OF THE PEAKON DYNAMICS
|
|
- Stephanie Turner
- 5 years ago
- Views:
Transcription
1 Pacific Journal of Applied Mathematics Volume 1 umber 4 pp ISS c 008 ova Science Publishers Inc. A OTE O r-matrix OF THE PEAKO DYAMICS Zhijun Qiao Department of Mathematics The University of Texas Pan-American 101 West University Drive Edinburg TX USA Abstract This paper deals with the r-matrix of the peakon dynamical systems. Our result shows that there does not exist constant r-matrix for the peakon dynamical system. Keywords: r-matrix Structure Peakon Systems. AMS Subject: 35Q53; 58F07; 35Q35 PACS: Gc; 03.40Kf; g In 1993 Camassa and Holm proposed a shallow water equation and discussed the peaked-soliton (peakon solution of the equation [1]. Later in 1996 Ragnisco and Bruschi [] showed the integrability of the finite-dimensional peakon system through constructing a constant r-matrix. Their starting point is the following Lax matrix (1. The r-matrix is usually dynamical in the framework of the r-matrix approach located in the fundamental Poisson bracket [3]. Ragnisco and Bruschi claimed that for a particular choice of the relevant parameters in the Hamiltonian (the one corresponding to the pure peakons case the r-matrix becomes essentially constant []. In ref. [4] Qiao extended the Camassa-Holm (CH equation to the whole integrable CH hierarchy including positive and negative members in the hierarchy and studied r-matrix structures of the constrained CH systems and algebraic-geometric solutions on a symplectic submanifold through using the constraint approach [5]. In this note what we want to show is no constant r-matrix for the CH peakon system. Let us discuss below. For the peakon system let us consider the Lax matrix which is given in ref. []: where L L i j E i j (1 i j1 address: qiao@utpa.edu L i j p i p j A i j ( A i j A(q i q j e 1 q i q j. (3
2 6 Zhijun Qiao In Eq. (3 A(x e 1 x (4 and A(x has the following properties: A (x 1 sgn(xa(x A i j A ji A ii 1 A i j A (q i q j A (q j q i A ji A ii 0 ( x + y A(xA(y A (xa(y + A(xA (y ( x + y A(xA(y y x 0. We work with the matrix basis E i j : 1 A(xA(y[sgn(x + sgn(y] (E i j kl δ ik δ jl i jkl 1... To have the r-matrix structure we consider the so-called fundamental Poisson bracket [3]: where {L 1 L } [r 1 L 1 ] [r 1 L ] (5 L 1 L 1 L 1 L r 1 r 1 {L 1 L } i j1 kl1 L i j E i j 1 L kl 1 E kl r lk E lk (E lk + E kl lk1 r lk (E lk + E kl E lk lk1 {L i j L kl }E i j E kl. i jkl1 Here {L i j L kl } is of sense under the standard Poisson bracket of two functions 1 is the unit matrix and r lk are to be determined. In Eq. (5 [ ] means the usual commutator of matrix.
3 A ote on r-matrix of the Peakon Dynamics 63 ow let us calculate the left hand side of Eq. (5. L i j q m p i p j A i j(δ im δ jm L kl p m A kl p k p l (p l δ km + p k δ lm ( Li j L {L i j L kl } kl L kl L i j m1 q m p m q m p m 1 [ pi p j A A kl i j (δ im δ jm (p l δ km + p k δ lm m1 pk p l ] p k p l A A i j kl (δ km δ lm (p j δ im + p i δ jm pi p j 1 ( pi p j p l δ ik A i j p A pk kl k 1 ( p j pi p l δ jk A i j p A kl + k i pk j where the supscript means A (x with the argument. Thus we obtain {L 1 L } 1 {L i j L kl }E i j E kl i jkl1 jkl1 + 1 ( pi p j p k δ il A i j p A pl kl + l i pl j 1 ( p j pi p k δ jl A i j p A kl l [ pk p j A jl (E jl E kl E kl E jl + p k p j A lka l j (E lk E l j E l j E lk + ] p k p j (A lk A jl (E lk E jl E jl E lk. ext we compute the right hand side of Eq. (5. Before doing that let us give some simple tensor product of the matrix basis E i j : So we have (E i j E st (E kl 1 δ jk E il E st (E kl 1(E i j E st δ il E k j E st (E i j E st (1 E kl δ tk E i j E sl (1 E kl (E i j E st δ ls E i j E kt E kl E st δ ls E kt. [r 1 L 1 ] [r 1 L ] r lk L i j ([E lk (E lk + E kl E i j 1] [(E lk + E kl E lk 1 E i j ] i jkl1
4 64 Zhijun Qiao r lk L i j (δ ik E l j (E lk + E kl δ ik (E lk + E kl E l j i jkl1 +δ jl (E lk + E kl E ik δ jl E ik (E lk + E kl ( r lk L k j E l j (E lk + E kl (E lk + E kl E l j jkl1 + jkl1 r kl L jk ( E jl (E lk + E kl + (E lk + E kl E jl r lk L jk ((E l j + E jl (E lk + E kl (E lk + E kl (E l j + E jl jkl1 where we set r lk r kl and used L jk L k j. After comparing both sides of the fundamental Poisson bracket (5 we should have the following equalities: r lk 1 A jl A jk (6 r l j r lk 1 (A lk A jl. A jk (7 In fact the nd one is a natural result derived from the 1st one. Thus for the CH peakons case we have r lk 1 A jl A jk 1 4 sgn(q k q l A kla jl A jk 1 4 sgn(q l q k e 1 ( q l q k + q j q l q j q k j Z +. (8 This equality holds for arbitrary j Z +. Obviously only in the cases of j > l > k or j < l < k Eq. (8 becomes constant namely ± 1 4. But for other j apparently Eq. (8 is OT constant. So we think that the constant matrix given in ref. [] r 1 a lk1 sgn(q l q k E lk (E lk + E kl a constant is not an r-matrix for the CH peakon dynamical system. Discussions for more general case of Lax matrix are seen in ref. [6]. Acknowledgments This work has been supported by the UTPA-FRC.
