A Short historical review Our goal The hierarchy and Lax... The Hamiltonian... The Dubrovin-type... Algebro-geometric... Home Page.
|
|
- Giles Wade
- 5 years ago
- Views:
Transcription
1 Page 1 of 46
2 Department of Mathematics,Shanghai The Hamiltonian Structure and Algebro-geometric Solution of a Dimensional Coupled Equations Xia Tiecheng and Pan Hongfei Page 2 of 46
3 Section One A Short Overview Section two Section three Lax pairs of the coupled dimens Section four The Hamiltonian structures Section five The Dubrovin-type equations Section six Algebro-geometric solutions Section seven Open problem Page 3 of 46
4 1 Algebraic-geometry solutions(also known as quasiperiodic solutions or finite-band solution) are an important exact solutions of nonlinear evolution equations and is a kind of a natural extension of soliton solution. It will be associated with the several branch of mathematics, such as the theory of differential equation, operator spectrum theory, complex variables functions, algebra geometry. And also many other types of solutions, such as soliton solution, and elliptic periodic solutions can be obtained by the degradation of algebraic geometry solutions. The most striking aspects of integrable system theory is one of nonlinear differential equation with cross application of algebraic geometry. It is considered to be an perfect example of today s modern mathematical physics. Page 4 of 46
5 In 1967 inverse scattering method of the solution of KdV equation with initial value problem have been found by Gardner, Greene, Kruskal, Miura. Last century,70 s Zakharov, Shabat, Ablowitz, Newell generalized inverse scattering method to a big class of nonlinear evolution equations. But as a result of the long-term behavior of time, there is no effective method to solve the inverse spectrum problem of periodic coefficient linear operator, so the inverse scattering method cannot be directly applied to the periodic initial value problem. Page 5 of 46
6 In 80 s Novikov Dubrovin Mckean Its Matveev Krichiev etc. successfully solve the periodic initial value problem by using inverse scattering method based on the algebraic geometry method. Thus these caused cross development between the integrable differential equation and algebraic geometry. Because of the periodic potential Schrodinger operator of the spectrum has band structure and the spectrum of absolutely continuous component is divided by a finite number of belt, and the corresponding potential is known as with finite-potential. Page 6 of 46
7 Analysis on KdV equations with finite potential discovered by Mr Novikov Matveev, Lax, Marchenko, independently. All with finite potential can be described by the high order KdV equations of steady state, and then Matveev noticed this with finite potential can be converted into a two sheets Riemann surface with genus N K N : y 2 = 2N+1 m=1 (z E m ) by using Jacobi inverse problem to describe. Page 7 of 46
8 Finally, the potential function of continuous spectrum can be expressed as the function based on Riemann surface N u(x, t) = E 0 + (E 2j 1 + E 2j 2λ j ) j=1 2 2 xlnθ(ψ Q0 A Q0 (P ) + αq 0 (D µ (x, t))) Page 8 of 46
9 In the last century, 90 s Gesztesy and Holden etc. developed an effective polynomial recursive method and succeeded to extend algebraic geometry solutions from a single equation to the entire equation hierarchy. Up to now, this method has been extended to the categories of integrable equation, including sine- Gordon equation, Camassa - Holm equation, Thirring equations, KP equation, discrete Toda equation, Ablowitz - Ladik equation, etc. Page 9 of 46
10 The concepts and mathematics knowledge used in algebraic geometry solutions: complex function (residue, Lagrange interpolation formula) Operator spectrum theory (gradual spectrum parameters, Baker - Akhiezer function) Algebra curve theory ( hyperelliptic curve with finite genus N, meromorphic function, in addition to the Abel maps and Riemann theta function) Integrable system (zero curvature, separation of variables, recursive method, polynomial Mateev - Its trace formula, Dubrovin equations) Page 10 of 46
11 Fritz Gesztesy Augest, 1957, Mathematician. Department of Mathematics, University of Missouri Columbia, USA Differential equations, Operator spectral theory, Integrable systems Works: Soliton Equations and Their Algebro-Geometric Solutions Vol I: (1+1)-Dimensional Continuous Models, Cambridge University Press, Cambridge 2003 Soliton Equations and Their Algebro-Geometric Solutions Volume II: (1+1)-Dimensional Discrete Models Cambridge University Press, Cambridge 2008 Solvable Models in Quantum Mechanics: American Mathematical Society, Providence, RI, USA, 2005 Page 11 of 46
12 2 In this talk, we will see how to construct the Hamiltonian structure and search for the algebro-geometric solution of the following coupled dimensional soliton equations as an example: q t = 1 2 (r xx 3q 2 r x + q x r 2 + 2qrr x ), r t = 1 2 (q xx + 3r 2 r x r x q 2 2qrq x ).. (1.1) Page 12 of 46
