121269622
A class of Neumann type systems and its application
A class of Neumann type systems are derived separating the spatial and temporal variables for the 2+1 dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada (CDGKS) equation and the modified Korteweg-de Vries (mKdV) hierarchy. The Lax-Moser matrix of Neumann type systems is worked out, which generates a sequence of integrals of motion and a hyperelliptic curve of KdV type. We deduce the constrained Hamiltonians to put Neumann type systems into canonical Hamiltonian equations and further complete the Liouville integrability for the Neumann type systems. We also specify the relationship between Neumann type systems and infinite dimensional integrable systems (IDISs), where the involutivity solutions of Neumann type systems yield the finite parametric solutions of IDISs. From the Abel-Jacobi variables, the evolution behavior of Neumann type flows are shown on the Jacobian of a Riemann surface. Finally, the Neumann type flows are applied to produce some explicit solutions expressed by Riemann theta functions for the 2+1 dimensional CDGKS equation and the mKdV hierarchy.
121979891
Hill’s Operator and Hyperelliptic Function Theory in the Presence of Infinitely Many Branch Points
\(C_{1}^{k},\,k \leq \infty \), is the class of k times continuously differentiable real-valued functions of period 1. Q denotes the Hill’s operator − d2∕dx2 + q(x) with a fixed q of class \(C_{1}^{\infty }\). The function y1(x, λ), respectively y2(x, λ), is the solution of Qy = λ y with \(y_{1}(0,\lambda ) = 1,\,y_{1}'(0,\lambda ) = 0\), respectively \(y_{2}(0,\lambda ) = 0,\,y_{2}'(0,\lambda ) = 1\).
119721745
Geometry of Quadrics and Spectral Theory
In this paper we are concerned with integrable Hamiltonian systems. This concept goes back to classical analytical dynamics of the last century. Briefly these are nonlinear systems of ordinary differential equations described by a Hamiltonian function and possessing sufficiently many integrals (or conserved quantities) so that they are more or less explicitly solvable by quadrature. Therefore these systems played a crucial role in the last century before more qualitative methods for differential equations were developed at the turn of the century. Subsequently interest in these systems decreased, partly due to the realization that the existence of global integrals can be established only for exceptional Hamiltonian systems.
117550519
Classical Integrable Systems Generated Through Nonlinearization of Eigenvalue Problems
It is a challenge for us to look for new finite-dimensional completely integrable systems. H. Flaschka [l] pointed out an important principle to obtain finite-dimensional integrable systems by constraining infinite-dimensional integrable systems on finite-dimensional invariant subset.
123096891
Dynamical r-matrices for the constrained Harry-Dym flows
Abstract The classical Poisson structures and r -matrices for the geodesic flow in R N and the G 0 -constrained Harry-Dym flow are studied. The integrabilities of the constrained flows are obtained in the r -matrix setting.
121363075
Neumann type integrable reduction for nonlinear evolution equations in 1+1 and 2+1 dimensions
A family of new Neumann type systems is given in view of the nonlinearization technique, realizing the variable separation of the modified Jaulent–Miodek hierarchy and a new coupled modified Kadomtsev–Petviashvili equation on the symplectic submanifold TSN−1. By two Casimir functions and a special solution of the Lenard eigenvalue equation, we deduce the Lax–Moser matrix of the Neumann type systems that yields integrals of motion and the constrained Hamiltonians whose vector fields are tangent to TSN−1. Based on the Dirac–Poisson bracket and a Lax equation on TSN−1, a new systematic way is proposed to prove the Liouville integrability of a family of Neumann type systems synchronously. The Dirac–Poisson bracket and the generating function are used to reveal the explicit relation between infinite dimensional integrable systems and Neumann type systems, and then we point out that compatible solutions of Neumann type systems yield the finite parametric solutions of 1+1 and 2+1 dimensional integrable nonlinear...
