The foundation: the algebraic integrability conditions

2015 ◽  
pp. 25-54
Author(s):  
Konrad Schöbel
Keyword(s):  
Algebraic Integrability ◽  
Integrability Conditions

scholarly journals Second-order integrable Lagrangians and WDVV equations

2021 ◽  
Vol 111 (2) ◽  
Author(s):  
E. V. Ferapontov ◽  
M. V. Pavlov ◽  
Lingling Xue
Keyword(s):  
Lagrangian Density ◽  
Theta Functions ◽  
Second Order ◽  
Elementary Functions ◽  
Lagrange Equations ◽  
Wdvv Equations ◽  
Integrability Conditions ◽  
Jacobi Theta Functions

AbstractWe investigate the integrability of Euler–Lagrange equations associated with 2D second-order Lagrangians of the form $$\begin{aligned} \int f(u_{xx},u_{xy},u_{yy})\ \mathrm{d}x\mathrm{d}y. \end{aligned}$$ ∫ f ( u xx , u xy , u yy ) d x d y . By deriving integrability conditions for the Lagrangian density f, examples of integrable Lagrangians expressible via elementary functions, Jacobi theta functions and dilogarithms are constructed. A link of second-order integrable Lagrangians to WDVV equations is established. Generalisations to 3D second-order integrable Lagrangians are also discussed.

The algebraic integrability of geodesic flow onSO(4)

1982 ◽  
Vol 67 (2) ◽  
pp. 297-331 ◽  
Author(s):  
M. Adler ◽  
P. van Moerbeke
Keyword(s):  
Geodesic Flow ◽  
Algebraic Integrability

scholarly journals Algebraic integrability of foliations of the plane

2006 ◽  
Vol 231 (2) ◽  
pp. 611-632 ◽  
Author(s):  
C. Galindo ◽  
F. Monserrat
Keyword(s):  
Algebraic Integrability

GEOMETRICAL CONSTRAINTS FOR D = 10, N = 1 SUPERGRAVITY

1989 ◽  
Vol 04 (15) ◽  
pp. 3819-3831 ◽  
Author(s):  
LING-LIE CHAU ◽  
CHONG-SA LIM
Keyword(s):  
Equations Of Motion ◽  
Bianchi Identities ◽  
Dynamical Consequence ◽  
Integrability Conditions ◽  
Geometrical Constraints ◽  
Additional Constraints

A set of geometrical constraints for D = 10, N = 1 supergravity is formulated. It has the meaning as integrability conditions on "hyperplanes" determined by light-like lines in the superspace. The dynamical consequence of these geometrical constraints is studied via Bianchi identities. Since no equations of motion have resulted, these geometrical constraints can form an off-shell set of constraints for the theory. We also discuss additional constraints that lead to Poincare supergravity equations of motion. The relation of the theory with D = 4 N = 4 supergravity is also illuminated.

On the spectrum and the distribution of singular values of Schrödinger operators with a complex potential

1983 ◽  
Vol 388 (1794) ◽  
pp. 195-218 ◽  
Keyword(s):  
Complex Potential ◽  
Spectral Properties ◽  
Polynomial Growth ◽  
Schrödinger Operators ◽  
Singular Values ◽  
Negative Real Part ◽  
Adjoint Problem ◽  
Schrodinger Operators ◽  
Asymptotic Estimates ◽  
Integrability Conditions

This paper is concerned with spectral properties of the Schrödinger operator ─ ∆+ q with a complex potential q which has non-negative real part and satisfies weak integrability conditions. The problem is dealt with as a genuine non-self-adjoint problem, not as a perturbation of a self-adjoint one, and global and asymptotic estimates are obtained for the corresponding singular values. From these estimates information is obtained about the eigenvalues of the problem. By way of illustration, detailed calculations are given for an example in which the potential has at most polynomial growth.

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