
Paolo Maria Santini
Research Interests
The theory of nonlinear (algebraic, differential, partial
differential, integro
differential, difference, functional) equations treatable by
spectral means. In
particular:
1. the study of the rich mathematical aspects of such a theory,
involving
several branches of Mathematics: Algebra, Complex Analysis,
Differential
Geometry, Discrete Geometry, and Algebraic Geometry;
2. the study of the interesting applications to the Natural Sciences.
The scientific production can be grouped into the following research
fields.
1. Longtime behaviour of solutions of nonlinear evolutionary systems,
like
the KadomtsevPetviashvili (KP) equation [4], describing the
evolution
of twodimensional waves in shallow water; its xdispersionless
version
[95], describing such evolution near the shore, when the wave breaks;
the
"heavenly" equation of Plebanski [88], describing selfdual Einstein
fields.
2. Multiple scale expansions for integrable and non integrable PDEs,
carried,
in principle, to all orders, and the nonlinear Schroedinger
hierarchy [56, 58,
59]. Application to the study of partial difference equations (PDEs)
in the
small lattice spacing regime, as i) a vicinity test between the PDE
and its
continuous limit and ii) integrability test for the PDE [100].
3. Use of the Inverse spectral transform for soliton equations in
1+1 and 2+1
dimensions, to solve the Cauchy problem for distinguished soliton
equations,
like the sineHilbert equation [10, 12, 13] and the DaveyStewartson
equation [38, 39, 40, 41, 42, 43], a universal model in the
description of
the amplitude modulation of strongly dispersive twodimensional
waves
in a weakly nonlinear medium, with the spectral characterization of
the
dromions, novel coherent structures on the plane.
4. Construction of the theory of recursion operators and bi 
Hamiltonian
structures for integrable PDEs in multidimensions [16, 17, 18, 19,
20, 21].
5. Construction of a theory of integrable discrete geometries based
on the
geometric notion of quadrilateral lattice [61, 63, 64, 65, 66, 67,
68, 69, 70].
6. Search and construction of novel and distinguished nonlinear
integrable
systems: algebraic and functional equations [34, 50], cellular
automata [47,
51], integrable multidimensional PDEs with constraints [48], the
socalled
SineHilbert equation [10, 12], integrable discrete geometries and
several
integrable generalizations of the Toda law on the square, triangular
and
honeycomb lattices [79, 94], a general matrix PDE in arbitrary
dimensions
integrable by the method of characteristics [91].
7. Development of variants of the Dressing method, to construct and
solve integrable
or quasi  integrable PDEs in multidimensions. In particular, the
construction of nonlinear PDEs in (n) dimensions possessing an
analytic
solution space of dimension (n2) [86].
8. Development of spectral methods to solve initial  boundary value
problems
for integrable PDE's [72, 73, 82].
9. Study of the transition from integrable to chaotic behavior of
dynamical
systems on the plane, explained as travel on Riemann surfaces [84,
97].
10. Development of the Inverse Spectral Transform for integrable
multidimensional
PDEs arising from the commutation of one parameter families
of multidimensional vector fields [88]. Applications of this theory
to the
heavenly equation of Plebanski (exact reduction of the Einstein
equations)
[88], to the dispersionless KP equation [90], with the analytic
description
of the gradient catastrophe of twodimensional shallow water waves
near
the shore [95], and to the socalled dispersionless Toda equation in
2+1
dimensions, relevant in field theory and in the study of ideal
HeleShaw
flows [98]. Study of the universal way in which weakly nonlinear
quasi
one dimensional waves break in Nature [101].