5 References A ote on r-matrix of the Peakon Dynamics 65 [1] R. Camassa and D. D. Holm An integrable shallow water equation with peaked solitons Phys. Rev. Lett. 71( [] O. Ragnisco and M. Bruschi Peakons r-matrix and Toda Lattice Physica A 8( [3] L. D. Faddeev and L. A. Takhtajan Hamiltonian Methods in the Theory of Solitons (Springer-Verlag Berlin [4] Zhijun Qiao The Camassa-Holm hierarchy -dimensional integrable systems and algebro-geometric solution on a symplectic submanifold Comm. Math. Phys [5] Zhijun Qiao Generalized r-matrix structure and algebro-geometric solutions for integrable systems Rev. Math. Phys. 13( [6] Zhijun Qiao r-matrix structure for general even functions in finite-dimensional Hamiltonian systems preprint 008.
arxiv:solv-int/ v1 28 Sep 1995
PEAKONS, R-MATRIX AND TODA LATTICE O.Ragnisco (a),m.bruschi (b) Istituto Nazionale di Fisica Nucleare, Sezione di Roma, P.le A.Moro 2, 00185 Roma,Italia (a) Dipartimento di Fisica E.Amaldi, III Università
More informationA Note on Poisson Symmetric Spaces
1 1 A Note on Poisson Symmetric Spaces Rui L. Fernandes Abstract We introduce the notion of a Poisson symmetric space and the associated infinitesimal object, a symmetric Lie bialgebra. They generalize
More informationA time-discretized version of the Calogero-Moser model
A time-discretized version of the Calogero-Moser model arxiv:hep-th/9403052v 8 Mar 994 Frank W. Nijhoff and Gen-Di Pang FB7 Mathematik-Informatik, Universität-Gesamthochschule Paderborn, D-33098 Paderborn,
More informationJournal of Geometry and Physics
Journal of Geometry and Physics 07 (206) 35 44 Contents lists available at ScienceDirect Journal of Geometry and Physics journal homepage: www.elsevier.com/locate/jgp Multi-component generalization of
More informationarxiv:nlin/ v1 [nlin.si] 25 Sep 2006
Remarks on the conserved densities of the Camassa-Holm equation Amitava Choudhuri 1, B. Talukdar 1a and S. Ghosh 1 Department of Physics, Visva-Bharati University, Santiniketan 73135, India Patha Bhavana,
More informationarxiv:math-ph/ v1 25 Feb 2002
FROM THE TODA LATTICE TO THE VOLTERRA LATTICE AND BACK arxiv:math-ph/0202037v1 25 Feb 2002 (1) PANTELIS A DAMIANOU AND RUI LOJA FERNANDES Abstract We discuss the relationship between the multiple Hamiltonian
More informationLax Representations and Zero Curvature Representations by Kronecker Product
Lax Representations and Zero Curvature Representations by Kronecker Product arxiv:solv-int/9610008v1 18 Oct 1996 Wen-Xiu Ma and Fu-Kui Guo Abstract It is showed that Kronecker product can be applied to
More informationCanonicity of Bäcklund transformation: r-matrix approach. I. arxiv:solv-int/ v1 25 Mar 1999
LPENSL-Th 05/99 solv-int/990306 Canonicity of Bäcklund transformation: r-matrix approach. I. arxiv:solv-int/990306v 25 Mar 999 E K Sklyanin Laboratoire de Physique 2, Groupe de Physique Théorique, ENS
More informationSOLVING ODE s NUMERICALLY WHILE PRESERVING ALL FIRST INTEGRALS
SOLVING ODE s NUMERICALLY WHILE PRESERVING ALL FIRST INTEGRALS G.R.W. QUISPEL 1,2 and H.W. CAPEL 3 Abstract. Using Discrete Gradient Methods (Quispel & Turner, J. Phys. A29 (1996) L341-L349) we construct
More informationCanonically conjugate variables for the periodic Camassa-Holm equation
arxiv:math-ph/21148v2 15 Mar 24 Canonically conjugate variables for the periodic Camassa-Holm equation Alexei V. Penskoi Abstract The Camassa-Holm shallow water equation is known to be Hamiltonian with
More informationarxiv: v1 [nlin.si] 19 Aug 2008
Manuscript submitted to AIMS Journals Volume X, Number 0X, XX 200X Website: http://aimsciences.org pp. X XX ON THE NON-INTEGRABILITY OF THE POPOWICZ PEAKON SYSTEM arxiv:0808.2617v1 [nlin.si] 19 Aug 2008
More informationOn Billiard Solutions of Nonlinear PDE s
On Billiard Solutions of Nonlinear PDE s and Mark S. Alber Department of Mathematics University of Notre Dame Notre Dame, IN 46556 Mark.S.Alber.1@nd.edu Roberto Camassa Center for Nonlinear Studies and
More informationA NONLINEAR GENERALIZATION OF THE CAMASSA-HOLM EQUATION WITH PEAKON SOLUTIONS
A NONLINEAR GENERALIZATION OF THE CAMASSA-HOLM EQUATION WITH PEAKON SOLUTIONS STEPHEN C. ANCO 1, ELENA RECIO 1,2, MARÍA L. GANDARIAS2, MARÍA S. BRUZÓN2 1 department of mathematics and statistics brock
More informationarxiv: v1 [nlin.si] 7 Oct 2013
A four-component Camassa-Holm type hierarchy arxiv:1310.1781v1 [nlin.si] 7 Oct 2013 Abstract Nianhua Li 1, Q. P. Liu 1, Z. Popowicz 2 1 Department of Mathematics China University of Mining and Technology