13 3 The hierarchy and Lax pairs of the coupled dimensional soliton equations Consider the spectral problem : ( ) λ q + r ψ x = Uψ, U = λ(q r) λ. (1.2) and the auxiliary problem: ( ψ tm = V (m) ψ, V (m) = V (m) 11 V (m) 12 V (m) 21 V (m) 11 ). (1.3) Page 13 of 46
14 Where V (m) 11 = V (m) 12 = V (m) 21 = m j=0 m j=0 m j=0 S (3) j λ m+1 j, S (2) j λ m j, S (1) j λ m+1 j. Page 14 of 46
15 Then the compatibility condition of Eq.(1.2) and Eq.(1.3) is U tm V x (m) +[U, V (m) ] = 0, which is equivalent to the hierarchy of nonlinear evolution equations q tm = 1 2 (S(2) mx + S mx), (1) In brief, r tm = 1 2 (S(2) mx S (1) mx). (q tm, r tm ) T = X m, m 0. (1.4) X m = 1 ( ) ( ) S m (2) 2 S m (1). Page 15 of 46
16 The first two nontrivial equations are q t0 = q x, r t0 = r x. (1.5) and q t1 = 1 2 (r xx 3q 2 r x + q x r 2 + 2qrr x ), r t1 = 1 2 (q xx + 3r 2 r x r x q 2 2qrq x ). (1.6) Page 16 of 46
17 4 The Hamiltonian structures Let ( ) V = V = V11 V 12 V 21 V 11 where V 11 = j 0 S (3) j λ j+1, V 12 = j 0 (2.1) S (2) j λ j, V 21 = j 0 S (1) j λ j+1. It is easy to calculate tr(v U λ ) = 2V 11 + (q r)v 12 = (2S (3) j + (q r)s (2) j 1 tr(v U q ) = λv 12 + V 21 = (S (2) j + S (1) j )λ j+1, j 0 tr(v U r ) = λv 12 + V 21 = j 0 ( S (2) j + S (1) j )λ j+1. (2.2) j 1 )λ j Page 17 of 46
18 According to the trace identity[tu paper,1989,jmp.], we have ) (2V 11 +(q r)v 12 ) = (λ s λ λs ) ( δ δq δ δr ( ) λv12 + V 21 λv 12 + V 21 Page 18 of 46
19 Comparing the coefficients of λ j+1, we obtain ) ( (2S (3) j +(q r)s (2) j 1 ) = ( j+2+s) ( δ δq δ δr S (2) j 1 + S(1) j 1 S (2) j 1 + S(1) j 1 ), Page 19 of 46
20 we set j = 1, and then get s = 1, and ( δ δq δ δr ) H j = ( S (2) j 1 + S(1) j 1 S (2) j 1 + S(1) j 1 where H j = 2S(3) j +(q r)s (2) j 1 j+1. ) = ( ) ( S (2) j 1 (2.3) S (1) j 1 ), Page 20 of 46
21 Thus the soliton equations (1.4) can be expressed the following form : ( ) qr = 1 ( ) ( ) ( S (2) δ ) m t m 2 S m (1) = J δq δ H m+1, m 0, δr (2.4) where J = 1 ( ) In speciality, the Hamiltonian structure of equation (1.1) is(m = 1): H 2 = 1 8 (q+r)2 (q r) 2 qq x 2rr x + qr x (q x r x )r x. 2 Page 21 of 46
22 5 The Dubrovin-type equations Let s consider the function det W which is a (2N + 2)th-order polynomial in λ with constant coefficients of the x- flow and t m -flow: det W = f 2 + gh = 2N+2 j=1 (λ λ j ) = R(λ), (3.1) Page 22 of 46
23 From (3.1) we see that: f λ=uk = R(u k ), f λ=vk = R(v k ). (3.2) Again we also obtain: g x λ=uk = (q+r)u kx h x λ=vk = (q r)v kx N j=1,j k N j=1,j k (u k u j ) = 2(q+r)f λ=uk, (v k v j ) = 2v k (q r)f λ=vk, Page 23 of 46
24 which together with (3.2) gives u kx = 2 R(u k ), 1 k N (3.3) N (u k u j ) j=1,j k v k,x = 2 R(v k ), 1 k N (3.4) N (v k v j ) j=1,j k Page 24 of 46
25 In a way similar to the above expression, by using (2.4)(m = 1, t 1 = t), we arrive at the evolution of {u k } and {v k } along the t m flow: Page 25 of 46
26 u k,t = 2f λ=u k V (1) (q+r) N j=1,j k 12 λ=u k (u k u j ) = 2 R(u k )[u k 1 2 (q+r)(q r)+1 2 ln(q+r)] = N j=1,j k (u k u j ) 2 R(u k )[u k ( N u j +α 1 )] N j=1,j k j=1 (u k u j ), (3.5) Page 26 of 46
27 v k,t = 2f λ=v k V (1) (q r)v k N j=1,j k 21 λ=v k (v k v j ) = 2 R(v k )[v k 1 2 (q+r)(q r) 1 2 ln(q r)] N (v k v j ) = j=1,j k 2 R(v k )[v k ( N v j +α 1 )] N j=1,j k j=1 (v k v j ). (3.6) Page 27 of 46
28 Therefore, if the (2N + 2) distinct parameters λ 1, λ 2,..., λ 2N+2 are given, and let u k (x, t) and v k (x, t) be distinct solutions of ordinary differential equations(3.3), (3.4), (3.5) and (3.6), then (q, r) determined by (A) is a solution of the coupled dimensional equations (1.1). Page 28 of 46
29 6 Algebro-geometric solutions In this section, we will give the algebro-geometric solutions of the coupled dimensional Equations (1.1). To this end, We first introduce the Riemann surface Γ of the hyperelliptic curve Γ : ζ 2 = R(λ), R(λ) = 2N+2 j=1 (λ λ j ), with genus N on Γ. On Γ there are two infinite points 1 and 2, which are not branch points of Γ. We equip Γ with a canonical basis of cycles: a 1, a 2,..., a N ; b 1, b 2,..., b N which are independent and have intersection numbers as follows: Page 29 of 46
30 a i a j = 0, b i b j = 0, a i b j = δ ij, i, j = 1, 2,..., N. We will choose the following set as our basis: ω l = λl 1 dλ R(λ), l = 1, 2,..., N, which are linearly independent from each other on Γ, and let A ij = ω i, B ij = ω i. a j b j It is possible to show that the matrices A = (A ij ) and B = (B ij ) are N N invertible matrices [?,?]. Page 30 of 46
31 Now we define the matrices C and τ by C = (C ij ) = A 1, τ = (τ ij ) = A 1 B. Then the matrix τ can be shown to symmetric (τ ij = τ ji ) and it has a positivedefinite imaginary part(im τ > 0).If we normalize ω j into the new basis ω j : ω j = Then we have: a j ω j = b j ω i = N C jl ω l, l = 1, 2,..., N. l=1 N C jl ω l = a j l=1 N C jl ω l = b j l=1 N C jl A li = δ ji l=1 N C jl B li = τ ji l=1 Page 31 of 46
