scholarly journals Spectral curve and Hamiltonian structure of isomonodromic SU(2) Calogero–Gaudin system

2003 ◽  
Vol 44 (9) ◽  
pp. 3979
Author(s):  
Kanehisa Takasaki
Keyword(s):  
Hamiltonian Structure ◽  
Spectral Curve ◽  
Gaudin System

scholarly journals Quasi-periodic solutions of the negative-order Jaulent–Miodek hierarchy

2019 ◽  
Vol 32 (03) ◽  
pp. 2050007 ◽  
Author(s):  
Jinbing Chen
Keyword(s):  
Periodic Solutions ◽  
Hamiltonian Structure ◽  
Spectral Curve ◽  
Type Systems ◽  
Finite Dimensional ◽  
Zero Curvature ◽  
Negative Order ◽  
Neumann Type ◽  
Jacobi Inversion ◽  
Negative Formula

A uniform construction of quasi-periodic solutions to the negative-order Jaulent–Miodek (nJM) hierarchy is presented by using a family of backward Neumann type systems. From the backward Lenard gradients, the nJM hierarchy is put into the zero-curvature setting and the bi-Hamiltonian structure displaying its integrability. The nonlinearization of Lax pair is generalized to the nJM hierarchy such that it can be reduced to a sequence of backward Neumann type systems, whose involutive solutions yield finite parametric solutions of the nJM hierarchy. The negative [Formula: see text]-order stationary JM equation is given to specify a finite-dimensional invariant subspace for the nJM flows. With a spectral curve determined by the Lax matrix, the nJM flows are linearized on the Jacobi variety of a Riemann surface. Finally, the Riemann–Jacobi inversion is applied to Abel–Jacobi solutions of the nJM flows, by which some quasi-periodic solutions are obtained for the nJM hierarchy.

scholarly journals Hamiltonian Aspects of Three-Layer Stratified Fluids

2021 ◽  
Vol 31 (4) ◽  
Author(s):  
R. Camassa ◽  
G. Falqui ◽  
G. Ortenzi ◽  
M. Pedroni ◽  
T. T. Vu Ho
Keyword(s):  
Hamiltonian Structure ◽  
Linear Equations ◽  
Hamiltonian Formalism ◽  
Reduction Process ◽  
Ideal Fluids ◽  
Upper Lid ◽  
Stratification Structure ◽  
Special Solutions ◽  
Dispersionless Limit ◽  
Unbounded Fluid

AbstractThe theory of three-layer density-stratified ideal fluids is examined with a view toward its generalization to the n-layer case. The focus is on structural properties, especially for the case of a rigid upper lid constraint. We show that the long-wave dispersionless limit is a system of quasi-linear equations that do not admit Riemann invariants. We equip the layer-averaged one-dimensional model with a natural Hamiltonian structure, obtained with a suitable reduction process from the continuous density stratification structure of the full two-dimensional equations proposed by Benjamin. For a laterally unbounded fluid between horizontal rigid boundaries, the paradox about the non-conservation of horizontal total momentum is revisited, and it is shown that the pressure imbalances causing it can be intensified by three-layer setups with respect to their two-layer counterparts. The generator of the x-translational symmetry in the n-layer setup is also identified by the appropriate Hamiltonian formalism. The Boussinesq limit and a family of special solutions recently introduced by de Melo Viríssimo and Milewski are also discussed.

scholarly journals Spectral curves are transcendental

2021 ◽  
Vol 111 (1) ◽  
Author(s):  
H. W. Braden
Keyword(s):  
Spectral Curve ◽  
Spectral Curves ◽  
Arithmetic Properties ◽  
A Charge

AbstractSome arithmetic properties of spectral curves are discussed: the spectral curve, for example, of a charge $$n\ge 2$$ n ≥ 2 Euclidean BPS monopole is not defined over $$\overline{\mathbb {Q}}$$ Q ¯ if smooth.

scholarly journals Integrability of Riemann-Type Hydrodynamical Systems and Dubrovin’s Integrability Classification of Perturbed KdV-Type Equations

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1077
Author(s):  
Yarema A. Prykarpatskyy
Keyword(s):  
Conservation Laws ◽  
Hamiltonian Structure ◽  
Necessary Condition ◽  
Type Representation ◽  
Infinite Hierarchy ◽  
Polynomial Coefficients ◽  
Kdv Type Equations ◽  
Special Case

Dubrovin’s work on the classification of perturbed KdV-type equations is reanalyzed in detail via the gradient-holonomic integrability scheme, which was devised and developed jointly with Maxim Pavlov and collaborators some time ago. As a consequence of the reanalysis, one can show that Dubrovin’s criterion inherits important parts of the gradient-holonomic scheme properties, especially the necessary condition of suitably ordered reduction expansions with certain types of polynomial coefficients. In addition, we also analyze a special case of a new infinite hierarchy of Riemann-type hydrodynamical systems using a gradient-holonomic approach that was suggested jointly with M. Pavlov and collaborators. An infinite hierarchy of conservation laws, bi-Hamiltonian structure and the corresponding Lax-type representation are constructed for these systems.

scholarly journals Classical spectral curve of the AdS5 × S5 lambda superstring

2020 ◽  
Vol 2020 (5) ◽  
Author(s):  
Timothy J. Hollowood ◽  
J. Luis Miramontes ◽  
Dafydd Price
Keyword(s):  
Spectral Curve

scholarly journals REDUCTION OF TODA LATTICE HIERARCHY TO GENERALIZED KdV HIERARCHIES AND THE TWO-MATRIX MODEL

1995 ◽  
Vol 10 (17) ◽  
pp. 2537-2577 ◽  
Author(s):  
H. ARATYN ◽  
E. NISSIMOV ◽  
S. PACHEVA ◽  
A.H. ZIMERMAN
Keyword(s):  
Poisson Bracket ◽  
Hamiltonian Structure ◽  
Toda Lattice ◽  
Free Field ◽  
String Model ◽  
Matrix Formulation ◽  
Hamiltonian Action ◽  
Kdv Hierarchy ◽  
Associated Matrix ◽  
Toda Lattice Hierarchy

Toda lattice hierarchy and the associated matrix formulation of the 2M-boson KP hierarchies provide a framework for the Drinfeld-Sokolov reduction scheme realized through Hamiltonian action within the second KP Poisson bracket. By working with free currents, which Abelianize the second KP Hamiltonian structure, we are able to obtain a unified formalism for the reduced SL (M+1, M−k) KdV hierarchies interpolating between the ordinary KP and KdV hierarchies. The corresponding Lax operators are given as superdeterminants of graded SL (M+1, M−k) matrices in the diagonal gauge and we describe their bracket structure and field content. In particular, we provide explicit free field representations of the associated W(M, M−k) Poisson bracket algebras generalizing the familiar nonlinear WM+1 algebra. Discrete Bäcklund transformations for SL (M+1, M−k) KdV are generated naturally from lattice translations in the underlying Toda-like hierarchy. As an application we demonstrate the equivalence of the two-matrix string model to the SL (M+1, 1) KdV hierarchy.

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