We study the critical behaviour of solutions to weakly dispersive Hamiltonian systems considered as perturbations of elliptic and hyperbolic systems of hydrodynamic type with two components. We argue that near the critical point of gradient catastrophe of the dispersionless system, the solutions to a suitable initial value problem for the perturbed equations are approximately described by particular solutions to the Painlev\'e-I (P$_I$) equation or its fourth order analogue P$_I^2$. As concrete examples we discuss nonlinear Schr\"odinger equations in the semiclassical limit. A numerical study of these cases provides strong evidence in support of the conjecture.

}, author = {Boris Dubrovin and Tamara Grava and Christian Klein and Antonio Moro} } @article {10979, title = {On the tritronqu{\'e}e solutions of P$_I^2$}, year = {2013}, institution = {SISSA}, abstract = {For equation P$_I^2$, the second member in the P$_I$ hierarchy, we prove existence of various degenerate solutions depending on the complex parameter $t$ and evaluate the asymptotics in the complex $x$ plane for $|x|\to\infty$ and $t=o(x^{2/3})$. Using this result, we identify the most degenerate solutions $u^{(m)}(x,t)$, $\hat u^{(m)}(x,t)$, $m=0,\dots,6$, called {\em tritronqu\'ee}, describe the quasi-linear Stokes phenomenon and find the large $n$ asymptotics of the coefficients in a formal expansion of these solutions. We supplement our findings by a numerical study of the tritronqu\'ee solutions.

}, author = {Tamara Grava and Andrey Kapaev and Christian Klein} } @article {2012, title = {The KdV hierarchy: universality and a Painleve transcendent}, journal = {International Mathematics Research Notices, vol. 22 (2012) , page 5063-5099}, number = {arXiv:1101.2602;}, year = {2012}, note = {This article was published in "International Mathematics Research Notices, vol. 22 (2012) , page 5063-5099}, publisher = {Oxford University Press}, abstract = {We study the Cauchy problem for the Korteweg-de Vries (KdV) hierarchy in the small dispersion limit where $\e\to 0$. For negative analytic initial data with a single negative hump, we prove that for small times, the solution is approximated by the solution to the hyperbolic transport equation which corresponds to $\e=0$. Near the time of gradient catastrophe for the transport equation, we show that the solution to the KdV hierarchy is approximated by a particular Painlev\'e transcendent. This supports Dubrovins universality conjecture concerning the critical behavior of Hamiltonian perturbations of hyperbolic equations. We use the Riemann-Hilbert approach to prove our results.}, keywords = {Small-Dispersion limit}, url = {http://hdl.handle.net/1963/6921}, author = {Tom Claeys and Tamara Grava} } @article {2012, title = {Numerical study of the small dispersion limit of the Korteweg-de Vries equation and asymptotic solutions}, journal = {Physica D 241, nr. 23-24 (2012): 2246-2264}, number = {arXiv:1202.0962;}, year = {2012}, publisher = {Elsevier}, abstract = {We study numerically the small dispersion limit for the Korteweg-de Vries (KdV) equation $u_t+6uu_x+\epsilon^{2}u_{xxx}=0$ for $\epsilon\ll1$ and give a quantitative comparison of the numerical solution with various asymptotic formulae for small $\epsilon$ in the whole $(x,t)$-plane. The matching of the asymptotic solutions is studied numerically.}, keywords = {Korteweg-de Vries equation}, doi = {10.1016/j.physd.2012.04.001}, author = {Tamara Grava and Christian Klein} } @article {2011, title = {Numerical Study of breakup in generalized Korteweg-de Vries and Kawahara equations}, journal = {SIAM J. Appl. Math. 71 (2011) 983-1008}, number = {arXiv:1101.0268;}, year = {2011}, publisher = {SIAM}, abstract = {This article is concerned with a conjecture in [B. Dubrovin, Comm. Math. Phys., 267 (2006), pp. 117{\textendash}139] on the formation of dispersive shocks in a class of Hamiltonian dispersive regularizations of the quasi-linear transport equation. The regularizations are characterized by two arbitrary functions of one variable, where the condition of integrability implies that one of these functions must not vanish. It is shown numerically for a large class of equations that the local behavior of their solution near the point of gradient catastrophe for the transport equation is described by a special solution of a Painlev{\'e}-type equation. This local description holds also for solutions to equations where blowup can occur in finite time. Furthermore, it is shown that a solution of the dispersive equations away from the point of gradient catastrophe is approximated by a solution of the transport equation with the same initial data, modulo terms of order $\\\\epsilon^2$, where $\\\\epsilon^2$ is the small dispersion parameter. Corrections up to order $\\\\epsilon^4$ are obtained and tested numerically.