Paolo Maria Santini
The theory of nonlinear (algebraic, differential, partial
differential, difference, functional) equations treatable by
spectral means. In
1. the study of the rich mathematical aspects of such a theory,
several branches of Mathematics: Algebra, Complex Analysis,
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
1. Longtime behaviour of solutions of nonlinear evolutionary systems,
the Kadomtsev-Petviashvili (KP) equation , describing the
of two-dimensional waves in shallow water; its x-dispersionless
, describing such evolution near the shore, when the wave breaks;
"heavenly" equation of Plebanski , describing self-dual Einstein
2. Multiple scale expansions for integrable and non integrable PDEs,
in principle, to all orders, and the nonlinear Schroedinger
hierarchy [56, 58,
59]. Application to the study of partial difference equations (PDEs)
small lattice spacing regime, as i) a vicinity test between the PDE
continuous limit and ii) integrability test for the PDE .
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
like the sine-Hilbert equation [10, 12, 13] and the Davey-Stewartson
equation [38, 39, 40, 41, 42, 43], a universal model in the
the amplitude modulation of strongly dispersive two-dimensional
in a weakly nonlinear medium, with the spectral characterization of
dromions, novel coherent structures on the plane.
4. Construction of the theory of recursion operators and bi -
structures for integrable PDEs in multidimensions [16, 17, 18, 19,
5. Construction of a theory of integrable discrete geometries based
geometric notion of quadrilateral lattice [61, 63, 64, 65, 66, 67,
68, 69, 70].
6. Search and construction of novel and distinguished nonlinear
systems: algebraic and functional equations [34, 50], cellular
51], integrable multidimensional PDEs with constraints , the
Sine-Hilbert equation [10, 12], integrable discrete geometries and
integrable generalizations of the Toda law on the square, triangular
honeycomb lattices [79, 94], a general matrix PDE in arbitrary
integrable by the method of characteristics .
7. Development of variants of the Dressing method, to construct and
or quasi - integrable PDEs in multidimensions. In particular, the
construction of nonlinear PDEs in (n) dimensions possessing an
solution space of dimension (n-2) .
8. Development of spectral methods to solve initial - boundary value
for integrable PDE's [72, 73, 82].
9. Study of the transition from integrable to chaotic behavior of
systems on the plane, explained as travel on Riemann surfaces [84,
10. Development of the Inverse Spectral Transform for integrable
PDEs arising from the commutation of one parameter families
of multidimensional vector fields . Applications of this theory
heavenly equation of Plebanski (exact reduction of the Einstein
, to the dispersionless KP equation , with the analytic
of the gradient catastrophe of two-dimensional shallow water waves
the shore , and to the so-called dispersionless Toda equation in
dimensions, relevant in field theory and in the study of ideal
flows . Study of the universal way in which weakly nonlinear
one dimensional waves break in Nature .