Given a vector field $\rho (1,\b) \in L^1_\loc(\R^+\times \R^{d},\R^{d+1})$ such that $\dive_{t,x} (\rho (1,\b))$ is a measure, we consider the problem of uniqueness of the representation $\eta$ of $\rho (1,\b) \mathcal L^{d+1}$ as a superposition of characteristics $\gamma : (t^-_\gamma,t^+_\gamma) \to \R^d$, $\dot \gamma (t)= \b(t,\gamma(t))$. We give conditions in terms of a local structure of the representation $\eta$ on suitable sets in order to prove that there is a partition of $\R^{d+1}$ into disjoint trajectories $\wp_\a$, $\a \in \A$, such that the PDE \begin{equation*} \dive_{t,x} \big( u \rho (1,\b) \big) \in \mathcal M(\R^{d+1}), \qquad u \in L^\infty(\R^+\times \R^{d}), \end{equation*} can be disintegrated into a family of ODEs along $\wp_\a$ with measure r.h.s.. The decomposition $\wp_\a$ is essentially unique. We finally show that $\b \in L^1_t(\BV_x)_\loc$ satisfies this local structural assumption and this yields, in particular, the renormalization property for nearly incompressible $\BV$ vector fields.

}, url = {http://preprints.sissa.it/handle/1963/35274}, author = {Stefano Bianchini and Paolo Bonicatto} } @article {MR3686488, title = {Universality of the matrix Airy and Bessel functions at spectral edges of unitary ensembles}, journal = {Random Matrices Theory Appl.}, volume = {6}, number = {3}, year = {2017}, pages = {1750010, 22}, issn = {2010-3263}, doi = {10.1142/S2010326317500101}, url = {http://dx.doi.org/10.1142/S2010326317500101}, author = {Marco Bertola and Mattia Cafasso} } @article {PhysRevLett.119.033901, title = {Universality of the Peregrine Soliton in the Focusing Dynamics of the Cubic Nonlinear Schr{\"o}dinger Equation}, journal = {Phys. Rev. Lett.}, volume = {119}, year = {2017}, month = {Jul}, pages = {033901}, publisher = {American Physical Society}, doi = {10.1103/PhysRevLett.119.033901}, url = {https://link.aps.org/doi/10.1103/PhysRevLett.119.033901}, author = {Tikan, Alexey and Billet, Cyril and Gennady El and Alexander Tovbis and Marco Bertola and Sylvestre, Thibaut and Gustave, Francois and Randoux, Stephane and Genty, Go{\"e}ry and Suret, Pierre and Dudley, John M.} } @article {Bertola-Bothner-MeijerG, title = {Universality Conjecture and Results for a Model of Several Coupled Positive-Definite Matrices}, journal = {Commun. Math. Phys.}, volume = {337}, number = {3}, year = {2015}, month = {08}, pages = {1077{\textendash}1141}, doi = {10.1007/s00220-015-2327-7}, url = {http://link.springer.com/article/10.1007/s00220-015-2327-7}, author = {Marco Bertola and Thomas Bothner} } @article {2014, title = {A uniqueness result for the continuity equation in two dimensions: dedicated to constantine dafermos on the occasion of his 70th birthday}, number = {Journal of the European Mathematical Society;Volume 16; issue 2; pp. 201-234;}, year = {2014}, publisher = {European Mathematical Society; Springer Verlag}, abstract = {We characterize the autonomous, divergence-free vector fields b on the plane such that the Cauchy problem for the continuity equation ∂tu +div(bu) = 0 admits a unique bounded solution (in the weak sense) for every bounded initial datum; the characterization is given in terms of a property of Sard type for the potential f associated to b. As a corollary we obtain uniqueness under the assumption that the curl of b is a measure. This result can be extended to certain nonautonomous vector fields b with bounded divergence.}, doi = {10.4171/JEMS/431}, url = {http://urania.sissa.it/xmlui/handle/1963/34692}, author = {Giovanni Alberti and Stefano Bianchini and Gianluca Crippa} } @article {BertolaTovbisNLS2, title = {Universality for the focusing nonlinear Schr{\"o}dinger equation at the gradient catastrophe point: rational breathers and poles of the \it Tritronqu{\'e}e solution to Painlev{\'e} I}, journal = {Comm. Pure Appl. Math.}, volume = {66}, number = {5}, year = {2013}, pages = {678{\textendash}752}, issn = {0010-3640}, doi = {10.1002/cpa.21445}, url = {http://dx.doi.org/10.1002/cpa.21445}, author = {Marco Bertola and Alexander Tovbis} } @article {SELVITELLA20111731, title = {Uniqueness and nondegeneracy of the ground state for a quasilinear Schr{\"o}dinger equation with a small parameter}, journal = {Nonlinear Analysis: Theory, Methods \& Applications}, volume = {74}, number = {5}, year = {2011}, pages = {1731 - 1737}, abstract = {We study least energy solutions of a quasilinear Schr{\"o}dinger equation with a small parameter. We prove that the ground state is nondegenerate and unique up to translations and phase shifts using bifurcation theory.