121543607
The Camassa-Holm Hierarchy, N-Dimensional Integrable Systems, and Algebro-Geometric Solution on a Symplectic Submanifold
This paper shows that the Camassa-Holm (CH) spectral problem yields two different integrable hierarchies of nonlinear evolution equations (NLEEs), one is of negative order CH hierachy while the other one is of positive order CH hierarchy. The two CH hierarchies possess the zero curvature representations through solving a key matrix equation. We see that the well-known CH equation is included in the negative order CH hierarchy while the Dym type equation is included in the positive order CH hierarchy. Furthermore, under two constraint conditions between the potentials and the eigenfunctions, the CH spectral problem is cast in: 1. a new Neumann-like N-dimensional system when it is restricted into a symplectic submanifold of ℝ2N which is proven to be integrable by using the Dirac-Poisson bracket and the r-matrix process; and 2. a new Bargmann-like N-dimensional system when it is considered in the whole ℝ2N which is proven to be integrable by using the standard Poisson bracket and the r-matrix process.
119367879
Integrable Hierarchy, 3×3 Constrained Systems, and Parametric Solutions
This paper provides a new integrable hierarchy. The DP equation: mt+umx+3mux=0, m=u−uxx, proposed recently by Degasperis and Procesi, is the first member in the negative order hierarchy while the first equation in the positive order hierarchy is: mt=4(m−2/3)x−5(m−2/3)xxx+(m−2/3)xxxxx. The whole hierarchy is shown Lax-integrable through solving a key matrix equation. To obtain the parametric solutions for the whole hierarchy, we separately discuss the negative order and the positive order hierarchies. For the negative order hierarchy, its 3×3 Lax pairs and corresponding adjoint representations are cast in Liouville-integrable Hamiltonian canonical systems under the Dirac–Poisson bracket defined on a symplectic submanifold of R6N. Based on the integrability of those finite-dimensional canonical Hamiltonian systems we give the parametric solutions of all equations in the negative order hierarchy. In particular, we obtain the parametric solution of the DP equation. Moreover, for the positive order hierarchy, we consider a different constraint and process a procedure similar to the negative case to obtain the parametric solutions of the positive order hierarchy. In a special case, we give the parametric solution of the 5th-order PDE mt=4(m−2/3)x−5(m−2/3)xxx+(m−2/3)xxxxx. Finally, we discuss the stationary solutions of the 5th-order PDE, which may be included in the parametric solution.
7503124
The Complex Geometry of Weak Piecewise Smooth Solutions of Integrable Nonlinear PDE's¶of Shallow Water and Dym Type
Abstract: An extension of the algebraic-geometric method for nonlinear integrable PDE's is shown to lead to new piecewise smooth weak solutions of a class of N-component systems of nonlinear evolution equations. This class includes, among others, equations from the Dym and shallow water equation hierarchies. The main goal of the paper is to give explicit theta-functional expressions for piecewise smooth weak solutions of these nonlinear PDE's, which are associated to nonlinear subvarieties of hyperelliptic Jacobians.The main results of the present paper are twofold. First, we exhibit some of the special features of integrable PDE's that admit piecewise smooth weak solutions, which make them different from equations whose solutions are globally meromorphic, such as the KdV equation. Second, we blend the techniques of algebraic geometry and weak solutions of PDE's to gain further insight into, and explicit formulas for, piecewise-smooth finite-gap solutions.The basic technique used to achieve these aims is rather different from earlier papers dealing with peaked solutions. First, profiles of the finite-gap piecewise smooth solutions are linked to certain finite dimensional billiard dynamical systems and ellipsoidal billiards. Second, after reducing the solution of certain finite dimensional Hamiltonian systems on Riemann surfaces to the solution of a nonstandard Jacobi inversion problem, this is resolved by introducing new parametrizations.Amongst other natural consequences of the algebraic-geometric approach, we find finite dimensional integrable Hamiltonian dynamical systems describing the motion of peaks in the finite-gap as well as the limiting (soliton) cases, and solve them exactly. The dynamics of the peaks is also obtained by using Jacobi inversion problems. Finally, we relate our method to the shock wave approach for weak solutions of wave equations by determining jump conditions at the peak location.
