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Physics and mathematics

of nonlinear processes and models

(University "La Sapienza" , Rome, Italy)

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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 Kadomtsev-Petviashvili (KP) equation [4], describing the evolution
of two-dimensional waves in shallow water; its x-dispersionless version
[95], describing such evolution near the shore, when the wave breaks; the
"heavenly" equation of Plebanski [88], describing self-dual 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 sine-Hilbert equation [10, 12, 13] and the Davey-Stewartson
equation [38, 39, 40, 41, 42, 43], a universal model in the description of
the amplitude modulation of strongly dispersive two-dimensional 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 so-called
Sine-Hilbert 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 (n-2) [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 two-dimensional shallow water waves near
the shore [95], and to the so-called dispersionless Toda equation in 2+1
dimensions, relevant in field theory and in the study of ideal Hele-Shaw
flows [98]. Study of the universal way in which weakly nonlinear quasi
one dimensional waves break in Nature [101].