}, keywords = {Bifurcation theory, Nonlinear Schr{\"o}dinger equations, Stationary solutions}, issn = {0362-546X}, doi = {https://doi.org/10.1016/j.na.2010.10.045}, url = {http://www.sciencedirect.com/science/article/pii/S0362546X10007613}, author = {Alessandro Selvitella} } @article {2011, title = {A uniqueness result for the continuity equation in two dimensions}, number = {SISSA;52/2011/M}, year = {2011}, institution = {SISSA}, url = {http://hdl.handle.net/1963/4663}, author = {Giovanni Alberti and Stefano Bianchini and Gianluca Crippa} } @article {2010, title = {Uhlenbeck-Donaldson compactification for framed sheaves on projective surfaces}, number = {SISSA;59/2010/FM}, year = {2010}, abstract = {We construct a compactification $M^{\\\\mu ss}$ of the Uhlenbeck-Donaldson type for the moduli space of slope stable framed bundles. This is a kind of a moduli space of slope semistable framed sheaves. We show that there exists a projective morphism $\\\\gamma \\\\colon M^s \\\\to M^{\\\\mu ss}$, where $M^s$ is the moduli space of S-equivalence classes of Gieseker-semistable framed sheaves. The space $M^{\\\\mu ss}$ has a natural set-theoretic stratification which allows one, via a Hitchin-Kobayashi correspondence, to compare it with the moduli spaces of framed ideal instantons.}, url = {http://hdl.handle.net/1963/4049}, author = {Ugo Bruzzo and Dimitri Markushevich and Alexander Tikhomirov} } @article {BertolaTovbis1, title = {Universality in the profile of the semiclassical limit solutions to the focusing nonlinear Schr{\"o}dinger equation at the first breaking curve}, journal = {Int. Math. Res. Not. IMRN}, number = {11}, year = {2010}, pages = {2119{\textendash}2167}, issn = {1073-7928}, doi = {10.1093/imrn/rnp196}, url = {http://0-dx.doi.org.mercury.concordia.ca/10.1093/imrn/rnp196}, author = {Marco Bertola and Alexander Tovbis} } @article {2009, title = {On universality of critical behaviour in the focusing nonlinear Schr{\"o}dinger equation, elliptic umbilic catastrophe and the {\\\\it tritronqu{\'e}e} solution to the Painlev{\'e}-I equation}, journal = {J. Nonlinear Sci. 19 (2009) 57-94}, number = {arXiv.org;0704.0501}, year = {2009}, abstract = {We argue that the critical behaviour near the point of {\textquoteleft}{\textquoteleft}gradient catastrophe\\\" of the solution to the Cauchy problem for the focusing nonlinear Schr\\\\\\\"odinger equation $ i\\\\epsilon \\\\psi_t +\\\\frac{\\\\epsilon^2}2\\\\psi_{xx}+ |\\\\psi|^2 \\\\psi =0$ with analytic initial data of the form $\\\\psi(x,0;\\\\epsilon) =A(x) e^{\\\\frac{i}{\\\\epsilon} S(x)}$ is approximately described by a particular solution to the Painlev\\\\\\\'e-I equation.}, doi = {10.1007/s00332-008-9025-y}, url = {http://hdl.handle.net/1963/2525}, author = {Boris Dubrovin and Tamara Grava and Christian Klein} } @article {2009, title = {Universality of the break-up profile for the KdV equation in the small dispersion limit using the Riemann-Hilbert approach}, journal = {Comm. Math. Phys. 286 (2009) 979-1009}, number = {arXiv.org;0801.2326}, year = {2009}, abstract = {We obtain an asymptotic expansion for the solution of the Cauchy problem for the Korteweg-de Vries (KdV) equation in the small dispersion limit near the point of gradient catastrophe (x_c,t_c) for the solution of the dispersionless equation.