scholarly journals ON THE INTEGRABILITY OF A CLASS OF MONGE–AMPÈRE EQUATIONS

2001 ◽  
Vol 13 (04) ◽  
pp. 529-543 ◽  
Author(s):  
J. C. BRUNELLI ◽  
M. GÜRSES ◽  
K. ZHELTUKHIN
Keyword(s):  
Sigma Models ◽  
Minimal Surfaces ◽  
Second Order ◽  
Lax Representation ◽  
Parameter Equation ◽  
Monge Ampere ◽  
Two Parameter

We give the Lax representations for the elliptic, hyperbolic and homogeneous second order Monge–Ampère equations. The connection between these equations and the equations of hydrodynamical type give us a scalar dispersionless Lax representation. A matrix dispersive Lax representation follows from the correspondence between sigma models, a two parameter equation for minimal surfaces and Monge–Ampère equations. Local as well nonlocal conserved densities are obtained.

scholarly journals Lax Pairs and First Integrals for Autonomous and Non-Autonomous Differential Equations Belonging to the Painlevé – Gambier List

2020 ◽  
Vol 16 (4) ◽  
pp. 637-650
Author(s):  
P. Guha ◽  
◽  
S. Garai ◽  
A.G. Choudhury ◽  
◽  
...  
Keyword(s):  
Differential Equations ◽  
First Integral ◽  
Lax Pair ◽  
First Integrals ◽  
Second Order ◽  
Lax Representation ◽  
Lax Pairs ◽  
Second Order Differential Equations ◽  
Autonomous Version ◽  
Autonomous Differential Equations

Recently Sinelshchikov et al. [1] formulated a Lax representation for a family of nonautonomous second-order differential equations. In this paper we extend their result and obtain the Lax pair and the associated first integral of a non-autonomous version of the Levinson – Smith equation. In addition, we have obtained Lax pairs and first integrals for several equations of the Painlevé – Gambier list, namely, the autonomous equations numbered XII, XVII, XVIII, XIX, XXI, XXII, XXIII, XXIX, XXXII, XXXVII, XLI, XLIII, as well as the non-autonomous equations Nos. XV and XVI in Ince’s book.

scholarly journals Second-order PDEs in four dimensions with half-flat conformal structure

2020 ◽  
Vol 476 (2233) ◽  
pp. 20190642
Author(s):  
S. Berjawi ◽  
E. V. Ferapontov ◽  
B. Kruglikov ◽  
V. Novikov
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Lax Pair ◽  
Conformal Structure ◽  
Second Order ◽  
Partial Differential ◽  
Characteristic Variety ◽  
Four Dimensions ◽  
Partial Classification ◽  
Monge Ampere

We study second-order partial differential equations (PDEs) in four dimensions for which the conformal structure defined by the characteristic variety of the equation is half-flat (self-dual or anti-self-dual) on every solution. We prove that this requirement implies the Monge–Ampère property. Since half-flatness of the conformal structure is equivalent to the existence of a non-trivial dispersionless Lax pair, our result explains the observation that all known scalar second-order integrable dispersionless PDEs in dimensions four and higher are of Monge–Ampère type. Some partial classification results of Monge–Ampère equations in four dimensions with half-flat conformal structure are also obtained.

Contigious hyperquadrics of coequipped hyperbands sHm

2019 ◽  
pp. 126-132
Author(s):  
Yu. Popov
Keyword(s):  
Projective Space ◽  
Second Order ◽  
Third Order ◽  
Base Surface ◽  
The Third ◽  
Two Parameter ◽  
Order Contact

We consider hyperquadrics that are internally connected to coequipped hyperbands in the projective space. Specifically, a hyperquadric Qn1 tangent to a hyperplane at the point is called a contiguous hyper quadric of a hyperband if it has a second-order contact with the base surface of the hyperband. In a the third order differential neighborhood of the forming element of the hyperband, two two-parameter bundles of fields of adjoining hyperquadrics are internally invariantly joined, their equations are given in a dot frame. The set of hyperquadrics such that the plane and the plane of Cartan are conjugate with respect to hyperquadric Qn1 is considered. The condition is shown under which the normal of the 2nd kind and the Cartan plane are conjugate with respect to the hyperquadric Qn1 . In addition, the following theorem is proved: normalization of a coequipped regular hyperband has a semi-internal equipment if and only if its normals of the first and second kind are polarly conjugate with respect to the hyperquadric.

Kinetics of the anionic coordination dimerization of isoprene

1982 ◽  
Vol 47 (2) ◽  
pp. 594-602 ◽  
Author(s):  
Jan Bartoň ◽  
Miroslav Kašpar ◽  
Vlastimil Růžička
Keyword(s):  
Kinetic Model ◽  
Time Dependence ◽  
Kinetic Measurements ◽  
Parameter Equation ◽  
Kinetics Of ◽  
Two Parameter

The kinetics of the anionic coordination dimerization of isoprene (i.e., 2-methyl-1,3-butadiene) to β-myrcene (i.e., 7-methyl-3-methylene-1,6-octadiene) was investigated. The reaction was initiated with sodium in the presence of dicyclohexyl amine in tetrahydrofuran. Kinetic measurements showed that ionic pairs of isoprenyl sodium were additionally solvated by two molecules of tetrahydrofuran. A kinetic model of the reaction enabled the time dependence of the isoprene concentration to be expressed in terms of a two-parameter equation.

An eigenfunction expansion associated with a two-parameter system of differential equations. III

1983 ◽  
Vol 93 (3-4) ◽  
pp. 189-195 ◽  
Author(s):  
M. Faierman
Keyword(s):  
Differential Equations ◽  
Uniform Convergence ◽  
Ordinary Differential Equations ◽  
Eigenfunction Expansion ◽  
Second Order ◽  
System Of Differential Equations ◽  
Parameter System ◽  
Two Parameter

SynopsisWe continue with the work of earlier papers concerning the use of partial dilferential equations to prove the uniform convergence of the eigenfunction expansion associated witha left definite two-parameter system of ordinary differential equations of the second order.

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