}, doi = {10.1137/100819783}, url = {http://hdl.handle.net/1963/4951}, author = {Boris Dubrovin and Tamara Grava and Christian Klein} } @article {2010, title = {Numerical Solution of the Small Dispersion Limit of the Camassa-Holm and Whitham Equations and Multiscale Expansions}, number = {SISSA;10/2010/FM}, year = {2010}, abstract = {The small dispersion limit of solutions to the Camassa-Holm (CH) equation is characterized by the appearance of a zone of rapid modulated oscillations. An asymptotic description of these oscillations is given, for short times, by the one-phase solution to the CH equation, where the branch points of the corresponding elliptic curve depend on the physical coordinates via the Whitham equations. We present a conjecture for the phase of the asymptotic solution. A numerical study of this limit for smooth hump-like initial data provides strong evidence for the validity of this conjecture....}, url = {http://hdl.handle.net/1963/3840}, author = {Simonetta Abenda and Tamara Grava and Christian Klein} } @article {2010, title = {Painlev{\'e} II asymptotics near the leading edge of the oscillatory zone for the Korteweg-de Vries equation in the small-dispersion limit}, journal = {Comm. Pure Appl. Math. 63 (2010) 203-232}, number = {arXiv.org;0812.4142v1}, year = {2010}, publisher = {Wiley}, abstract = {In the small dispersion limit, solutions to the Korteweg-de Vries equation develop an interval of fast oscillations after a certain time. We obtain a universal asymptotic expansion for the Korteweg-de Vries solution near the leading edge of the oscillatory zone up to second order corrections. This expansion involves the Hastings-McLeod solution of the Painlev\\\\\\\'e II equation. We prove our results using the Riemann-Hilbert approach.}, doi = {10.1002/cpa.20277}, url = {http://hdl.handle.net/1963/3799}, author = {Tom Claeys and Tamara Grava} } @article {2010, title = {Solitonic asymptotics for the Korteweg-de Vries equation in the small dispersion limit}, journal = {SIAM J. Math. Anal. 42 (2010) 2132-2154}, number = {SISSA;09/2010/FM}, year = {2010}, abstract = {We study the small dispersion limit for the Korteweg-de Vries (KdV) equation $u_t+6uu_x+\\\\epsilon^{2}u_{xxx}=0$ in a critical scaling regime where $x$ approaches the trailing edge of the region where the KdV solution shows oscillatory behavior. Using the Riemann-Hilbert approach, we obtain an asymptotic expansion for the KdV solution in a double scaling limit, which shows that the oscillations degenerate to sharp pulses near the trailing edge. Locally those pulses resemble soliton solutions of the KdV equation.}, doi = {10.1137/090779103}, url = {http://hdl.handle.net/1963/3839}, author = {Tamara Grava and Tom Claeys} } @article {2009, title = {Initial value problem of the Whitham equations for the Camassa-Holm equation}, journal = {Physica D 238 (2009) 55-66}, number = {arXiv.org;0805.2558v1}, year = {2009}, publisher = {Elsevier}, abstract = {We study the Whitham equations for the Camassa-Holm equation. The equations are neither strictly hyperbolic nor genuinely nonlinear. We are interested in the initial value problem of the Whitham equations. When the initial values are given by a step function, the Whitham solution is self-similar. When the initial values are given by a smooth function, the Whitham solution exists within a cusp in the x-t plane. On the boundary of the cusp, the Whitham equation matches the Burgers solution, which exists outside the cusp.