A Short historical review Our goal The hierarchy and Lax... The Hamiltonian... The Dubrovin-type... Algebro-geometric... Home Page.

Size: px
Start display at page:

Download "A Short historical review Our goal The hierarchy and Lax... The Hamiltonian... The Dubrovin-type... Algebro-geometric... Home Page."

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

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 information

Reductions to Korteweg-de Vries Soliton Hierarchy

Reductions 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 information

ALGEBRO-GEOMETRIC CONSTRAINTS ON SOLITONS WITH RESPECT TO QUASI-PERIODIC BACKGROUNDS

ALGEBRO-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 information

ON 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 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 information

S.Novikov. Singular Solitons and Spectral Theory

S.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 information

Numerical 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 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 information

A 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 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 information

Lax Representations and Zero Curvature Representations by Kronecker Product

Lax 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 information

Inverse scattering technique in gravity

Inverse 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 information

A Finite Genus Solution of the Veselov s Discrete Neumann System

A 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 information

Non-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. 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 information

A new integrable system: The interacting soliton of the BO

A 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 information

Systems of MKdV equations related to the affine Lie algebras

Systems 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 information

DUAL HAMILTONIAN STRUCTURES IN AN INTEGRABLE HIERARCHY

DUAL 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 information

On Hamiltonian perturbations of hyperbolic PDEs

On 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 information

The generalized Kupershmidt deformation for integrable bi-hamiltonian systems

The 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 information

Systematic construction of (boundary) Lax pairs

Systematic 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 information

Canonically conjugate variables for the periodic Camassa-Holm equation

Canonically 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 information

Periodic finite-genus solutions of the KdV equation are orbitally stable

Periodic 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 information

Numerical 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 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 information

Nonlinear Fourier Analysis

Nonlinear 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 information

Math 575-Lecture 26. KdV equation. Derivation of KdV

Math 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 information

On universality of critical behaviour in Hamiltonian PDEs

On 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 information

A Method for Obtaining Darboux Transformations

A 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 information

Bi-Integrable and Tri-Integrable Couplings and Their Hamiltonian Structures

Bi-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 information

Exact Solutions to the Focusing Discrete Nonlinear Schrödinger Equation

Exact 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 information

GLASGOW Paolo Lorenzoni

GLASGOW 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 information

An inverse numerical range problem for determinantal representations

An 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 information

Continuous 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 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 information

A 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

A 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 information

THE SECOND PAINLEVÉ HIERARCHY AND THE STATIONARY KDV HIERARCHY

THE 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 information

INVERSE SCATTERING TRANSFORM, KdV, AND SOLITONS

INVERSE 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 information

Baker-Akhiezer functions and configurations of hyperplanes

Baker-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 information

Bernard Deconinck and Harvey Segur Here }(z) denotes the Weierstra elliptic function. It can be dened by its meromorphic expansion }(z) =! X z + (z +

Bernard 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 information

arxiv: v2 [math.fa] 19 Oct 2014

arxiv: 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 information

Fourier Transform, Riemann Surfaces and Indefinite Metric

Fourier 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 information

Classical 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 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 information

The Weierstrass Theory For Elliptic Functions. The Burn 2007

The 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 information

A New Integrable Couplings of Classical-Boussinesq Hierarchy with Self-Consistent Sources

A 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 information

Breathers 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 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 information

Determinant Expressions for Discrete Integrable Maps

Determinant 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 information

Prolongation structure for nonlinear integrable couplings of a KdV soliton hierarchy

Prolongation 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 information

Continuous and Discrete Homotopy Operators with Applications in Integrability Testing. Willy Hereman

Continuous 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 information

Hyperelliptic Jacobians in differential Galois theory (preliminary report)

Hyperelliptic 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 information

Recursion Systems and Recursion Operators for the Soliton Equations Related to Rational Linear Problem with Reductions

Recursion 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 information

Elliptic Functions. Introduction

Elliptic 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 information

arxiv: v2 [nlin.si] 27 Jun 2015

arxiv: 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 information

Bäcklund transformation and special solutions for Drinfeld Sokolov Satsuma Hirota system of coupled equations

Bäcklund 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 information

Infinite Sequence Soliton-Like Exact Solutions of (2 + 1)-Dimensional Breaking Soliton Equation

Infinite 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 information

Partition 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 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 information

NUMERICAL METHODS FOR SOLVING NONLINEAR EVOLUTION EQUATIONS

NUMERICAL 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 information

Diagonalization of the Coupled-Mode System.