120495313
The finite-band solution of the Jaulent–Miodek equation
A method to construct the finite-band solution to the soliton equation is presented. We take the Jaulent–Miodek equation as an example. With the use of the nonlinearization of Lax pair, the Jaulent–Miodek equation is decomposed into two finite-dimensional integrable systems, whose properties are studied from the view of an r-matrix. Then through solving these finite-dimensional integrable systems, the finite-band solution of the Jaulent–Miodek equation is obtained.
122242623
Relation between the Kadometsev–Petviashvili equation and the confocal involutive system
The special quasiperiodic solution of the (2+1)-dimensional Kadometsev–Petviashvili equation is separated into three systems of ordinary differential equations, which are the second, third, and fourth members in the well-known confocal involutive hierarchy associated with the nonlinearized Zakharov–Shabat eigenvalue problem. The explicit theta function solution of the Kadometsev–Petviashvili equation is obtained with the help of this separation technique. A generating function approach is introduced to prove the involutivity and the functional independence of the conserved integrals which are essential for the Liouville integrability.
121440603
The spectrum of Hill's equation
AbstractLetq be an infinitely differentiable function of period 1. Then the spectrum of Hill's operatorQ=−d2/dx2+q(x) in the class of functions of period 2 is a discrete series - ∞<λ0<λ1≦λ2<λ3≦λ4<...<λ2i−1≦λ2i↑∞. Let the numer of simple eigenvalues be 2n+1<=∞. Borg [1] proved thatn=0 if and only ifq is constant. Hochstadt [21] proved thatn=1 if and only ifq=c+2p with a constantc and a Weierstrassian elliptic functionp. Lax [29] notes thatn=m if1q=4k2K2m(m+1)sn2(2Kx,k). The present paper studies the casen<∞, continuing investigations of Borg [1], Buslaev and Faddeev [2], Dikii [3, 4], Flaschka [10], Gardneret al. [12], Gelfand [13], Gelfand and Levitan [14], Hochstadt [21], and Lax [28–30] in various directions. The content may be summed up in the statement thatq is an abelian function; in fact, from the present standpoint, the whole subject appears as a part of the classical function theory of the hyperelliptic irrationality
$$\ell (\lambda ) = \sqrt { - (\lambda  - \lambda _0 )(\lambda  - \lambda _1 )...(\lambda  - \lambda _{2n} )} .$$
 The casen=∞ requires the development of the theory of abelian and theta functions for infinite genus; this will be reported upon in another place. Some of the results have been obtained independently by Novikov [34], Dubrovin and Novikov [6] and A. R. Its and V. B. Matveev [22].
121918216
Canonically Conjugate Variables for the Korteweg-de Vries Equation and the Toda Lattice with Periodic Boundary Conditions*)
A new set of canonically conjugate variables 1s introduced for the penoclic Korteweg-de Vries equation and the periodic Toda lattice. These vanables are used for reducing both equations to a nonlinear system which can be integrated in terms of theta functions. It becomes clear that the discrete and the continuous problems are, in a sense, Jsomorph1c. Action variables are defined by loop integrals, and the basic oscillation frequencies are computed. In the infinite-period limit, these action variables tend to the ones used in the canonical description of the inverse-scattering solution method.
118249455
Soliton Equations and Hamiltonian Systems
Integrable Systems Generated by Linear Differential nth Order Operators Hamiltonian Structures Hamiltonian Structure of the GD Hierarchies Modified KdV and GD. The Kupershmidt-Wilson Theorem The KP Hierarchy Baker Function, t-Function Additional Symmetries, String Equation Grassmannian. Algebraic-Geometrical Krichever Solutions Matrix First-Order Operator, AKNS-D Hierarchy Generalization of the AKNS-D Hierarchy: Single-Pole and Multi-Pole Matrix Hierarchies Isomonodromic Deformations and the Most General Matrix Hierarchy Tau Functions of Matrix Hierarchies KP, Modified KP, Constrained KP, Discrete KP, and q-KP Another Chain of KP Hierarchies and Integrals Over Matrix Varieties Transformational Properties of a Differential Operator under Diffeomorphisms and Classical W-Algebras Further Restrictions of the KP, Stationary Equations Stationary Equations of the Matrix Hierarchy Field Lagrangian and Hamiltonian Formalism Further Examples and Applications.