32 Using B = (B jk ) g g is a symmetry matrix and ImB > 0(Positive definite) We can define theta function in the algebraic curve as follows: θ(ζ, B) = m Z g expπ 1(< Bm, m > +2 < ζ, m >), ζ C g For arbitrary m Z g, we have (1) θ( ζ, B) = θ(ζ, B); (2) θ(ζ + m, B) = θ(ζ, B) (3) θ(ζ + mb, B) = θ(ζ, B)e 1π(Bm,m) 2 1π(m,ζ) Page 32 of 46
33 As a series in the above definition, θ function of the compact set of C g is an uniform convergence and analytical function. Unit matrix (δ jk ) g g and symmetry matrix (B jk ) g g can be called Quasi periodic of θ function. Page 33 of 46
34 Now we introduce the Abel-Jacobi coordinates as follows: N p(uk ρ (1) (x,t)) N N uk λ l 1 dλ j (x, t) = ω j = C jl, k=1 p 0 k=1 l=1 λ(p 0 ) R(λ) (5.1) N p(vk ρ (2) (x,t)) N N vk (x,t) λ l 1 dλ j (x, t) = ω j = C jl, k=1 p 0 k=1 l=1 λ(p 0 ) R(λ) (5.2) where p(u k (x, t)) = (u k, R(u k )), p(v k (x, t)) = (v k, R(v k )), and λ(p 0 ) is the local coordinate of p 0. Page 34 of 46
35 From above, we get N N x ρ (1) u l 1 k u N kx j = C jl R(uk ) = k=1 which implies l=1 k=1 N l=1 C jl u l 1 k, N (u k u j ) j=1,j k x ρ (1) j = 2C jn = Ω (1) j, j = 1, 2,..., N. (5.3) With the help of the following equality: Page 35 of 46
36 N k=1 N j=1,j k u l 1 k (u k u j ) = δ ln, l = 1, 2,..., N. In a similar way, we obtain from (5.1), (5.2), (4.9), (4.10), (4.11) and (4.12): t ρ (1) j = 2C j,n 1 + 2α 1 C j,n = Ω (2) j, (5.4) x ρ (2) j = Ω (1) j, j = 1, 2,..., N. (5.5) t ρ (2) j = Ω (2) j, j = 1, 2,..., N. (5.6) Page 36 of 46
37 On the basis of these results, we obtain the following: ρ (1) j (x, t) = Ω (1) j x + Ω (2) j t + γ (1) j (5.7) ρ (2) j (x, t) = Ω (1) j x Ω (2) j t + γ (2) j (5.8) Where γ (i) (i = 1, 2) are constants, and γ (1) j = j N k=1 p(ũk (0,0)) p 0 ω j, γ (2) j = N k=1 p(ṽk (0,0)) ω j ρ (1) = (ρ ((1) 1, ρ (1) 2..., ρ (1) N )T, ρ (2) = (ρ ((2) 1, ρ (2) 2,..., ρ(2) N )T. Ω (m) = (Ω (m) 1, Ω (m) 2,..., Ω (m) )T, γ (m) = (γ (m) 1, γ (m) N p 0 2,..., γ (m) Page 37 of 46 N )T, m
38 Now we introduce the Abel map A(p): A(p) = p p 0 ω, ω = (ω 1, ω 2,..., ω N ) T, A( p k ) = n k A(p k ), k and Abel-Jacobi coordinates: N N p(uk ρ (1) ) = A( p(u k )) = ω, p 0 ρ (2) = A( k=1 N p(v k )) = k=1 k=1 N k=1 p(vk ) p 0 ω, According to the Riemann theorem[?,?], there exists a Riemann constant vector M C N such that the function: Page 38 of 46
39 F (m) (λ) = θ(a(p(λ)) ρ (m) M (m) ), m = 1, 2. has exactly N zeros at u 1, u 2,..., u N for m = 1 or v 1, v 2,..., v N for m = 2. To make the function single valued, the surface Γ is cut along all a k, b k to form a simple connected region, whose boundary is denoted by γ. By Refs.[1.2], the integrals I(Γ) = 1 2πi γ λd ln F (m), m = 1, 2 are constants independent of ρ (1) and ρ (2) with N I = I(Γ) = λω j a j j=1 Page 39 of 46
40 By the residue theorem, we have: N 2 u j = I Res λ= s λd ln F (1) (λ) (5.9) j=1 N v j = I j=1 s=1 2 Res λ= s λd ln F (2) (λ) (5.10) s=1 Here we need only compute the residues in (5.9) and (5.10). In a way similar to calculations in [?], we arrive at Page 40 of 46
41 Res λ= s λd ln F (m) (λ) = ( 1) s+m ln θ (m) Where s, m = 1, 2; s = 1, 2. (5.11) θ s (1) = θ(ω (1) x+ω (2) t+ξ s ), θ s (2) = θ( Ω (1) x Ω (2) t+η s ), ξ s and η s are constants. Thus from (5.9)-(5.11), we arrive at N j=1 u j = I ln θ(1) 1 θ (1) 2, N j=1 v j = I ln θ(2) 2 θ (2) 1. (5.12) Page 41 of 46
42 Substituting (5.12) into (4.4), then we get an algebrogeometric solution for the coupled dimensional soliton equations(1.1): q = A(t) 2 exp( 1 2N+2 j=1 λ j I 2 ln θ(2) 2 + A(t) 2 exp( 1 2N+2 θ (2) 1 j=1 λ j I 2 ln θ(2) 1 θ (1) 2 1 exp( 2 ln θ(2) exp( 2 ln θ(2) 2 Page 42 of 46 θ (2) 1 θ (2) 1
43 r = A(t) 2 exp( 1 2N+2 A(t) 2 exp( 1 2N+2 j=1 λ j I 2 ln θ(2) 1 θ (1) 2 j=1 λ j I 2 ln θ(2) exp( 2 ln θ(2) 2 θ(1) 2 θ (2) 1 where A(t) is arbitrary complex functions about variable t. 1 exp( 2 ln θ 2 Page 43 of 46 θ (2) 1 θ(1) 1 θ 1
44 Arbitrary really 3 3 spectral problem, it is very difficult to calculate its algebraic geometry solutions. Page 44 of 46
45 [1].C. L. Siegel, Topics in Complex Function Theory, vol. 2, John Wiley and Sons, New York, NY,USA, [14] P.Griffiths and J.Harris, [2].Principles of Algebraic Geometry,Wiley Classics Library, John Wiley and Sons, New York, NY, USA, Page 45 of 46
46 Thank you for your attention! Page 46 of 46
2. Examples of Integrable Equations
Integrable equations A.V.Mikhailov and V.V.Sokolov 1. Introduction 2. Examples of Integrable Equations 3. Examples of Lax pairs 4. Structure of Lax pairs 5. Local Symmetries, conservation laws and the
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 informationALGEBRO-GEOMETRIC CONSTRAINTS ON SOLITONS WITH RESPECT TO QUASI-PERIODIC BACKGROUNDS
ALGEBRO-GEOMETRIC CONSTRAINTS ON SOLITONS WITH RESPECT TO QUASI-PERIODIC BACKGROUNDS GERALD TESCHL Abstract. We investigate the algebraic conditions that have to be satisfied by the scattering data of
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 informationS.Novikov. Singular Solitons and Spectral Theory
