G. Soliani

Bio: G. Soliani is an academic researcher from Istituto Nazionale di Fisica Nucleare. The author has contributed to research in topics: Nonlinear system & Differential equation. The author has an hindex of 15, co-authored 84 publications receiving 651 citations.

Bright solitons as black holes

Abstract: 2D Jackiw-Teitelboim gravity is represented as a completely integrable nonlinear reaction-diffusion system, whose Euclidean version leads to the nonlinear Schr\"odinger equation. The solitonlike solutions, called dissipatons, to such systems characterize completely the black holes of the considered gravity model (the black hole horizon, the Hawking temperature, and the causal structure). The collision of black holes is described in terms of elastic scattering of dissipatons, which shows a novel transmissionless character, creating a metastable state with a specific lifetime. Finally, alternative descriptions of the model in terms of other completely integrable systems are overlooked.

Lie-Bäcklund symmetries for the Harry-Dym equation

Abstract: A recursion operator (strong symmetry) for the Harry-Dym equation is found. It is also hereditary, and can be used to generate infinitely many Lie-B\"acklund symmetries.

Deformation of surfaces, integrable systems, and Chern–Simons theory

Abstract: A few years ago, some of us devised a method to obtain integrable systems in (2+1)-dimensions from the classical non-Abelian pure Chern–Simons action via the reduction of the gauge connection in Hermitian symmetric spaces. In this article we show that the methods developed in studying classical non-Abelian pure Chern–Simons actions can be naturally implemented by means of a geometrical interpretation of such systems. The Chern–Simons equation of motion turns out to be related to time evolving two-dimensional surfaces in such a way that these deformations are both locally compatible with the Gauss–Mainardi–Codazzi equations and completely integrable. The properties of these relationships are investigated together with the most relevant consequences. Explicit examples of integrable surface deformations are displayed and discussed.

Geometry of 2D topological field theories

Abstract: These lecture notes are devoted to the theory of equations of associativity describing geometry of moduli spaces of 2D topological field theories. Introduction. Lecture 1. WDVV equations and Frobenius manifolds. {Appendix A.} Polynomial solutions of WDVV. {Appendix B.} Symmetriies of WDVV. Twisted Frobenius manifolds. {Appendix C.} WDVV and Chazy equation. Affine connections on curves with projective structure. Lecture 2. Topological conformal field theories and their moduli. Lecture 3. Spaces of isomonodromy deformations as Frobenius manifolds. {Appendix D.} Geometry of flat pencils of metrics. {Appendix E.} WDVV and Painlev\'e-VI. {Appendix F.} Branching of solutions of the equations of isomonodromic deformations and braid group. {Appendix G.} Monodromy group of a Frobenius manifold. {Appendix H.} Generalized hypergeometric equation associated to a Frobenius manifold and its monodromy. {Appendix I.} Determination of a superpotential of a Frobenius manifold. Lecture 4. Frobenius structure on the space of orbits of a Coxeter group. {Appendix J.} Extended complex crystallographic groups and twisted Frobenius manifolds. Lecture 5. Differential geometry of Hurwitz spaces. Lecture 6. Frobenius manifolds and integrable hierarchies. Coupling to topological gravity.

Solitons and the Inverse Scattering Transform

Abstract: Dispersion and nonlinearity play a fundamental role in wave motions in nature. The nonlinear shallow water equations that neglect dispersion altogether lead to breaking phenomena of the typical hyperbolic kind with the development of a vertical profile. In particular, the linear dispersive term in the Korteweg–de Vries equation prevents this from ever happening in its solution. In general, breaking can be prevented by including dispersive effects in the shallow water theory. The nonlinear theory provides some insight into the question of how nonlinearity affects dispersive wave motions. Another interesting feature is the instability and subsequent modulation of an initially uniform wave profile.

Localized excitations in (2+1)-dimensional systems.

Abstract: By means of a special variable separation approach, a common formula with some arbitrary functions has been obtained for some suitable physical quantities of various (2+1)-dimensional models such as the Davey-Stewartson (DS) model, the Nizhnik-Novikov-Veselov (NNV) system, asymmetric NNV equation, asymmetric DS equation, dispersive long wave equation, Broer-Kaup-Kupershmidt system, long wave-short wave interaction model, Maccari system, and a general (N+M)-component Ablowitz-Kaup-Newell-Segur (AKNS) system. Selecting the arbitrary functions appropriately, one may obtain abundant stable localized interesting excitations such as the multidromions, lumps, ring soliton solutions, breathers, instantons, etc. It is shown that some types of lower dimensional chaotic patterns such as the chaotic-chaotic patterns, periodic-chaotic patterns, chaotic line soliton patterns, chaotic dromion patterns, fractal lump patterns, and fractal dromion patterns may be found in higher dimensional soliton systems. The interactions between the traveling ring type soliton solutions are completely elastic. The traveling ring solitons pass through each other and preserve their shapes, velocities, and phases. Some types of localized weak solutions, peakons, are also discussed. Especially, the interactions between two peakons are not completely elastic. After the interactions, the traveling peakons also pass through each other and preserve their velocities and phases, however, they completely exchange their shapes.