117520249
Soliton Equations and their Algebro-Geometric Solutions
As a partner to Volume 1: Dimensional Continuous Models, this monograph provides a self-contained introduction to algebro-geometric solutions of completely integrable, nonlinear, partial differential-difference equations, also known as soliton equations. The systems studied in this volume include the Toda lattice hierarchy, the Kac-van Moerbeke hierarchy, and the Ablowitz-Ladik hierarchy. An extensive treatment of the class of algebro-geometric solutions in the stationary as well as time-dependent contexts is provided. The theory presented includes trace formulas, algebro-geometric initial value problems, Baker-Akhiezer functions, and theta function representations of all relevant quantities involved. The book uses basic techniques from the theory of difference equations and spectral analysis, some elements of algebraic geometry and especially, the theory of compact Riemann surfaces. The presentation is constructive and rigorous, with ample background material provided in various appendices. Detailed notes for each chapter, together with an exhaustive bibliography, enhance understanding of the main results.
121574604
Constraints of the 2+1 dimensional integrable soliton systems
The authors show that the linear systems associated with some integrable hierarchies of the soliton equations in 2+1 dimensions can be constrained to integrable hierarchies in 1+1 dimensions such that submanifolds solutions of the given systems in 2+1 can be obtained by solving the resulting integrable systems in 1+1 dimensions. The constraints of the KP hierarchy to the AKNS and Burgers hierarchies respectively are shown in detail and the results of these for the modified KP and 2+1 dimensional analogue of the Caudrey-Dodd-Gibbon-Kotera-Sawata equations to several integrable systems in 1+1 are given.
122007329
Korteweg-de Vries Equation and Generalizations. I. A Remarkable Explicit Nonlinear Transformation
An explicit nonlinear transformation relating solutions of the Korteweg‐de Vries equation and a similar nonlinear equation is presented. This transformation is generalized to solutions of a one‐parameter family of similar nonlinear equations. A transformation is given which relates solutions of a ``forced'' Korteweg‐de Vries equation to those of the Korteweg‐de Vries equation.
17916722
Integrals of Nonlinear Equations of Evolution and Solitary Waves
In Section 1 we present a general principle for associating nonlinear equations evolutions with linear operators so that the eigenvalues of the linear operator integrals of the nonlinear equation. A striking instance of such a procedure discovery by Gardner, Miura and Kruskal that the eigenvalues of the Schrodinger operator are integrals of the Korteweg-de Vries equation. 
 
In Section 2 we prove the simplest case of a conjecture of Kruskal and Zabusky concerning the existence of double wave solutions of the Korteweg-de Vries equation, i.e., of solutions which for |I| large behave as the superposition of two solitary waves travelling at different speeds. The main tool used is the first of remarkable series of integrals discovered by Kruskal and Zabusky.
122099454
An explicit symmetry constraint for the Lax pairs and the adjoint Lax pairs of AKNS systems
Abstract An explicit symmetry constraint is proposed for the Lax pairs and the adjoint Lax pairs of AKNS systems. The corresponding Lax pairs and adjoint Lax pairs are nonlinearized into a hierarchy of commutative, finite-dimensional integrable Hamiltonian systems in the Liouville sense and thus an involutive representation of solutions of AKNS systems is obtained. The purpose of this Letter is to elucidate that the nonlinearization method (i.e. a kind of symmetry constraint method) of integrable systems can be applied to the Lax pairs and the adjoint Lax pairs associated with integrable systems.
121993179
Symmetry constraint of MKdV equations by binary nonlinearization
A symmetry constraint for MKdV integrable hierarchy is presented by binary nonlinearization. The spatial part and the temporal parts of the Lax pairs and the adjoint Lax pairs of MKdV equations are all constrained as finite dimensional Liouville integrable Hamiltonian systems, whose integrals of motion are explicitly given. In terms of the proposed symmetry constraint, MKdV equations are decomposed into two finite-dimensional Liouville integrable constrained systems and thus a kind of separation of variables for MKdV equations is established.