S.Novikov Singular Solitons and Spectral Theory Moscow, August 2014 Collaborators: P.Grinevich References: Novikov s Homepage www.mi.ras.ru/ snovikov click Publications, items 175,176,182, 184. New Results
More informationNumerical computation of the finite-genus solutions of the Korteweg-de Vries equation via Riemann Hilbert problems
Numerical computation of the finite-genus solutions of the Korteweg-de Vries equation via Riemann Hilbert problems Thomas Trogdon 1 and Bernard Deconinck Department of Applied Mathematics University of
More informationA PERIODICITY PROBLEM FOR THE KORTEWEG DE VRIES AND STURM LIOUVILLE EQUATIONS. THEIR CONNECTION WITH ALGEBRAIC GEOMETRY
A PERIODICITY PROBLEM FOR THE KORTEWEG DE VRIES AND STURM LIOUVILLE EQUATIONS. THEIR CONNECTION WITH ALGEBRAIC GEOMETRY B. A. DUBROVIN AND S. P. NOVIKOV 1. As was shown in the remarkable communication
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 informationInverse scattering technique in gravity
1 Inverse scattering technique in gravity The purpose of this chapter is to describe the Inverse Scattering Method ISM for the gravitational field. We begin in section 1.1 with a brief overview of the
More informationA Finite Genus Solution of the Veselov s Discrete Neumann System
Commun. Theor. Phys. 58 202 469 474 Vol. 58, No. 4, October 5, 202 A Finite Genus Solution of the Veselov s Discrete Neumann System CAO Ce-Wen and XU Xiao-Xue Æ Department of Mathematics, Zhengzhou University,
More informationNon-linear Equations of Korteweg de Vries Type, Finite-Zone Linear Operators, and Abelian Varieties. B.A.Dubrovin V.B.Matveev S.P.
Non-linear Equations of Korteweg de Vries Type, Finite-Zone Linear Operators, and Abelian Varieties B.A.Dubrovin V.B.Matveev S.P.Novikov Abstract. The basic concept of this survey is an exposition of a
More informationA new integrable system: The interacting soliton of the BO
Phys. Lett., A 204, p.336-342, 1995 A new integrable system: The interacting soliton of the BO Benno Fuchssteiner and Thorsten Schulze Automath Institute University of Paderborn Paderborn & Germany Abstract
More informationSystems of MKdV equations related to the affine Lie algebras
Integrability and nonlinearity in field theory XVII International conference on Geometry, Integrability and Quantization 5 10 June 2015, Varna, Bulgaria Systems of MKdV equations related to the affine
More informationDUAL HAMILTONIAN STRUCTURES IN AN INTEGRABLE HIERARCHY
DUAL HAMILTONIAN STRUCTURES IN AN INTEGRABLE HIERARCHY RAQIS 16, University of Geneva Somes references and collaborators Based on joint work with J. Avan, A. Doikou and A. Kundu Lagrangian and Hamiltonian
More informationOn Hamiltonian perturbations of hyperbolic PDEs
Bologna, September 24, 2004 On Hamiltonian perturbations of hyperbolic PDEs Boris DUBROVIN SISSA (Trieste) Class of 1+1 evolutionary systems w i t +Ai j (w)wj x +ε (B i j (w)wj xx + 1 2 Ci jk (w)wj x wk
More informationThe generalized Kupershmidt deformation for integrable bi-hamiltonian systems
The generalized Kupershmidt deformation for integrable bi-hamiltonian systems Yuqin Yao Tsinghua University, Beijing, China (with Yunbo Zeng July, 2009 catalogue 1 Introduction 2 KdV6 3 The generalized
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 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 informationPeriodic finite-genus solutions of the KdV equation are orbitally stable
Periodic finite-genus solutions of the KdV equation are orbitally stable Michael Nivala and Bernard Deconinck Department of Applied Mathematics, University of Washington, Campus Box 352420, Seattle, WA,
More informationNumerical solutions of the small dispersion limit of KdV, Whitham and Painlevé equations
Numerical solutions of the small dispersion limit of KdV, Whitham and Painlevé equations Tamara Grava (SISSA) joint work with Christian Klein (MPI Leipzig) Integrable Systems in Applied Mathematics Colmenarejo,
More informationNonlinear Fourier Analysis
Nonlinear Fourier Analysis The Direct & Inverse Scattering Transforms for the Korteweg de Vries Equation Ivan Christov Code 78, Naval Research Laboratory, Stennis Space Center, MS 99, USA Supported by
More informationMath 575-Lecture 26. KdV equation. Derivation of KdV
Math 575-Lecture 26 KdV equation We look at the KdV equations and the so-called integrable systems. The KdV equation can be written as u t + 3 2 uu x + 1 6 u xxx = 0. The constants 3/2 and 1/6 are not
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 informationA Method for Obtaining Darboux Transformations
Journal of Nonlinear Mathematical Physics 998, V.5, N, 40 48. Letter A Method for Obtaining Darbou Transformations Baoqun LU,YongHE and Guangjiong NI Department of Physics, Fudan University, 00433, Shanghai,
More informationBi-Integrable and Tri-Integrable Couplings and Their Hamiltonian Structures