More informationarxiv: v1 [math-ph] 13 Feb 2008
Bi-Hamiltonian nature of the equation u tx = u xy u y u yy u x V. Ovsienko arxiv:0802.1818v1 [math-ph] 13 Feb 2008 Abstract We study non-linear integrable partial differential equations naturally arising
More informationA New Integrable Couplings of Classical-Boussinesq Hierarchy with Self-Consistent Sources
Commun. Theor. Phys. Beijing, China 54 21 pp. 1 6 c Chinese Physical Society and IOP Publishing Ltd Vol. 54, No. 1, July 15, 21 A New Integrable Couplings of Classical-Boussinesq Hierarchy with Self-Consistent
More informationA Short historical review Our goal The hierarchy and Lax... The Hamiltonian... The Dubrovin-type... Algebro-geometric... Home Page.
Page 1 of 46 Department of Mathematics,Shanghai The Hamiltonian Structure and Algebro-geometric Solution of a 1 + 1-Dimensional Coupled Equations Xia Tiecheng and Pan Hongfei Page 2 of 46 Section One A
More informationThe Toda Lattice. Chris Elliott. April 9 th, 2014
The Toda Lattice Chris Elliott April 9 th, 2014 In this talk I ll introduce classical integrable systems, and explain how they can arise from the data of solutions to the classical Yang-Baxter equation.
More informationAlgebraic Spectral Relations forelliptic Quantum Calogero-MoserProblems
Journal of Nonlinear Mathematical Physics 1999, V.6, N 3, 263{268. Letter Algebraic Spectral Relations forelliptic Quantum Calogero-MoserProblems L.A. KHODARINOVA yz and I.A. PRIKHODSKY z y Wessex Institute
More informationProlongation structure for nonlinear integrable couplings of a KdV soliton hierarchy
Prolongation structure for nonlinear integrable couplings of a KdV soliton hierarchy Yu Fa-Jun School of Mathematics and Systematic Sciences, Shenyang Normal University, Shenyang 110034, China Received
More informationThe Erwin Schrodinger International Pasteurgasse 6/7. Institute for Mathematical Physics A-1090 Wien, Austria
ESI The Erwin Schrodinger International Pasteurgasse 6/7 Institute for Mathematical Physics A-1090 Wien, Austria On an Extension of the 3{Dimensional Toda Lattice Mircea Puta Vienna, Preprint ESI 165 (1994)
More informationGLASGOW Paolo Lorenzoni
GLASGOW 2018 Bi-flat F-manifolds, complex reflection groups and integrable systems of conservation laws. Paolo Lorenzoni Based on joint works with Alessandro Arsie Plan of the talk 1. Flat and bi-flat
More informationBifurcations of Traveling Wave Solutions for a Generalized Camassa-Holm Equation
Computational and Applied Mathematics Journal 2017; 3(6): 52-59 http://www.aascit.org/journal/camj ISSN: 2381-1218 (Print); ISSN: 2381-1226 (Online) Bifurcations of Traveling Wave Solutions for a Generalized
More informationarxiv:solv-int/ v1 31 May 1993
ILG-TMP-93-03 May, 993 solv-int/9305004 Alexander A.Belov #, Karen D.Chaltikian $ LATTICE VIRASORO FROM LATTICE KAC-MOODY arxiv:solv-int/9305004v 3 May 993 We propose a new version of quantum Miura transformation
More informationChisholm-Caianiello-Fubini Identities for S = 1 Barut-Muzinich-Williams Matrices
Chisholm-Caianiello-Fubini Identities for S = 1 Barut-Muzinich-Williams Matrices M. de G. Caldera Cabral and V. V. Dvoeglazov UAF Universidad Autónoma de Zacatecas Apartado Postal 636 Suc. 3 Cruces Zacatecas
More informationDerivation of general Maxwell type equations for open quantum systems. Abstract
Derivation of general Maxwell type equations for open quantum systems A.V. Khugaev 1,2 G.G. Adamian 1, N.V. Antonenko 1 1 Joint Institute for Nuclear Research, 141980 Dubna, Russia arxiv:1901.04332v2 [physics.class-ph]
More informationInvertible Matrices over Idempotent Semirings
Chamchuri Journal of Mathematics Volume 1(2009) Number 2, 55 61 http://www.math.sc.chula.ac.th/cjm Invertible Matrices over Idempotent Semirings W. Mora, A. Wasanawichit and Y. Kemprasit Received 28 Sep
More informationPoisson Manifolds Bihamiltonian Manifolds Bihamiltonian systems as Integrable systems Bihamiltonian structure as tool to find solutions
The Bi hamiltonian Approach to Integrable Systems Paolo Casati Szeged 27 November 2014 1 Poisson Manifolds 2 Bihamiltonian Manifolds 3 Bihamiltonian systems as Integrable systems 4 Bihamiltonian structure
More informationQuasigraded Lie Algebras and Matrix Generalization of Landau Lifshitz Equation
Proceedings of Institute of Mathematics of NAS of Ukraine 2004, Vol. 50, Part 1, 462 469 Quasigraded Lie Algebras and Matrix Generalization of Landau Lifshitz Equation Taras V. SKRYPNYK Bogoliubov Institute