\\nThe sub-leading term in this expansion is described by the smooth solution of a fourth order ODE, which is a higher order analogue to the Painleve I equation. This is in accordance with a conjecture of Dubrovin, suggesting that this is a universal phenomenon for any Hamiltonian perturbation of a hyperbolic equation. Using the Deift/Zhou steepest descent method applied on the Riemann-Hilbert problem for the KdV equation, we are able to prove the asymptotic expansion rigorously in a double scaling limit.}, doi = {10.1007/s00220-008-0680-5}, url = {http://hdl.handle.net/1963/2636}, author = {Tamara Grava and Tom Claeys} } @article {2007, title = {Uniqueness and continuous dependence on boundary data for integro-extremal minimizers of the functional of the gradient}, journal = {J. Convex Anal. 14 (2007) 705-727}, year = {2007}, abstract = {We study some qualitative properties of the integro-extremal minimizers of the functional of the gradient defined on Sobolev spaces with Dirichlet boundary conditions. We discuss their use in the non-convex case via viscosity methods and give conditions under which they are unique and depend continuously on boundary data.}, url = {http://hdl.handle.net/1963/2762}, author = {Sandro Zagatti} } @inbook {2006, title = {On universality of critical behaviour in Hamiltonian PDEs}, booktitle = {Geometry, topology, and mathematical physics : S.P. Novikov\\\'s seminar : 2006-2007 / V.M. Buchstaber, I.M. Krichever, editors. - Providence, R.I. : American Mathematical Society, 2008. - pages : 59-109}, number = {American Mathematical Society translations, ISSN 0065-9290;ser. 2;v. 224}, year = {2006}, publisher = {American Mathematical Society}, organization = {American Mathematical Society}, abstract = {Our main goal is the comparative study of singularities of solutions to\\r\\nthe systems of rst order quasilinear PDEs and their perturbations containing higher\\r\\nderivatives. The study is focused on the subclass of Hamiltonian PDEs with one\\r\\nspatial dimension. For the systems of order one or two we describe the local structure\\r\\nof singularities of a generic solution to the unperturbed system near the point of\\r\\n\\\\gradient catastrophe\\\" in terms of standard objects of the classical singularity theory;\\r\\nwe argue that their perturbed companions must be given by certain special solutions\\r\\nof Painlev e equations and their generalizations.}, isbn = {978-0-8218-4674-2}, url = {http://hdl.handle.net/1963/6491}, author = {Boris Dubrovin} } @article {2001, title = {Uniqueness of classical and nonclassical solutions for nonlinear hyperbolic systems}, journal = {J. Differential Equations 172 (2001) 59-82}, year = {2001}, publisher = {Elsevier}, doi = {10.1006/jdeq.2000.3869}, url = {http://hdl.handle.net/1963/3113}, author = {Paolo Baiti and Philippe G. LeFloch and Benedetto Piccoli} } @article {2001, title = {Uniqueness of solutions to Hamilton-Jacobi equations arising in the Calculus of Variations}, journal = {Optimal control and partial differential equations : in honour of professor Alain Bensoussan\\\'s 60th birthday / edited by José Luis Menaldi, Edmundo Rofman, and Agnès Sulem.,Amsterdam : IOS Press, 2001, p. 335-345}, number = {SISSA;57/00/M}, year = {2001}, publisher = {SISSA Library}, abstract = {We prove the uniqueness of the viscosity solution to the Hamilton-Jacobi equation associated with a Bolza problem of the Calculus of Variations, assuming that the Lagrangian is autonomous, continuous, superlinear, and satisfies the usual convexity hypothesis. Under the same assumptions we prove also the uniqueness, in a class of lower semicontinuous functions, of a slightly different notion of solution, where classical derivatives are replaced only by subdifferentials. These results follow from a new comparison theorem for lower semicontinuous viscosity supersolutions of the Hamilton-Jacobi equation, that is proved in the general case of lower semicontinuous Lagrangians.