University of South Florida Scholar Commons Graduate Theses and Dissertations Graduate School January 2012 Bi-Integrable and Tri-Integrable Couplings and Their Hamiltonian Structures Jinghan Meng University
More informationExact Solutions to the Focusing Discrete Nonlinear Schrödinger Equation
Exact Solutions to the Focusing Discrete Nonlinear Schrödinger Equation Francesco Demontis (based on a joint work with C. van der Mee) Università degli Studi di Cagliari Dipartimento di Matematica e Informatica
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 informationAn inverse numerical range problem for determinantal representations
An inverse numerical range problem for determinantal representations Mao-Ting Chien Soochow University, Taiwan Based on joint work with Hiroshi Nakazato WONRA June 13-18, 2018, Munich Outline 1. Introduction
More informationContinuous limits and integrability for a semidiscrete system Zuo-nong Zhu Department of Mathematics, Shanghai Jiao Tong University, P R China
Continuous limits and integrability for a semidiscrete system Zuo-nong Zhu Department of Mathematics, Shanghai Jiao Tong University, P R China the 3th GCOE International Symposium, Tohoku University, 17-19
More informationA i A j = 0. A i B j = δ i,j. A i B j = 1. A i. B i B j = 0 B j
Review of Riemann Surfaces Let X be a Riemann surface (one complex dimensional manifold) of genus g. Then (1) There exist curves A 1,, A g, B 1,, B g with A i A j = 0 A i B j = δ i,j B i B j = 0 B j A
More informationTHE SECOND PAINLEVÉ HIERARCHY AND THE STATIONARY KDV HIERARCHY
THE SECOND PAINLEVÉ HIERARCHY AND THE STATIONARY KDV HIERARCHY N. JOSHI Abstract. It is well known that soliton equations such as the Korteweg-de Vries equation are members of infinite sequences of PDEs
More informationINVERSE SCATTERING TRANSFORM, KdV, AND SOLITONS
INVERSE SCATTERING TRANSFORM, KdV, AND SOLITONS Tuncay Aktosun Department of Mathematics and Statistics Mississippi State University Mississippi State, MS 39762, USA Abstract: In this review paper, the
More informationBaker-Akhiezer functions and configurations of hyperplanes
Baker-Akhiezer functions and configurations of hyperplanes Alexander Veselov, Loughborough University ENIGMA conference on Geometry and Integrability, Obergurgl, December 2008 Plan BA function related
More informationBernard Deconinck and Harvey Segur Here }(z) denotes the Weierstra elliptic function. It can be dened by its meromorphic expansion }(z) =! X z + (z +
Pole Dynamics for Elliptic Solutions of the Korteweg-deVries Equation Bernard Deconinck and Harvey Segur March, 999 Abstract The real, nonsingular elliptic solutions of the Korteweg-deVries equation are
More informationarxiv: v2 [math.fa] 19 Oct 2014
P.Grinevich, S.Novikov 1 Spectral Meromorphic Operators and Nonlinear Systems arxiv:1409.6349v2 [math.fa] 19 Oct 2014 LetusconsiderordinarydifferentiallinearoperatorsL = n x + n i 2 a i n i x with x-meromorphic
More informationFourier Transform, Riemann Surfaces and Indefinite Metric
Fourier Transform, Riemann Surfaces and Indefinite Metric P. G. Grinevich, S.P.Novikov Frontiers in Nonlinear Waves, University of Arizona, Tucson, March 26-29, 2010 Russian Math Surveys v.64, N.4, (2009)
More informationClassical and quantum aspects of ultradiscrete solitons. Atsuo Kuniba (Univ. Tokyo) 2 April 2009, Glasgow
Classical and quantum aspects of ultradiscrete solitons Atsuo Kuniba (Univ. Tokyo) 2 April 29, Glasgow Tau function of KP hierarchy ( N τ i (x) = i e H(x) exp j=1 ) c j ψ(p j )ψ (q j ) i (e H(x) = time
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 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 informationBreathers and solitons of generalized nonlinear Schrödinger equations as degenerations of algebro-geometric solutions
Breathers and solitons of generalized nonlinear Schrödinger equations as degenerations of algebro-geometric solutions Caroline Kalla To cite this version: Caroline Kalla. Breathers and solitons of generalized
More informationDeterminant Expressions for Discrete Integrable Maps
Typeset with ps2.cls Full Paper Determinant Expressions for Discrete Integrable Maps Kiyoshi Sogo Department of Physics, School of Science, Kitasato University, Kanagawa 228-8555, Japan Explicit
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 informationContinuous and Discrete Homotopy Operators with Applications in Integrability Testing. Willy Hereman
Continuous and Discrete Homotopy Operators with Applications in Integrability Testing Willy Hereman Department of Mathematical and Computer Sciences Colorado School of Mines Golden, Colorado, USA http://www.mines.edu/fs
More informationHyperelliptic Jacobians in differential Galois theory (preliminary report)
Hyperelliptic Jacobians in differential Galois theory (preliminary report) Jerald J. Kovacic Department of Mathematics The City College of The City University of New York New York, NY 10031 jkovacic@member.ams.org
More informationRecursion Systems and Recursion Operators for the Soliton Equations Related to Rational Linear Problem with Reductions