More informationOn generating discrete integrable systems via Lie algebras and commutator equations
Hong Kong Baptist University HKBU Institutional Repository Department of Computer Science Journal Articles Department of Computer Science 2016 On generating discrete integrable systems via Lie algebras
More informationarxiv: v1 [math-ph] 15 Sep 2009
On the superintegrability of the rational Ruijsenaars-Schneider model V. Ayadi a and L. Fehér a,b arxiv:0909.2753v1 [math-ph] 15 Sep 2009 a Department of Theoretical Physics, University of Szeged Tisza
More informationA.1 Appendix on Cartesian tensors
1 Lecture Notes on Fluid Dynamics (1.63J/2.21J) by Chiang C. Mei, February 6, 2007 A.1 Appendix on Cartesian tensors [Ref 1] : H Jeffreys, Cartesian Tensors; [Ref 2] : Y. C. Fung, Foundations of Solid
More informationOn Expected Gaussian Random Determinants
On Expected Gaussian Random Determinants Moo K. Chung 1 Department of Statistics University of Wisconsin-Madison 1210 West Dayton St. Madison, WI 53706 Abstract The expectation of random determinants whose
More informationarxiv: v2 [math-ph] 24 Feb 2016
ON THE CLASSIFICATION OF MULTIDIMENSIONALLY CONSISTENT 3D MAPS MATTEO PETRERA AND YURI B. SURIS Institut für Mathemat MA 7-2 Technische Universität Berlin Str. des 17. Juni 136 10623 Berlin Germany arxiv:1509.03129v2
More informationFaculty of Engineering, Mathematics and Science School of Mathematics
Faculty of Engineering, Mathematics and Science School of Mathematics JS & SS Mathematics SS Theoretical Physics SS TSM Mathematics Module MA3429: Differential Geometry I Trinity Term 2018???, May??? SPORTS
More informationPHYS 705: Classical Mechanics. Rigid Body Motion Introduction + Math Review
1 PHYS 705: Classical Mechanics Rigid Body Motion Introduction + Math Review 2 How to describe a rigid body? Rigid Body - a system of point particles fixed in space i r ij j subject to a holonomic constraint:
More informationTwo-variable Wilson polynomials and the generic superintegrable system on the 3-sphere
Two-variable Wilson polynomials and the generic superintegrable system on the 3-sphere Willard Miller, [Joint with E.G. Kalnins (Waikato) and Sarah Post (CRM)] University of Minnesota Special Functions
More informationCanonical Forms for BiHamiltonian Systems
Canonical Forms for BiHamiltonian Systems Peter J. Olver Dedicated to the Memory of Jean-Louis Verdier BiHamiltonian systems were first defined in the fundamental paper of Magri, [5], which deduced the
More informationarxiv: v2 [nlin.si] 3 Aug 2017
A generalised multicomponent system of Camassa-Holm-Novikov equations arxiv:1608.04604v2 [nlin.si] 3 Aug 2017 1. Introduction Diego Catalano Ferraioli 1 and Igor Leite Freire 2 Departamento de Matemática
More informationOn Generating Discrete Integrable Systems via Lie Algebras and Commutator
Commun. Theor. Phys. 65 (216 335 34 Vol. 65, No. 3, March 1, 216 On Generating Discrete Integrable Systems via Lie Algebras and Commutator Equations Yu-Feng Zhang ( ô 1 and Honwah Tam 2 1 College of Sciences,
More informationNEW PERIODIC WAVE SOLUTIONS OF (3+1)-DIMENSIONAL SOLITON EQUATION
Liu, J., et al.: New Periodic Wave Solutions of (+)-Dimensional Soliton Equation THERMAL SCIENCE: Year 7, Vol., Suppl., pp. S69-S76 S69 NEW PERIODIC WAVE SOLUTIONS OF (+)-DIMENSIONAL SOLITON EQUATION by
More informationPOISSON-LIE T-DUALITY AND INTEGRABLE SYSTEMS
REVISTA DE LA UNIÓN MATEMÁTICA ARGENTINA Volumen 49, Número 1, 2008, Páginas 71 82 POISSON-LIE T-DUALITY AND INTEGRABLE SYSTEMS H. MONTANI Abstract. We describe a hamiltonian approach to Poisson-Lie T-duality
More informationSystematic construction of (boundary) Lax pairs
Thessaloniki, October 2010 Motivation Integrable b.c. interesting for integrable systems per ce, new info on boundary phenomena + learn more on bulk behavior. Examples of integrable b.c. that modify the
More informationarxiv:hep-th/ v1 16 Jul 1992
IC-92-145 hep-th/9207058 Remarks on the Additional Symmetries and W-constraints in the Generalized KdV Hierarchy arxiv:hep-th/9207058v1 16 Jul 1992 Sudhakar Panda and Shibaji Roy International Centre for
More informationElectromagnetic energy and momentum
Electromagnetic energy and momentum Conservation of energy: the Poynting vector In previous chapters of Jackson we have seen that the energy density of the electric eq. 4.89 in Jackson and magnetic eq.