}, url = {http://hdl.handle.net/1963/1515}, author = {Gianni Dal Maso and Helene Frankowska} } @article {2000, title = {A Uniqueness Condition for Hyperbolic Systems of Conservation Laws}, journal = {Discrete Contin. Dynam. Systems 6 (2000) 673-682}, number = {SISSA;88/98/M}, year = {2000}, publisher = {American Institute of Mathematical Sciences}, abstract = {Consider the Cauchy problem for a hyperbolic $n\\\\times n$ system of conservation laws in one space dimension: $$u_t+f(u)_x=0, u(0,x)=\\\\bar u(x).\\\\eqno(CP)$$ Relying on the existence of a continuous semigroup of solutions, we prove that the entropy admissible solution of (CP) is unique within the class of functions $u=u(t,x)$ which have bounded variation along a suitable family of space-like curves.}, url = {http://hdl.handle.net/1963/3195}, author = {Alberto Bressan and Marta Lewicka} } @article {1998, title = {Uniqueness for discontinuous ODE and conservation laws}, journal = {Nonlinear Analysis 34 (1998) 637-652}, number = {SISSA;26/97/M}, year = {1998}, publisher = {Elsevier}, abstract = {Consider a scalar O.D.E. of the form $\\\\dot x=f(t,x),$ where $f$ is possibly discontinuous w.r.t. both variables $t,x$. Under suitable assumptions, we prove that the corresponding Cauchy problem admits a unique solution, which depends H\\\\\\\"older continuously on the initial data.\\nOur result applies in particular to the case where $f$ can be written in the form $f(t,x)\\\\doteq g\\\\big( u(t,x)\\\\big)$, for some function $g$ and some solution $u$ of a scalar conservation law, say $u_t+F(u)_x=0$. In turn, this yields the uniqueness and continuous dependence of solutions to a class of $2\\\\times 2$ strictly hyperbolic systems, with initial data in $\\\\L^\\\\infty$.}, doi = {10.1016/S0362-546X(97)00590-7}, url = {http://hdl.handle.net/1963/3699}, author = {Alberto Bressan and Wen Shen} } @article {1995, title = {Unique solutions of 2x2 conservation laws with large data}, journal = {Indiana Univ. Math. J. 44 (1995), no. 3, 677-725}, number = {SISSA;73/95/M}, year = {1995}, publisher = {Indiana University Mathematics Journal}, abstract = {For a 2x2 hyperbolic system of conservation laws, we first consider a Riemann problem with arbitrarily large data. A stability assumption is introduced, which yields the existence of a Lipschitz semigroup of solutions, defined on a domain containing all suitably small BV perturbations of the Riemann data. We then establish a uniqueness result for large BV solutions, valid within the same class of functions where a local existence theorem can be proved.}, doi = {10.1512/iumj.1995.44.2004}, url = {http://hdl.handle.net/1963/975}, author = {Alberto Bressan and Rinaldo M. Colombo} } @article {1989, title = {Upper semicontinuous differential inclusions without convexity}, journal = {Proc. Amer. Math. Soc. 106 (1989), no. 3, 771-775}, number = {SISSA;74/88/M}, year = {1989}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/670}, author = {Alberto Bressan and Arrigo Cellina and Giovanni Colombo} } @article {9979, title = {Uniqueness and multiplicity of periodic solutions to first order ordinary differential equations}, journal = {Not Found}, number = {SISSA;18/84/M}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/321}, author = {Giovanni Vidossich} }