122498536
Separation of variables for soliton equations via their binary constrained flows
Binary constrained flows of soliton equations admitting 2×2 Lax matrices have 2N degrees of freedom, which is twice as many degrees of freedom than in the case of monoconstrained flows. By using the normal method, their Lax matrices directly give rise to first N pairs of canonical separated variables for their separation of variables. We propose a new method to introduce the other N pairs of canonical separated variables and additional separated equations. The Jacobi inversion problems for binary constrained flows are established. Finally, the factorization of soliton equations by two commuting binary constrained flows and the separability of binary constrained flows enable us to construct the Jacobi inversion problems for some soliton hierarchies.
18959477
Binary constrained flows and separation of variables for soliton equations
Abstract In contrast to mono-constrained flows with N degrees of freedom, binary constrained flows of soliton equations, admitting 2 × 2 Lax matrices, have 2N degrees of freedom. Currently existing methods only enable Lax matrices to yield the first N pairs of canonical separated variables. An approach for constructing the second N pairs of canonical separated variables with N additional separated equations is introduced. The Jacobi inversion problems for binary constrained flows are then established. Finally, the separability of binary constrained flows together with the factorization of soliton equations by the spatial and temporal binary constrained flows leads to the Jacobi inversion problems for soliton equations.
119759856
Hamiltonian structures and Lax equations
Abstract We show that any hamiltonian system, which is integrable in the sense of Liouville, admits a Lax representation, at least locally at generic points in phase space. We introduce the most general Poisson bracket ensuring the involution property of the integrals of motion and existence of a Lax pair. We give examples of the structure we describe.
120384799
Lax representation, r-matrix method, and separation of variables for the Neumann-type restricted flow
The r-matrix method and the method of separation of variables for restricted flows with Neumann constraints are considered. Restricted Jaulent–Miodek flows are taken as illustrative examples. Lax representations are obtained from the adjoint representation of the corresponding soliton hierarchy; the r-matrix formula, based on this Lax representation, is derived; and the separation of variables for the Hamilton–Jacobi equation are performed.
54084849
The C. Neumann problem as a completely integrable system on an adjoint orbit
It is shown by purely Lie algebraic methods that the C. Neumann problem-the motion of a material point on a sphere under the influence of a quadratic potential-is a completely integrable system of Euler-Poisson equations on a minimal-dimensional orbit of a semidirect product of Lie algebras. 1. The C. Neumann problem. The motion of a point on the sphere S`1 under the influence of a quadratic potential U(x) = 'Ax * x, x E R', A = diag(al,... , an) is a completely integrable Hamiltonian system. For n = 3 this has been shown by C. Neumann in 1859 [12] and for arbitrary n by K. Uhlenbeck [16], R. Devaney [3], J. Moser [10], [11], M. Adler, and P. van Moerbeke [2]. In this paper we show how this problem fits naturally in the framework of Euler-Poisson equations [41, [5], [14], [17] proving that the C. Neumann problem is a Hamiltonian system on a minimaldimensional adjoint orbit in a semidirect product of Lie algebras. Thus its complete integrability will follow entirely from Lie algebraic considerations. The equations of motion are i= -ax1 + AXx, i = 1,...,n, (1.1) where the Lagrange multiplier A = Ax' x 11x112 is chosen such that x E Sn-I during the motion. Set x = y and get the equivalent system to (1.1) xi = yi, yi = -a1xi + (Ax. x 11y112)xi, llxll = 1, x y = 0. (1.2) The following crucial remark that motivated the present investigation is due to K. Uhlenbeck and can be verified without any difficulties. LEMMA 1.1. Put X = (xixj), P = (yixj xiyj). System (1.2) is equivalent to X=[P,X, P =[X,A], lIxl = 1, x y = O. (1.3) Remark that if one replaces X and A by X Id/n and A (Tr(A))Id/n respectively, where Id is the n X n identity matrix, equations (1.3) remain unchanged. From now on we shall assume that in (1.3) this change has been made so that X, P, A E sl(n). The next section gives a Lie algebraic interpretation to these equations. Received by the editors March 12, 1980. 1980 Mathematics Subject Classification. Primary 58F07, 70H05; Secondary 53C15, 17B99.