GMV The s Systems and for the Soliton Equations Related to Rational Linear Problem with Reductions Department of Mathematics & Applied Mathematics University of Cape Town XIV th International Conference
More informationElliptic Functions. Introduction
Elliptic Functions Introduction 1 0.1 What is an elliptic function Elliptic function = Doubly periodic meromorphic function on C. Too simple object? Indeed, in most of modern textbooks on the complex analysis,
More informationarxiv: v2 [nlin.si] 27 Jun 2015
On auto and hetero Bäcklund transformations for the Hénon-Heiles systems A. V. Tsiganov St.Petersburg State University, St.Petersburg, Russia e mail: andrey.tsiganov@gmail.com arxiv:1501.06695v2 [nlin.si]
More informationBäcklund transformation and special solutions for Drinfeld Sokolov Satsuma Hirota system of coupled equations
arxiv:nlin/0102001v1 [nlin.si] 1 Feb 2001 Bäcklund transformation and special solutions for Drinfeld Sokolov Satsuma Hirota system of coupled equations Ayşe Karasu (Kalkanli) and S Yu Sakovich Department
More informationInfinite Sequence Soliton-Like Exact Solutions of (2 + 1)-Dimensional Breaking Soliton Equation
Commun. Theor. Phys. 55 (0) 949 954 Vol. 55, No. 6, June 5, 0 Infinite Sequence Soliton-Like Exact Solutions of ( + )-Dimensional Breaking Soliton Equation Taogetusang,, Sirendaoerji, and LI Shu-Min (Ó
More informationPartition function of one matrix-models in the multi-cut case and Painlevé equations. Tamara Grava and Christian Klein
Partition function of one matrix-models in the multi-cut case and Painlevé equations Tamara Grava and Christian Klein SISSA IMS, Singapore, March 2006 Main topics Large N limit beyond the one-cut case
More informationNUMERICAL METHODS FOR SOLVING NONLINEAR EVOLUTION EQUATIONS
NUMERICAL METHODS FOR SOLVING NONLINEAR EVOLUTION EQUATIONS Thiab R. Taha Computer Science Department University of Georgia Athens, GA 30602, USA USA email:thiab@cs.uga.edu Italy September 21, 2007 1 Abstract
More informationDiagonalization of the Coupled-Mode System.
Diagonalization of the Coupled-Mode System. Marina Chugunova joint work with Dmitry Pelinovsky Department of Mathematics, McMaster University, Canada Collaborators: Mason A. Porter, California Institute
More informationOF THE PEAKON DYNAMICS
Pacific Journal of Applied Mathematics Volume 1 umber 4 pp. 61 65 ISS 1941-3963 c 008 ova Science Publishers Inc. A OTE O r-matrix OF THE PEAKO DYAMICS Zhijun Qiao Department of Mathematics The University
More informationPublic-key Cryptography: Theory and Practice
Public-key Cryptography Theory and Practice Department of Computer Science and Engineering Indian Institute of Technology Kharagpur Chapter 2: Mathematical Concepts Divisibility Congruence Quadratic Residues
More informationRandom Eigenvalue Problems Revisited
Random Eigenvalue Problems Revisited S Adhikari Department of Aerospace Engineering, University of Bristol, Bristol, U.K. Email: S.Adhikari@bristol.ac.uk URL: http://www.aer.bris.ac.uk/contact/academic/adhikari/home.html
More informationKnot types, Floquet spectra, and finite-gap solutions of the vortex filament equation
Mathematics and Computers in Simulation 55 (2001) 341 350 Knot types, Floquet spectra, and finite-gap solutions of the vortex filament equation Annalisa M. Calini, Thomas A. Ivey a Department of Mathematics,
More informationTranscendents defined by nonlinear fourth-order ordinary differential equations
J. Phys. A: Math. Gen. 3 999) 999 03. Printed in the UK PII: S0305-447099)9603-6 Transcendents defined by nonlinear fourth-order ordinary differential equations Nicolai A Kudryashov Department of Applied
More information2 Soliton Perturbation Theory
Revisiting Quasistationary Perturbation Theory for Equations in 1+1 Dimensions Russell L. Herman University of North Carolina at Wilmington, Wilmington, NC Abstract We revisit quasistationary perturbation
More informationNew Analytical Solutions For (3+1) Dimensional Kaup-Kupershmidt Equation
International Conference on Computer Technology and Science (ICCTS ) IPCSIT vol. 47 () () IACSIT Press, Singapore DOI:.776/IPCSIT..V47.59 New Analytical Solutions For () Dimensional Kaup-Kupershmidt Equation
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 informationZeta Functions and Regularized Determinants for Elliptic Operators. Elmar Schrohe Institut für Analysis
Zeta Functions and Regularized Determinants for Elliptic Operators Elmar Schrohe Institut für Analysis PDE: The Sound of Drums How Things Started If you heard, in a dark room, two drums playing, a large
More informationH = d 2 =dx 2 +q(x) in L 2 (R). Our construction is new in the present general complexvalued periodic nite-gap case. Before describing our approach in
PICARD POTENTIALS AND HILL'S EQUATION ON A TORUS F. GESZTESY 1 AND R. WEIKARD 2 Abstract. An explicit characterization of all elliptic (algebro-geometric) nitegap solutions of the KdV hierarchy is presented.