More informationTravelling wave solutions for a CBS equation in dimensions
AMERICAN CONFERENCE ON APPLIED MATHEMATICS (MATH '8), Harvard, Massachusetts, USA, March -6, 8 Travelling wave solutions for a CBS equation in + dimensions MARIA LUZ GANDARIAS University of Cádiz Department
More informationSYMMETRY AND REDUCTIONS OF INTEGRABLE DYNAMICAL SYSTEMS: PEAKON AND THE TODA CHAIN SYSTEMS
Dublin Institute of Technology ARROW@DIT Articles School of Mathematics 014 SYMMETRY AND REDUCTIONS OF INTEGRABLE DYNAMICAL SYSTEMS: PEAKON AND THE TODA CHAIN SYSTEMS Vladimir Gerdjikov INRNE, Sofia, Bulgaria,
More informationON POISSON BRACKETS COMPATIBLE WITH ALGEBRAIC GEOMETRY AND KORTEWEG DE VRIES DYNAMICS ON THE SET OF FINITE-ZONE POTENTIALS
ON POISSON BRACKETS COMPATIBLE WITH ALGEBRAIC GEOMETRY AND KORTEWEG DE VRIES DYNAMICS ON THE SET OF FINITE-ZONE POTENTIALS A. P. VESELOV AND S. P. NOVIKOV I. Some information regarding finite-zone potentials.
More informationarxiv:nlin/ v1 [nlin.si] 23 Jul 2003
Deformed Harry Dym and Hunter-Zheng Equations J. C. Brunelli a, Ashok Das b and Ziemowit Popowicz c arxiv:nlin/0307043v1 [nlin.si] 3 Jul 003 a Departamento de Física, CFM Universidade Federal de Santa
More informationTailoring BKL to Loop Quantum Cosmology
Tailoring BKL to Loop Quantum Cosmology Adam D. Henderson Institute for Gravitation and the Cosmos Pennsylvania State University LQC Workshop October 23-25 2008 Introduction Loop Quantum Cosmology: Big
More informationOn Camassa Holm Equation with Self-Consistent Sources and Its Solutions
Commun. Theor. Phys. Beiing China 53 2010 pp. 403 412 c Chinese Physical Society and IOP Publishing Ltd Vol. 53 No. 3 March 15 2010 On Camassa Holm Equation with Self-Consistent Sources and Its Solutions
More informationA MULTI-COMPONENT LAX INTEGRABLE HIERARCHY WITH HAMILTONIAN STRUCTURE
Pacific Journal of Applied Mathematics Volume 1, Number 2, pp. 69 75 ISSN PJAM c 2008 Nova Science Publishers, Inc. A MULTI-COMPONENT LAX INTEGRABLE HIERARCHY WITH HAMILTONIAN STRUCTURE Wen-Xiu Ma Department
More informationReductions to Korteweg-de Vries Soliton Hierarchy
Commun. Theor. Phys. (Beijing, China 45 (2006 pp. 23 235 c International Academic Publishers Vol. 45, No. 2, February 5, 2006 Reductions to Korteweg-de Vries Soliton Hierarchy CHEN Jin-Bing,,2, TAN Rui-Mei,
More informationCartesian Tensors. e 2. e 1. General vector (formal definition to follow) denoted by components
Cartesian Tensors Reference: Jeffreys Cartesian Tensors 1 Coordinates and Vectors z x 3 e 3 y x 2 e 2 e 1 x x 1 Coordinates x i, i 123,, Unit vectors: e i, i 123,, General vector (formal definition to
More informationA Kupka-Smale Theorem for Hamiltonian systems. systems from Mañé s viewpoint, a control approach. Ludovic Rifford PICOF 12
A Kupka-Smale Theorem for Hamiltonian systems from Mañé s viewpoint, a control approach Université de Nice - Sophia Antipolis & Institut Universitaire de France PICOF 12 The Kupka-Smale Theorem for vector
More informationarxiv:nlin/ v1 [nlin.si] 17 Jun 2006
Integrable dispersionless KdV hierarchy with sources arxiv:nlin/0606047v [nlin.si] 7 Jun 2006 Zhihua Yang, Ting Xiao and Yunbo Zeng Department of Mathematical Sciences, Tsinghua University, Beijing 00084,
More informationKeywords: Exp-function method; solitary wave solutions; modified Camassa-Holm
International Journal of Modern Mathematical Sciences, 2012, 4(3): 146-155 International Journal of Modern Mathematical Sciences Journal homepage:www.modernscientificpress.com/journals/ijmms.aspx ISSN:
More informationarxiv:physics/ v1 [physics.class-ph] 3 Apr 2000
Dirac monopole with Feynman brackets Alain Bérard arxiv:physics/0004008v1 [physicsclass-ph] 3 Apr 2000 LPLI-Institut de Physique, 1 blvd DArago, F-57070 Metz, France Y Grandati LPLI-Institut de Physique,
More informationEXACT SOLITARY WAVE AND PERIODIC WAVE SOLUTIONS OF THE KAUP-KUPERSCHMIDT EQUATION