}, doi = {10.1016/j.physd.2008.08.016}, url = {http://hdl.handle.net/1963/3429}, author = {Tamara Grava and Virgil U. Pierce and Fei-Ran Tian} } @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 {2008, title = {Numerical study of a multiscale expansion of the Korteweg-de Vries equation and Painlev{\'e}-II equation}, journal = {Proc. R. Soc. A 464 (2008) 733-757}, number = {arXiv.org;0708.0638v3}, year = {2008}, abstract = {The Cauchy problem for the Korteweg de Vries (KdV) equation with small dispersion of order $\\\\e^2$, $\\\\e\\\\ll 1$, is characterized by the appearance of a zone of rapid modulated oscillations. These oscillations are approximately described by the elliptic solution of KdV where the amplitude, wave-number and frequency are not constant but evolve according to the Whitham equations. Whereas the difference between the KdV and the asymptotic solution decreases as $\\\\epsilon$ in the interior of the Whitham oscillatory zone, it is known to be only of order $\\\\epsilon^{1/3}$ near the leading edge of this zone. To obtain a more accurate description near the leading edge of the oscillatory zone we present a multiscale expansion of the solution of KdV in terms of the Hastings-McLeod solution of the Painlev\\\\\\\'e-II equation. We show numerically that the resulting multiscale solution approximates the KdV solution, in the small dispersion limit, to the order $\\\\epsilon^{2/3}$.}, doi = {10.1098/rspa.2007.0249}, url = {http://hdl.handle.net/1963/2592}, author = {Tamara Grava and Christian Klein} } @article {2007, title = {Numerical solution of the small dispersion limit of Korteweg de Vries and Whitham equations}, number = {SISSA;91/2005/FM}, year = {2007}, abstract = {The Cauchy problem for the Korteweg de Vries (KdV) equation with small dispersion of order $\\\\epsilon^2$, is characterized by the appearance of a zone of rapid modulated oscillations of wave-length of order $\\\\epsilon$. These oscillations are approximately described by the elliptic solution of KdV where the amplitude, wave-number and frequency are not constant but evolve according to the Whitham equations. In this manuscript we give a quantitative analysis of the discrepancy between the numerical solution of the KdV equation in the small dispersion limit and the corresponding approximate solution for values of $\\\\epsilon$ between $10^{-1}$ and $10^{-3}$. The numerical results are compatible with a difference of order $\\\\epsilon$ within the {\textquoteleft}interior\\\' of the Whitham oscillatory zone, of order $\\\\epsilon^{1/3}$ at the left boundary outside the Whitham zone and of order $\\\\epsilon^{1/2}$ at the right boundary outside the Whitham zone.}, doi = {10.1002/cpa.20183}, url = {http://hdl.handle.net/1963/1788}, author = {Tamara Grava and Christian Klein} } @article {2007, title = {Numerical study of a multiscale expansion of KdV and Camassa-Holm equation}, number = {arXiv.org;math-ph/0702038v1}, year = {2007}, abstract = {We study numerically solutions to the Korteweg-de Vries and Camassa-Holm equation close to the breakup of the corresponding solution to the dispersionless equation. The solutions are compared with the properly rescaled numerical solution to a fourth order ordinary differential equation, the second member of the Painlev\\\\\\\'e I hierarchy. It is shown that this solution gives a valid asymptotic description of the solutions close to breakup. We present a detailed analysis of the situation and compare the Korteweg-de Vries solution quantitatively with asymptotic solutions obtained via the solution of the Hopf and the Whitham equations. We give a qualitative analysis for the Camassa-Holm equation}, url = {http://hdl.handle.net/1963/2527}, author = {Tamara Grava and Christian Klein} } @article {2007, title = {Reciprocal transformations and flat metrics on Hurwitz spaces}, journal = {J. Phys. A 40 (2007) 10769-10790}, number = {arXiv.org;0704.1779v2}, year = {2007}, abstract = {We consider hydrodynamic systems which possess a local Hamiltonian structure of Dubrovin-Novikov type. To such a system there are also associated an infinite number of nonlocal Hamiltonian structures. We give necessary and sufficient conditions so that, after a nonlinear transformation of the independent variables, the reciprocal system still possesses a local Hamiltonian structure of Dubrovin-Novikov type. We show that, under our hypotheses, bi-hamiltonicity is preserved by the reciprocal transformation. Finally we apply such results to reciprocal systems of genus g Whitham-KdV modulation equations.