Mengshu Zhang University of South Florida, [email protected] Follow this and additional works at: http://scholarcommons.usf.edu/etd Part of the American Studies Commons, and the Mathematics Commons Scholar Commons Citation Zhang, Mengshu, "Integrable Couplings of the Kaup-Newell Soliton Hierarchy" (2012). Graduate Theses and Dissertations. http://scholarcommons.usf.edu/etd/4267 This Thesis is brought to you for free and open access by the Graduate School at Scholar Commons. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Scholar Commons. For more information, please contact [email protected]. Integrable Couplings of the Kaup-Newell Soliton Hierarchy by Mengshu Zhang A thesis submitted in partial fulfillment of the requirements for the degree of Master of Arts Department of Mathematics & Statistics College of Arts and Sciences University of South Florida Major Professor: Wen-Xiu Ma, Ph.D. Arthur Danielyan, Ph.D. Sherwin Kouchekian, Ph.D. Date of Approval: June 08, 2012 Keywords: Lax pairs, Zero curvature representation, Enlarged systems, Recursion operator, Infinitely many symmetries Copyright © 2012, Mengshu Zhang Acknowledgments First and foremost, I would like to offer my sincerest gratitude to my advisor, Dr. Wen-Xiu Ma, who has helped me enormously during my study these two years, and has given much guidance with his patience and knowledge throughout my thesis writing, I would never have been able to complete my thesis without his help. And many thanks to my committee members: Dr.
    [Show full text]