Journal of Applied Analysis and Computation Volume 5, Number 3, August 015, 485 495 Website:http://jaac-online.com/ doi:10.11948/015039 EXACT SOLITARY WAVE AND PERIODIC WAVE SOLUTIONS OF THE KAUP-KUPERSCHMIDT
More informationE.., (2) g t = e 2' g E. g t = g ij (t u k )du i du j, i j k =1 2. (u 1 0 0) u2 2 U, - v, w, g 0 (v w) = g ij (0 u k 0)v i w j = 0, (t) = g ij (t u k
2007 10 (545) 517.929..,.. 1. g t M, d dt g t = ;2 Ric(g t ) (1) Ric(g) g.,, -, - (., [1], [2]).,,.,, f t - (M G), g t = ft G,, (1)., -, -, (, ), - (,, [3]). t - E 3, g t, t E 3, (1). t -., -. (M g) Ric
More informationarxiv: v1 [math.ra] 10 May 2013
ON FINITE-DIMENSIONAL SUBALGEBRAS OF DERIVATION ALGEBRAS arxiv:1305.2285v1 [math.ra] 10 May 2013 Abstract. Let K be a field, R be an associative and commutative K-algebra and L be a Lie algebra over K.
More informationTravelling Wave Solutions for the Gilson-Pickering Equation by Using the Simplified G /G-expansion Method
ISSN 1749-3889 (print, 1749-3897 (online International Journal of Nonlinear Science Vol8(009 No3,pp368-373 Travelling Wave Solutions for the ilson-pickering Equation by Using the Simplified /-expansion
More informationHAMILTONIAN STRUCTURE OF PEAKONS AS WEAK SOLUTIONS FOR THE MODIFIED CAMASSA-HOLM EQUATION
HAMILTONIAN STUCTUE OF PEAKONS AS WEAK SOLUTIONS FO THE MODIFIED CAMASSA-HOLM EQUATION STEPHEN C. ANCO and DANIEL KAUS arxiv:1708.02520v2 [nlin.si] 31 Jan 2018 department of mathematics and statistics
More informationLECTURE 9: MOVING FRAMES IN THE NONHOMOGENOUS CASE: FRAME BUNDLES. 1. Introduction
LECTURE 9: MOVING FRAMES IN THE NONHOMOGENOUS CASE: FRAME BUNDLES 1. Introduction Until now we have been considering homogenous spaces G/H where G is a Lie group and H is a closed subgroup. The natural
More informationIntegrable Rosochatius Deformations for an Integrable Couplings of CKdV Equation Hierarchy
Commun. Theor. Phys. Beiing, China 54 00 pp. 609 64 c Chinese Physical Society and IOP Publishing Ltd Vol. 54, No. 4, October 5, 00 Integrable Rosochatius Deformations for an Integrable Couplings of CKdV
More informationSOME OBSERVATIONS REGARDING BRACKETS AND DISSIPATION. Philip J.Morrison of Mathematics University of California Berkeley, CA 94720
SOME OBSERVATIONS REGARDING BRACKETS AND DISSIPATION + Philip J.Morrison of Mathematics University of California Berkeley, CA 94720 D~partment Abstract Some ideas relating to a bracket formulation for
More informationOn construction of recursion operators from Lax representation
On construction of recursion operators from Lax representation Metin Gürses, Atalay Karasu, and Vladimir V. Sokolov Citation: J. Math. Phys. 40, 6473 (1999); doi: 10.1063/1.533102 View online: http://dx.doi.org/10.1063/1.533102
More informationQuantum Field Theory III
Quantum Field Theory III Prof. Erick Weinberg January 19, 2011 1 Lecture 1 1.1 Structure We will start with a bit of group theory, and we will talk about spontaneous symmetry broken. Then we will talk
More informationOSCILLATION-INDUCED BLOW-UP TO THE MODIFIED CAMASSA HOLM EQUATION WITH LINEAR DISPERSION
OSCILLATION-INDUCED BLOW-UP TO THE MODIFIED CAMASSA HOLM EQUATION WITH LINEAR DISPERSION ROBIN MING CHEN, YUE LIU, CHANGZHENG QU, AND SHUANGHU ZHANG Abstract. In this paper, we provide a blow-up mechanism
More informationarxiv:hep-th/ v1 8 Mar 1995
GALILEAN INVARIANCE IN 2+1 DIMENSIONS arxiv:hep-th/9503046v1 8 Mar 1995 Yves Brihaye Dept. of Math. Phys. University of Mons Av. Maistriau, 7000 Mons, Belgium Cezary Gonera Dept. of Physics U.J.A Antwerpen,
More informationOn universality of critical behaviour in Hamiltonian PDEs
Riemann - Hilbert Problems, Integrability and Asymptotics Trieste, September 23, 2005 On universality of critical behaviour in Hamiltonian PDEs Boris DUBROVIN SISSA (Trieste) 1 Main subject: Hamiltonian
More informationThe quasiclassical limit of the symmetry constraint of the KP hierarchy and the dispersionless KP hierarchy with self-consistent sources
