Alexander Odesski, Brock University
“Integrable Lagrangians and modular forms”
We
investigate
non-degenerate
Lagrangians
of
the
form
f(ux,uy,ut)dxdydt
such
that
the
corresponding
Euler-Lagrange
equations
(fux
)x
+(fuy
)y
+(fut
)t
=
0
are
integrable
by
the
method
of
hydrodynamic
reductions.
We
demonstrate
that
the
integrability
conditions,
which
constitute
an
involutive
over-determined
system
of
fourth
order
PDEs
for
the
Lagrangian
density
f,
are
invariant
under
a
20-parameter
group
of
Lie-point
symmetries
whose
action
on
the
moduli
space
of
integrable
Lagrangians
has
an
open
orbit.
The
density
of
the
‘master-Lagrangian’
corresponding
to
this
orbit
is
shown
to
be
a
modular
form
in
three
variables
defined
on
a
complex
hyperbolic
ball.
We
demonstrate
how
the
knowledge
of
the
symmetry
group
allows
one
to
linearise
the
integrability
conditions.