More informationInverse Scattering Transform and the Theory of Solitons
Inverse Scattering Transform and the Theory of Solitons TUNCAY AKTOSUN ab a University of Texas at Arlington Arlington Texas USA b Supported in part by the National Science Foundation under grant DMS-0610494
More informationUniversidad del Valle. Equations of Lax type with several brackets. Received: April 30, 2015 Accepted: December 23, 2015
Universidad del Valle Equations of Lax type with several brackets Raúl Felipe Centro de Investigación en Matemáticas Raúl Velásquez Universidad de Antioquia Received: April 3, 215 Accepted: December 23,
More informationTHE PERIODIC PROBLEM FOR THE KORTEWEG DE VRIES EQUATION
THE PERIODIC PROBLEM FOR THE KORTEWEG DE VRIES EQUATION S. P. NOVIKOV Introduction The Korteweg de Vries equation (KV arose in the nineteenth century in connection with the theory of waves in shallow water.
More informationPeriodic and almost-periodic potentials in inverse problems
Inverse Problems 15 (1999) R117 R144. Printed in the UK PII: S0266-5611(99)77336-3 TOPICAL REVIEW Periodic and almost-periodic potentials in inverse problems I Krichever andspnovikov Columbia University,
More informationComputable Integrability. Chapter 2: Riccati equation
Computable Integrability. Chapter 2: Riccati equation E. Kartashova, A. Shabat Contents 1 Introduction 2 2 General solution of RE 2 2.1 a(x) = 0.............................. 2 2.2 a(x) 0..............................
More informationFinding Eigenvalue Problems for Solving Nonlinear Evolution Equations*>
72 Progress of Theoretical Physics. Vol. 54, No. 1, July 1975 Finding Eigenvalue Problems for Solving Nonlinear Evolution Equations*> D. ]. KAUP Department of Physics, Clarkson College of Technology, Potsdam
More informationarxiv:math/ v2 [math.ca] 28 Oct 2008
THE HERMITE-KRICHEVER ANSATZ FOR FUCHSIAN EQUATIONS WITH APPLICATIONS TO THE SIXTH PAINLEVÉ EQUATION AND TO FINITE-GAP POTENTIALS arxiv:math/0504540v [math.ca] 8 Oct 008 KOUICHI TAKEMURA Abstract. Several
More informationPutzer s Algorithm. Norman Lebovitz. September 8, 2016
Putzer s Algorithm Norman Lebovitz September 8, 2016 1 Putzer s algorithm The differential equation dx = Ax, (1) dt where A is an n n matrix of constants, possesses the fundamental matrix solution exp(at),
More informationIntegrable Couplings of the Kaup-Newell Soliton Hierarchy
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,
More informationFreedom in the Expansion and Obstacles to Integrability in Multiple-Soliton Solutions of the Perturbed KdV Equation
Freedom in the Expansion and Obstacles to Integrability in Multiple-Soliton Solutions of the Perturbed KdV Equation Alex Veksler 1 and Yair Zarmi 1, Ben-Gurion University of the Negev, Israel 1 Department
More informationTHE LAX PAIR FOR THE MKDV HIERARCHY. Peter A. Clarkson, Nalini Joshi & Marta Mazzocco
Séminaires & Congrès 14, 006, p. 53 64 THE LAX PAIR FOR THE MKDV HIERARCHY by Peter A. Clarkson, Nalini Joshi & Marta Mazzocco Abstract. In this paper we give an algorithmic method of deriving the Lax
More informationExact Solutions of the Generalized- Zakharov (GZ) Equation by the Infinite Series Method
Available at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 193-9466 Vol. 05, Issue (December 010), pp. 61 68 (Previously, Vol. 05, Issue 10, pp. 1718 175) Applications and Applied Mathematics: An International
More informationRandom Matrix Eigenvalue Problems in Probabilistic Structural Mechanics
Random Matrix Eigenvalue Problems in Probabilistic Structural Mechanics S Adhikari Department of Aerospace Engineering, University of Bristol, Bristol, U.K. URL: http://www.aer.bris.ac.uk/contact/academic/adhikari/home.html
More informationComputational Problems Using Riemann Theta Functions in Sage
Computational Problems Using Riemann Theta Functions in Sage Chris Swierczewski University of Washington Department of Applied Mathematics cswiercz@uw.edu 2011 AMS Western Fall Sectional Salt Lake City,
More informationMACSYMA PROGRAM FOR THE PAINLEVÉ TEST FOR NONLINEAR ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS
. MACSYMA PROGRAM FOR THE PAINLEVÉ TEST FOR NONLINEAR ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS Willy Hereman Mathematics Department and Center for the Mathematical Sciences University of Wisconsin at
More informationLecture 10: Whitham Modulation Theory
Lecture 10: Whitham Modulation Theory Lecturer: Roger Grimshaw. Write-up: Andong He June 19, 2009 1 Introduction The Whitham modulation theory provides an asymptotic method for studying slowly varying
More informationReductions and New types of Integrable models in two and more dimensions
0-0 XIV International conference Geometry Integrabiliy and Quantization Varna, 8-13 June 2012, Bulgaria Reductions and New types of Integrable models in two and more dimensions V. S. Gerdjikov Institute
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 informationThe Higher Dimensional Bateman Equation and Painlevé Analysis of Nonintegrable Wave Equations
Symmetry in Nonlinear Mathematical Physics 1997, V.1, 185 192. The Higher Dimensional Bateman Equation and Painlevé Analysis of Nonintegrable Wave Equations Norbert EULER, Ove LINDBLOM, Marianna EULER
More informationVariational Discretization of Euler s Elastica Problem