Journal of Nonlinear Mathematical Physics Volume 13, Number 2 (2006), 193 204 Article The quasiclassical limit of the symmetry constraint of the KP hierarchy and the dispersionless KP hierarchy with self-consistent
More informationChapter 2 Introduction to Phenomenological Crystal Structure
Chapter 2 Introduction to Phenomenological Crystal Structure 2.1 Crystal Structure An ideal crystal represents a periodic pattern generated by infinite, regular repetition of identical microphysical structural
More informationAppendix A Spin-Weighted Spherical Harmonic Function
Appendix A Spin-Weighted Spherical Harmonic Function Here, we review the properties of the spin-weighted spherical harmonic function. In the past, this was mainly applied to the analysis of the gravitational
More informationHamiltonian flow in phase space and Liouville s theorem (Lecture 5)
Hamiltonian flow in phase space and Liouville s theorem (Lecture 5) January 26, 2016 90/441 Lecture outline We will discuss the Hamiltonian flow in the phase space. This flow represents a time dependent
More informationA Realization of Yangian and Its Applications to the Bi-spin System in an External Magnetic Field
Commun. Theor. Phys. Beijing, China) 39 003) pp. 1 5 c International Academic Publishers Vol. 39, No. 1, January 15, 003 A Realization of Yangian and Its Applications to the Bi-spin System in an External
More informationProduct Zero Derivations on Strictly Upper Triangular Matrix Lie Algebras
Journal of Mathematical Research with Applications Sept., 2013, Vol.33, No. 5, pp. 528 542 DOI:10.3770/j.issn:2095-2651.2013.05.002 Http://jmre.dlut.edu.cn Product Zero Derivations on Strictly Upper Triangular
More informationNew Homoclinic and Heteroclinic Solutions for Zakharov System
Commun. Theor. Phys. 58 (2012) 749 753 Vol. 58, No. 5, November 15, 2012 New Homoclinic and Heteroclinic Solutions for Zakharov System WANG Chuan-Jian ( ), 1 DAI Zheng-De (à ), 2, and MU Gui (½ ) 3 1 Department
More informationGeometric control and dynamical systems
Université de Nice - Sophia Antipolis & Institut Universitaire de France 9th AIMS International Conference on Dynamical Systems, Differential Equations and Applications Control of an inverted pendulum
More informationSEMI-INNER PRODUCTS AND THE NUMERICAL RADIUS OF BOUNDED LINEAR OPERATORS IN HILBERT SPACES
SEMI-INNER PRODUCTS AND THE NUMERICAL RADIUS OF BOUNDED LINEAR OPERATORS IN HILBERT SPACES S.S. DRAGOMIR Abstract. The main aim of this paper is to establish some connections that exist between the numerical
More informationNonlinear Analysis. Variational identities and applications to Hamiltonian structures of soliton equations
Nonlinear Analysis 71 (2009) e1716 e1726 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na Variational identities and applications to Hamiltonian
More informationTHE TODA LATTICE. T (y) = α(e βy 1)
THE TODA LATTICE Background The Toda lattice is named after Morikazu Toda, who discovered in the 1960 s that the differential equations for a lattice with internal force T (y) = α(e βy 1) (with α, β constant)
More informationarxiv: v1 [math.ac] 11 Dec 2013
A KOSZUL FILTRATION FOR THE SECOND SQUAREFREE VERONESE SUBRING arxiv:1312.3076v1 [math.ac] 11 Dec 2013 TAKAYUKI HIBI, AYESHA ASLOOB QURESHI AND AKIHIRO SHIKAMA Abstract. The second squarefree Veronese
More informationCANONICAL METRICS AND STABILITY OF PROJECTIVE VARIETIES
CANONICAL METRICS AND STABILITY OF PROJECTIVE VARIETIES JULIUS ROSS This short survey aims to introduce some of the ideas and conjectures relating stability of projective varieties to the existence of
More informationProblem Set 4. Eshelby s Inclusion
ME34B Elasticity of Microscopic Structures Wei Cai Stanford University Winter 24 Problem Set 4. Eshelby s Inclusion Chris Weinberger and Wei Cai c All rights reserved Problem 4. (5 Spherical inclusion.