Diagonalization 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 information

OF THE PEAKON DYNAMICS

OF 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 information

Public-key Cryptography: Theory and Practice

Public-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 information

Random Eigenvalue Problems Revisited

Random 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 information

Knot types, Floquet spectra, and finite-gap solutions of the vortex filament equation

Knot 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 information

Transcendents defined by nonlinear fourth-order ordinary differential equations

Transcendents 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 information

2 Soliton Perturbation Theory

2 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 information

New Analytical Solutions For (3+1) Dimensional Kaup-Kupershmidt Equation

New 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 information

Journal of Geometry and Physics

Journal 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 information

Zeta 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 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 information

H = 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

H = 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 information

Inverse Scattering Transform and the Theory of Solitons

Inverse 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 information

Universidad 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. 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 information

THE PERIODIC PROBLEM FOR THE KORTEWEG DE VRIES EQUATION

THE 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 information

Periodic and almost-periodic potentials in inverse problems

Periodic 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 information

Computable Integrability. Chapter 2: Riccati equation

Computable 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 information

Finding Eigenvalue Problems for Solving Nonlinear Evolution Equations*>

Finding 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 information

arxiv:math/ v2 [math.ca] 28 Oct 2008

arxiv: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 information

Putzer s Algorithm. Norman Lebovitz. September 8, 2016

Putzer 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 information

Integrable Couplings of the Kaup-Newell Soliton Hierarchy

Integrable 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 information

Freedom 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 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 information

THE LAX PAIR FOR THE MKDV HIERARCHY. Peter A. Clarkson, Nalini Joshi & Marta Mazzocco

THE 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 information

Exact Solutions of the Generalized- Zakharov (GZ) Equation by the Infinite Series Method

Exact 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 information

Random Matrix Eigenvalue Problems in Probabilistic Structural Mechanics

Random 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 information

Computational Problems Using Riemann Theta Functions in Sage

Computational 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 information

MACSYMA 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 . 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 information

Lecture 10: Whitham Modulation Theory

Lecture 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 information

Reductions and New types of Integrable models in two and more dimensions

Reductions 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 information

A MULTI-COMPONENT LAX INTEGRABLE HIERARCHY WITH HAMILTONIAN STRUCTURE

A 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 information

The Higher Dimensional Bateman Equation and Painlevé Analysis of Nonintegrable Wave Equations

The 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 information

Variational Discretization of Euler s Elastica Problem

Variational 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 information

Exact Travelling Wave Solutions of the Coupled Klein-Gordon Equation by the Infinite Series Method

Exact 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 information

Hamiltonian and quasi-hamiltonian structures associated with semi-direct sums of Lie algebras

Hamiltonian 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 information

arxiv: v2 [nlin.si] 9 Aug 2017

arxiv: 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 information

New Integrable Decomposition of Super AKNS Equation

New 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 information

The (G'/G) - Expansion Method for Finding Traveling Wave Solutions of Some Nonlinear Pdes in Mathematical Physics

The (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 information

New Exact Solutions for the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli Equation

New 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 information

THE SELBERG TRACE FORMULA OF COMPACT RIEMANN SURFACES

THE 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 information

Monopoles, Periods and Problems

Monopoles, 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 information

Kinetic equation for a dense soliton gas

Kinetic 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 information

A Numerical Solution of the Lax s 7 th -order KdV Equation by Pseudospectral Method and Darvishi s Preconditioning

A 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 information

TWO SOLITON SOLUTION OF TZITZEICA EQUATION. Bulgarian Academy of Sciences, Sofia, Bulgaria

TWO 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 information

Initial-Boundary Value Problem for Two-Component Gerdjikov Ivanov Equation with 3 3 Lax Pair on Half-Line

Initial-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 information

Fermionic coherent states in infinite dimensions

Fermionic 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 information

Breaking soliton equations and negative-order breaking soliton equations of typical and higher orders

Breaking 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 information

The elliptic sinh-gordon equation in the half plane

The 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 information

On Local Time-Dependent Symmetries of Integrable Evolution Equations

On 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 information

Dispersive 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 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 information

Iterative Methods for Ax=b

Iterative 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