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.
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.
We consider the 3D Mikhalev system,
u
t
=
w
x
,
u
y
=
w
t
-
u
w
x
+
w
u
x
,
which has first appeared in the context of KdV-type hierarchies. Under the reduction
w
=
f
(
u
)
, one obtains a pair of ...commuting first-order equations,
u
t
=
f
′
u
x
,
u
y
=
(
f
′
2
-
u
f
′
+
f
)
u
x
,
which govern simple wave solutions of the Mikhalev system. In this paper we study
higher-order
reductions of the form
w
=
f
(
u
)
+
ϵ
a
(
u
)
u
x
+
ϵ
2
b
1
(
u
)
u
xx
+
b
2
(
u
)
u
x
2
+
⋯
,
which turn the Mikhalev system into a pair of commuting higher-order equations. Here the terms at
ϵ
n
are assumed to be differential polynomials of degree
n
in the
x
-derivatives of
u
. We will view
w
as an (infinite) formal series in the deformation parameter
ϵ
. It turns out that for such a reduction to be non-trivial, the function
f
(
u
) must be quadratic,
f
(
u
)
=
λ
u
2
, furthermore, the value of the parameter
λ
(which has a natural interpretation as an eigenvalue of a certain second-order operator acting on an infinite jet space), is quantised. There are only two positive allowed eigenvalues,
λ
=
1
and
λ
=
3
/
2
, as well as infinitely many negative rational eigenvalues. Two-component reductions of the Mikhalev system are also discussed. We emphasise that the existence of higher-order reductions of this kind is a reflection of
linear degeneracy
of the Mikhalev system, in particular, such reductions do not exist for most of the known 3D dispersionless integrable systems such as the dispersionless KP and Toda equations.
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.
We prove that integrability of a dispersionless Hirota-type equation implies the symplectic Monge–Ampère property in any dimension $\geq 4$. In 4D, this yields a complete classification of integrable ...dispersionless partial differential equations (PDEs) of Hirota type through a list of heavenly type equations arising in self-dual gravity. As a by-product of our approach, we derive an involutive system of relations characterizing symplectic Monge–Ampère equations in any dimension. Moreover, we demonstrate that in 4D the requirement of integrability is equivalent to self-duality of the conformal structure defined by the characteristic variety of the equation on every solution, which is in turn equivalent to the existence of a dispersionless Lax pair. We also give a criterion of linearizability of a Hirota-type equation via flatness of the corresponding conformal structure and study symmetry properties of integrable equations.
Hydrodynamic type systems in Riemann invariants arise in a whole range of applications in fluid dynamics, Whitham averaging procedure, differential geometry and the theory of Frobenius manifolds. In ...this paper we discuss parabolic (Jordan block) analogues of diagonalisable systems. Our main observation is that integrable quasilinear systems of Jordan block type are parametrised by solutions of the modified Kadomtsev-Petviashvili hierarchy. Such systems appear naturally as degenerations of quasilinear systems associated with multi-dimensional hypergeometric functions, in the context of parabolic regularisation of the Riemann equation, as finite-component reductions of hydrodynamic chains, and as hydrodynamic reductions of linearly degenerate dispersionless integrable PDEs in multi-dimensions.
Einstein–Weyl geometry is a triple
(
D
,
g
,
ω
)
where
D
is a symmetric connection,
g
is a conformal structure and
ω
is a covector such that
∙
connection
D
preserves the conformal class
g
, that ...is,
D
g
=
ω
g
;
∙
trace-free part of the symmetrised Ricci tensor of
D
vanishes. Three-dimensional Einstein–Weyl structures naturally arise on solutions of second-order dispersionless integrable PDEs in 3D. In this context,
g
coincides with the characteristic conformal structure and is therefore uniquely determined by the equation. On the contrary, covector
ω
is a somewhat more mysterious object, recovered from the Einstein–Weyl conditions. We demonstrate that, for generic second-order PDEs (for instance, for all equations not of Monge–Ampère type), the covector
ω
is also expressible in terms of the equation, thus providing an efficient ‘dispersionless integrability test’. The knowledge of
g
and
ω
provides a dispersionless Lax pair by an explicit formula which is apparently new. Some partial classification results of PDEs with Einstein–Weyl characteristic conformal structure are obtained. A rigidity conjecture is proposed according to which for any generic second-order PDE with Einstein–Weyl property, all dependence on the 1-jet variables can be eliminated via a suitable contact transformation.
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