More informationMax-Planck-Institut für Mathematik in den Naturwissenschaften Leipzig
Max-Planck-Institut für Mathematik in den aturwissenschaften Leipzig Bell inequality for multipartite qubit quantum system and the maximal violation by Ming Li and Shao-Ming Fei Preprint no.: 27 2013 Bell
More informationDerivation of Generalized Camassa-Holm Equations from Boussinesq-type Equations
Derivation of Generalized Camassa-Holm Equations from Boussinesq-type Equations H. A. Erbay Department of Natural and Mathematical Sciences, Faculty of Engineering, Ozyegin University, Cekmekoy 34794,
More informationSymmetry reductions and travelling wave solutions for a new integrable equation
Symmetry reductions and travelling wave solutions for a new integrable equation MARIA LUZ GANDARIAS University of Cádiz Department of Mathematics PO.BOX 0, 50 Puerto Real, Cádiz SPAIN marialuz.gandarias@uca.es
More information1 Introduction , 1995, Protvino, Russia, eds. V.A. Petrov, A.P. Samokhin, R.N. Rogalyov (IHEP, Protvino, 1996) pp
FUNCTIONAL APPROACH TO PHASE SPACE FORMULATION OF QUANTUM MECHANICS 1 A.K.Aringazin Department of Theoretical Physics, Karaganda State University, Karaganda 470074, Kazakstan We present BRST gauge fixing
More informationMaslov indices and monodromy
Loughborough University Institutional Repository Maslov indices and monodromy This item was submitted to Loughborough University's Institutional Repository by the/an author. Additional Information: This
More informationClassical Mechanics in Hamiltonian Form
Classical Mechanics in Hamiltonian Form We consider a point particle of mass m, position x moving in a potential V (x). It moves according to Newton s law, mẍ + V (x) = 0 (1) This is the usual and simplest
More informationWeak discrete logarithms in non-abelian groups
Weak discrete logarithms in non-abelian groups Ivana Ilić, Spyros S. Magliveras Department of Mathematical Sciences, Florida Atlantic University 777 Glades Road, Boca Raton, FL 33431, U.S.A. iilic@fau.edu,
More informationGENERALIZED r-matrix STRUCTURE AND ALGEBRO-GEOMETRIC SOLUTION FOR INTEGRABLE SYSTEM
Reviews in Mathematical Physics, Vol. 13, No. 5 2001 545 586 c World Scientific Publishing Company GENERALIZED r-matrix STRUCTURE AND ALGEBRO-GEOMETRIC SOLUTION FOR INTEGRABLE SYSTEM ZHIJUN QIAO Institute
More informationThe Weierstrass Theory For Elliptic Functions. The Burn 2007
The Weierstrass Theory For Elliptic Functions Including The Generalisation To Higher Genus Department of Mathematics, MACS Heriot Watt University Edinburgh The Burn 2007 Outline Elliptic Function Theory
More information1 Introduction The search for constants of motion in the Lagrangian approach has been traditionally related with the existence of one{parameter subgro
Helmholtz conditions and Alternative Lagrangians: Study of an integrable Henon-Heiles system Jose F. Cari~nena and Manuel F. Ra~nada Departamento de Fsica Teorica, Facultad de Ciencias Universidad de Zaragoza,
More informationReflectionless Analytic Difference Operators (A Os): Examples, Open Questions and Conjectures
Journal of Nonlinear Mathematical Physics 2001, V.8, Supplement, 240 248 Proceedings: NEEDS 99 Reflectionless Analytic Difference Operators (A Os): Examples, Open Questions and Conjectures S N M RUIJSENAARS
More informationarxiv: v1 [math.dg] 1 Jul 2014
Constrained matrix Li-Yau-Hamilton estimates on Kähler manifolds arxiv:1407.0099v1 [math.dg] 1 Jul 014 Xin-An Ren Sha Yao Li-Ju Shen Guang-Ying Zhang Department of Mathematics, China University of Mining
More informationBreaking soliton equations and negative-order breaking soliton equations of typical and higher orders
Pramana J. Phys. (2016) 87: 68 DOI 10.1007/s12043-016-1273-z c Indian Academy of Sciences Breaking soliton equations and negative-order breaking soliton equations of typical and higher orders ABDUL-MAJID
More informationarxiv:math-ph/ v2 29 Mar 2003
THE ATIYAH-HITCHIN BRACKET AND 1D INTEGRABLE SYSTEMS arxiv:math-ph/0111017v2 29 Mar 2003 KL Vaninsky Michigan State University, East Lansing, MI 48824 Abstract All fashionable integrable equations analyzed
More informationThe Riccati equation with variable coefficients expansion algorithm to find more exact solutions of nonlinear differential equations
MM Research Preprints, 275 284 MMRC, AMSS, Academia Sinica, Beijing No. 22, December 2003 275 The Riccati equation with variable coefficients expansion algorithm to find more exact solutions of nonlinear
More information