342996
Mathematical Methods of Classical Mechanics
Part 1 Newtonian mechanics: experimental facts investigation of the equations of motion. Part 2 Lagrangian mechanics: variational principles Lagrangian mechanics on manifolds oscillations rigid bodies. Part 3 Hamiltonian mechanics: differential forms symplectic manifolds canonical formalism introduction to pertubation theory.
118963833
Principles of Algebraic Geometry
A comprehensive, self-contained treatment presenting general results of the theory. Establishes a geometric intuition and a working facility with specific geometric practices. Emphasizes applications through the study of interesting examples and the development of computational tools. Coverage ranges from analytic to geometric. Treats basic techniques and results of complex manifold theory, focusing on results applicable to projective varieties, and includes discussion of the theory of Riemann surfaces and algebraic curves, algebraic surfaces and the quadric line complex as well as special topics in complex manifolds.
17413356
Binary Symmetry Constraints of N -wave Interaction Equations in 1 + 1 and 2 + 1 Dimensions
Binary symmetry constraints of the N-wave interaction equations in 1+1 and 2+1 dimensions are proposed to reduce the N-wave interaction equations into finite-dimensional Liouville integrable systems. A new involutive and functionally independent system of polynomial functions is generated from an arbitrary order square matrix Lax operator and used to show the Liouville integrability of the constrained flows of the N-wave interaction equations. The constraints on the potentials resulting from the symmetry constraints give rise to involutive solutions to the N-wave interaction equations, and thus the integrability by quadratures are shown for the N-wave interaction equations by the constrained flows.
123252110
A nonconfocal involutive system and constrained flows associated with the MKdV− equation
By symmetry constraints, new finite-dimensional integrable systems are deduced from a Lax representation of the MKdV− equation, whose two terms containing spatial derivatives have the same sign. Lax representations are presented for the resulting finite-dimensional integrable systems and an r-matrix formulation is established for the corresponding Lax operator. From the Lax operator, a nonconfocal involutive system of functionally independent polynomial functions is constructed. Solutions of the MKdV− can be obtained by the method of separation of variables.
117168011
Tata Lectures on Theta I
and motivation: theta functions in one variable.- Basic results on theta functions in several variables.
120883400
A (2+1)-dimensional derivative Toda equation in the context of the Kaup-Newell spectral problem
A (2+1)-dimensional derivative Toda equation is derived from the Lax triad composed of the Kaup–Newell, the negative Kaup–Newell and a discrete spectral problem. Two integrable Hamiltonian systems and an integrable symplectic map are derived from the Lax triad. They are straightened out in the Jacobi variety of the associated hyperelliptic curve. A finite genus solution is obtained through a technique based on the Riemann–Jacobi inversion theorem. Besides, explicit solutions are calculated for other associated integrable models, including the mKP equation and the nKN equation.
15458632
Energy dependent Schrödinger operators and complex Hamiltonian systems on Riemann surfaces
We use so-called energy-dependent Schrodinger operators to establish a link between special classes of solutions on N-component systems of evolution equations and finite dimensional Hamiltonian systems on the moduli spaces of Riemann surfaces. We also investigate the phase-space geometry of these Hamiltonian systems and introduce deformations of the level sets associated to conserved quantities, which results in a new class of solutions with monodromy for N-component systems of PDEs. 
After constructing a variety of mechanical systems related to the spatial flows of nonlinear evolution equations, we investigate their semiclassical limits. In particular, we obtain semicalssical asymptotics for the Bloch eigenfunctions of the energy dependent Schrodinger operators, which is of importance in investigating zero-dispersion limits of N-component systems of PDEs.