Applying symmetry reduction to a class of SL(2,R)$\mathrm{SL}(2,\mathbb {R})$‐invariant third‐order ordinary differential equations (ODEs), we obtain Abel equations whose general solution can be ...parameterized by hypergeometric functions. Particular case of this construction provides a general parametric solution to the Kudashev equation, an ODE arising in the Gurevich–Pitaevskii problem, thus giving the first term of a large‐time asymptotic expansion of its solution in the oscillatory (Whitham) zone.
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.
Macroscopic dynamics of soliton gases can be analytically described by the thermodynamic limit of the Whitham equations, yielding an integro-differential kinetic equation for the density of states. ...Under a delta-functional ansatz, the kinetic equation for soliton gas reduces to a non-diagonalisable system of hydrodynamic type whose matrix consists of several
2
×
2
Jordan blocks. Here we demonstrate the integrability of this system by showing that it possesses a hierarchy of commuting hydrodynamic flows and can be solved by an extension of the generalised hodograph method. Our approach is a generalisation of Tsarev’s theory of diagonalisable systems of hydrodynamic type to quasilinear systems with non-trivial Jordan block structure.
Equations of dispersionless Hirota type
F
(
u
x
i
x
j
)
=
0
have been thoroughly investigated in mathematical physics and differential geometry. It is known that the parameter space of integrable ...Hirota type equations in 3D is 21-dimensional, and that the action of the natural equivalence group
Sp
(
6
,
R
)
on the parameter space has an open orbit. However the structure of the generic equation corresponding to the open orbit remained elusive. Here we prove that the generic 3D Hirota equation is given by the remarkable formula
ϑ
m
(
τ
)
=
0
,
τ
=
i
Hess
(
u
)
where
ϑ
m
is any genus 3 theta constant with even characteristics and
Hess
(
u
)
is the
3
×
3
Hessian matrix of a (real-valued) function
u
(
x
1
,
x
2
,
x
3
)
. Thus, generic Hirota equation coincides with the equation of the genus 3 hyperelliptic divisor (to be precise, its intersection with the imaginary part of the Siegel upper half space
H
3
). The rich geometry of integrable Hirota type equations sheds new light on local differential geometry of the genus 3 hyperelliptic divisor, in particular, the integrability conditions can be viewed as local differential-geometric constraints that characterise the hyperelliptic divisor uniquely modulo
Sp
(
6
,
C
)
-equivalence.
It was observed by Tod (1995 Class. Quantum Grav.12 1535-47) and later by Dunajski and Tod (2002 Phys. Lett. A 303 253-64) that the Boyer-Finley (BF) and the dispersionless Kadomtsev-Petviashvili ...(dKP) equations possess solutions whose level surfaces are central quadrics in the space of independent variables (the so-called central quadric ansatz). It was demonstrated that generic solutions of this type are described by Painlevé equations PIII and PII, respectively. The aim of our paper is threefold: (1) Based on the method of hydrodynamic reductions, we classify integrable models possessing the central quadric ansatz. This leads to the five canonical forms (including BF and dKP). (2) Applying the central quadric ansatz to each of the five canonical forms, we obtain all Painlevé equations PI-PVI, with PVI corresponding to the generic case of our classification. (3) We argue that solutions coming from the central quadric ansatz constitute a subclass of two-phase solutions provided by the method of hydrodynamic reductions.
On a class of integrable Hamiltonian equations in 2+1 dimensions Gormley, Ben; Ferapontov, Eugene V; Novikov, Vladimir S
Proceedings of the Royal Society. A, Mathematical, physical, and engineering sciences,
05/2021, Letnik:
477, Številka:
2249
Journal Article
Recenzirano
Odprti dostop
We classify integrable Hamiltonian equations of the form
where the Hamiltonian density
(
,
) is a function of two variables: dependent variable
and the non-locality
. Based on the method of ...hydrodynamic reductions, the integrability conditions are derived (in the form of an involutive PDE system for the Hamiltonian density
). We show that the generic integrable density is expressed in terms of the Weierstrass
-function:
(
,
) =
(
) e
. Dispersionless Lax pairs, commuting flows and dispersive deformations of the resulting equations are also discussed.
It was shown in Ferapontov et al. (Lett Math Phys 108(6):1525–1550, 2018) that the classification of
n
-component systems of conservation laws possessing a third-order Hamiltonian structure reduces ...to the following algebraic problem: classify
n
-planes
H
in
∧
2
(
V
n
+
2
)
such that the induced map
S
y
m
2
H
⟶
∧
4
V
n
+
2
has 1-dimensional kernel generated by a non-degenerate quadratic form on
H
∗
. This problem is trivial for
n
=
2
,
3
and apparently wild for
n
≥
5
. In this paper we address the most interesting borderline case
n
=
4
. We prove that the variety
V
parametrizing those 4-planes
H
is an irreducible 38-dimensional
P
G
L
(
V
6
)
-invariant subvariety of the Grassmannian
G
(
4
,
∧
2
V
6
)
. With every
H
∈
V
we associate a
characteristic
cubic surface
S
H
⊂
P
H
, the locus of rank 4 two-forms in
H
. We demonstrate that the induced characteristic map
σ
:
V
/
P
G
L
(
V
6
)
⤏
M
c
,
where
M
c
denotes the moduli space of cubic surfaces in
P
3
, is dominant, hence generically finite. Based on Manivel and Mezzetti (Manuscr Math 117:319–331, 2005), a complete classification of 4-planes
H
∈
V
with the reducible characteristic surface
S
H
is given.
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