Typeset with jpsj.cls Full Paper Variational Discretization of Euler s Elastica Problem Kiyoshi Sogo Department of Physics, School of Science, Kitasato University, Kanagawa 8-8555, Japan A discrete
More informationExact Travelling Wave Solutions of the Coupled Klein-Gordon Equation by the Infinite Series Method
Available at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 93-9466 Vol. 6, Issue (June 0) pp. 3 3 (Previously, Vol. 6, Issue, pp. 964 97) Applications and Applied Mathematics: An International Journal (AAM)
More informationHamiltonian and quasi-hamiltonian structures associated with semi-direct sums of Lie algebras
INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND GENERAL J. Phys. A: Math. Gen. 39 (2006) 10787 10801 doi:10.1088/0305-4470/39/34/013 Hamiltonian and quasi-hamiltonian structures
More informationarxiv: v2 [nlin.si] 9 Aug 2017
Whitham modulation theory for the Kadomtsev-Petviashvili equation Mark J. Ablowitz 1 Gino Biondini 23 and Qiao Wang 2 1 Department of Applied Mathematics University of Colorado Boulder CO 80303 2 Department
More informationNew Integrable Decomposition of Super AKNS Equation
Commun. Theor. Phys. (Beijing, China) 54 (2010) pp. 803 808 c Chinese Physical Society and IOP Publishing Ltd Vol. 54, No. 5, November 15, 2010 New Integrable Decomposition of Super AKNS Equation JI Jie
More informationThe (G'/G) - Expansion Method for Finding Traveling Wave Solutions of Some Nonlinear Pdes in Mathematical Physics
Vol.3, Issue., Jan-Feb. 3 pp-369-376 ISSN: 49-6645 The ('/) - Expansion Method for Finding Traveling Wave Solutions of Some Nonlinear Pdes in Mathematical Physics J.F.Alzaidy Mathematics Department, Faculty
More informationNew Exact Solutions for the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli Equation
Applied Mathematical Sciences, Vol. 6, 2012, no. 12, 579-587 New Exact Solutions for the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli Equation Ying Li and Desheng Li School of Mathematics and System Science
More informationTHE SELBERG TRACE FORMULA OF COMPACT RIEMANN SURFACES
THE SELBERG TRACE FORMULA OF COMPACT RIEMANN SURFACES IGOR PROKHORENKOV 1. Introduction to the Selberg Trace Formula This is a talk about the paper H. P. McKean: Selberg s Trace Formula as applied to a
More informationMonopoles, Periods and Problems
Bath 2010 Monopole Results in collaboration with V.Z. Enolskii, A.D Avanzo. Spectral curve programs with T.Northover. Overview Equations Zero Curvature/Lax Spectral Curve C S Reconstruction Baker-Akhiezer
More informationKinetic equation for a dense soliton gas
Loughborough University Institutional Repository Kinetic equation for a dense soliton gas This item was submitted to Loughborough University's Institutional Repository by the/an author. Additional Information:
More informationA Numerical Solution of the Lax s 7 th -order KdV Equation by Pseudospectral Method and Darvishi s Preconditioning
Int. J. Contemp. Math. Sciences, Vol. 2, 2007, no. 22, 1097-1106 A Numerical Solution of the Lax s 7 th -order KdV Equation by Pseudospectral Method and Darvishi s Preconditioning M. T. Darvishi a,, S.
More informationTWO SOLITON SOLUTION OF TZITZEICA EQUATION. Bulgarian Academy of Sciences, Sofia, Bulgaria
TWO SOLITON SOLUTION OF TZITZEICA EQUATION Corina N. Babalic 1,2, Radu Constantinescu 1 and Vladimir S. Gerdjikov 3 1 Dept. of Physics, University of Craiova, Romania 2 Dept. of Theoretical Physics, NIPNE,
More informationInitial-Boundary Value Problem for Two-Component Gerdjikov Ivanov Equation with 3 3 Lax Pair on Half-Line
Commun. Theor. Phys. 68 27 425 438 Vol. 68, No. 4, October, 27 Initial-Boundary Value Problem for Two-Component Gerdjiov Ivanov Equation with 3 3 Lax Pair on Half-Line Qiao-Zhen Zhu 朱巧珍,, En-Gui Fan 范恩贵,,
More informationFermionic coherent states in infinite dimensions
Fermionic coherent states in infinite dimensions Robert Oeckl Centro de Ciencias Matemáticas Universidad Nacional Autónoma de México Morelia, Mexico Coherent States and their Applications CIRM, Marseille,
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 informationThe elliptic sinh-gordon equation in the half plane
Available online at www.tjnsa.com J. Nonlinear Sci. Appl. 8 25), 63 73 Research Article The elliptic sinh-gordon equation in the half plane Guenbo Hwang Department of Mathematics, Daegu University, Gyeongsan
More informationOn Local Time-Dependent Symmetries of Integrable Evolution Equations
Proceedings of Institute of Mathematics of NAS of Ukraine 2000, Vol. 30, Part 1, 196 203. On Local Time-Dependent Symmetries of Integrable Evolution Equations A. SERGYEYEV Institute of Mathematics of the
More informationDispersive and soliton perturbations of finite-genus solutions of the KdV equation: computational results
Dispersive and soliton perturbations of finite-genus solutions of the KdV equation: computational results Thomas Trogdon and Bernard Deconinck Department of Applied Mathematics University of Washington
More informationIterative Methods for Ax=b
1 FUNDAMENTALS 1 Iterative Methods for Ax=b 1 Fundamentals consider the solution of the set of simultaneous equations Ax = b where A is a square matrix, n n and b is a right hand vector. We write the iterative
More information