}, doi = {10.1088/1751-8113/40/35/004}, url = {http://hdl.handle.net/1963/2210}, author = {Simonetta Abenda and Tamara Grava} } @article {2006, title = {Large Parameter Behavior of Equilibrium Measures}, number = {SISSA;92/2005/FM}, year = {2006}, abstract = {We study the equilibrium measure for a logarithmic potential in the presence of an external field V*(x) + tp(x), where t is a parameter, V*(x) is a smooth function and p(x) a monic polynomial. When p(x) is of an odd degree, the equilibrium measure is shown to be supported on a single interval as |t| is sufficiently large. When p(x) is of an even degree, the equilibrium measure is supported on two disjoint intervals as t is negatively large; it is supported on a single interval for convex p(x) as t is positively large and is likely to be supported on multiple disjoint intervals for non-convex p(x).}, url = {http://hdl.handle.net/1963/1789}, author = {Tamara Grava and Fei-Ran Tian} } @article {2006, title = {Thomae type formulae for singular Z_N curves}, journal = {Lett. Math. Phys. 76 (2006) 187-214}, number = {arXiv.org;math-ph/0602017v1}, year = {2006}, abstract = {We give an elementary and rigorous proof of the Thomae type formula for singular $Z_N$ curves. To derive the Thomae formula we use the traditional variational method which goes back to Riemann, Thomae and Fuchs. An important step of the proof is the use of the Szego kernel computed explicitly in algebraic form for non-singular 1/N-periods. The proof inherits principal points of Nakayashiki\\\'s proof [31], obtained for non-singular ZN curves.}, doi = {10.1007/s11005-006-0073-7}, url = {http://hdl.handle.net/1963/2125}, author = {Victor Z. Enolski and Tamara Grava} } @article {2005, title = {Modulation of the Camassa-Holm equation and reciprocal transformations}, journal = {Ann. Inst. Fourier (Grenoble) 55 (2005) 1803-1834}, number = {SISSA;107/2004/FM}, year = {2005}, abstract = {We derive the modulation equations or Whitham equations for the Camassa-Holm (CH) equation. We show that the modulation equations are hyperbolic and admit bi-Hamiltonian structure. Furthermore they are connected by a reciprocal transformation to the modulation equations of the first negative flow of the Korteweg de Vries (KdV) equation. The reciprocal transformation is generated by the Casimir of the second Poisson bracket of the KdV averaged flow. We show that the geometry of the bi-Hamiltonian structure of the KdV and CH modulation equations is quite different: indeed the KdV averaged bi-Hamiltonian structure can always be related to a semisimple Frobenius manifold while the CH one cannot.}, url = {http://hdl.handle.net/1963/2305}, author = {Simonetta Abenda and Tamara Grava} } @article {2004, title = {Singular Z_N curves, Riemann-Hilbert problem and modular solutions of the Schlesinger equation}, journal = {Int. Math. Res. Not. 2004, no. 32, 1619-1683}, number = {arXiv.org;math-ph/0306050}, year = {2004}, abstract = {We are solving the classical Riemann-Hilbert problem of rank N>1 on the extended complex plane punctured in 2m+2 points, for NxN quasi-permutation monodromy matrices. Following Korotkin we solve the Riemann-Hilbert problem in terms of the Szego kernel of certain Riemann surfaces branched over the given 2m+2 points. These Riemann surfaces are constructed from a permutation representation of the symmetric group S_N to which the quasi-permutation monodromy representation has been reduced. The permutation representation of our problem generates the cyclic subgroup Z_N. For this reason the corresponding Riemann surfaces of genus N(m-1) have Z_N symmetry. This fact enables us to write the matrix entries of the solution of the NxN Riemann-Hilbert problem as a product of an algebraic function and theta-function quotients. The algebraic function turns out to be related to the Szego kernel with zero characteristics. From the solution of the Riemann- Hilbert problem we automatically obtain a particular solution of the Schlesinger system. The tau-function of the Schlesinger system is computed explicitly. The rank 3 problem with four singular points (0,t,1,\\\\infty) is studied in detail. The corresponding solution of the Riemann-Hilbert problem and the Schlesinger system is given in terms of Jacobi\\\'s theta-function with modulus T=T(t), Im(T)>0. The function T=T(t) is invertible if it belongs to the Siegel upper half space modulo the subgroup \\\\Gamma_0(3) of the modular group. The inverse function t=t(T) generates a solution of a general Halphen system.}, doi = {10.1155/S1073792804132625}, url = {http://hdl.handle.net/1963/2540}, author = {Victor Z. Enolski and Tamara Grava} } @mastersthesis {1998, title = {On the Cauchy Problem for the Whitham Equations}, year = {1998}, school = {SISSA}, keywords = {Korteweg de Vries equation}, url = {http://hdl.handle.net/1963/5555}, author = {Tamara Grava} }