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Tamara Grava

Publications and source records attributed to Tamara Grava.

35 records · Page 2Linked to original sources
arXiv2021

Rigorous asymptotics of a KdV soliton gas

We analytically study the long time and large space asymptotics of a new broad class of solutions of the KdV equation introduced by Dyachenko, Zakharov, and Zakharov. These solutions are characterized by a Riemann--Hilbert problem which we show arises as the limit $N\to \infty$ of a gas of $N$-solitons. We show that this gas of solitons in the limit $N \to \infty$ is slowly approaching a cnoidal wave solution for $x \to - \infty$ (up to terms of order $\mathcal{O} (1/x)$), while approaching zero exponentially fast for $x\to+\infty$. We establish an asymptotic description of the gas of solitons for large times that is valid over the entire spatial domain, in terms of Jacobi elliptic functions.

math-ph↗
arXiv2021

On the long time asymptotic behavior of the modified Korteweg de Vries equation with step-like initial data

We study the long time asymptotic behaviour of the solution $q(x,t) $, to the modified Korteweg de Vries equation (MKDV) $q_t+6q^2q_x+q_{xxx}=0$ with step-like initial datum q(x,t=0)->c_- for x->-infinity and q(x,t=0)->c_+ for x-> +infinity. For the exact shock initial data q(x,t=0)=c_- for x<0 and q(x,t=0)=c_+ for x>0 the solution develops an oscillatory regions called dispersive shock wave that connects the two constant regions c_+ and c_-. We show that the dispersive shock wave is described by a modulated periodic travelling wave solution of the MKDV equation where the modulation parameters evolve according to the Whitham modulation equation. The oscillatory region is expanding within a cone in the $(x,t) plane defined as -6c_{-}^2+12c_{+}^2<x/t<4c_{-}^2+2c_{+}^2, with t\gg 1. For step like initial data we show that the solution decomposes for long times in three main regions: - a region where solitons and breathers travel with positive velocities on a constant background c_+; - an expanding oscillatory region that can contain generically breathers; - a region of breathers travelling with negative velocities on the constant background c_-. When the oscillatory region does not contain breathers, it coincides up to a phase shift with the dispersive shock wave solution obtained for the exact step initial data. The phase shift depends on the solitons, breathers and the radiation of the initial data. This shows that the dispersive shock wave is a coherent structure that interacts in an elastic way with solitons, breathers and radiation.

math-ph↗
arXiv2020

Correlation Functions for a Chain of Short Range Oscillators

We consider a system of harmonic oscillators with short range interactions and we study their correlation functions when the initial data is sampled with respect to the Gibbs measure. Such correlation functions display rapid oscillations that travel through the chain. We show that the correlation functions always have two fastest peaks which move in opposite directions and decay at rate $t^{-\frac{1}{3}}$ for position and momentum correlations and as $t^{-\frac{2}{3}}$ for energy correlations. The shape of these peaks is asymptotically described by the Airy function. Furthermore, the correlation functions have some non generic peaks with lower decay rates. In particular, there are peaks which decay at rate $t^{-\frac{1}{4}}$ for position and momentum correlators and with rate $t^{-\frac{1}{2}}$ for energy correlators. The shape of these peaks is described by the Pearcey integral. Crucial for our analysis is an appropriate generalisation of spacings, i.e. differences of the positions of neighbouring particles, that are used as spatial variables in the case of nearest neighbour interactions. Using the theory of circulant matrices we are able to introduce a quantity that retains both localisation and analytic viability. This also allows us to define and analyse some additional quantities used for nearest neighbour chains. Finally, we study numerically the evolution of the correlation functions after adding nonlinear perturbations to our model. Within the time range of our numerical simulations the asymptotic description of the linear case seems to persist for small nonlinear perturbations while stronger nonlinearities change shape and decay rates of the peaks significantly.

math-ph↗
arXiv2019

A representation of joint moments of CUE characteristic polynomials in terms of Painleve functions

We establish a representation of the joint moments of the characteristic polynomial of a CUE random matrix and its derivative in terms of a solution of the sigma-Painleve V equation. The derivation involves the analysis of a formula for the joint moments in terms of a determinant of generalised Laguerre polynomials using the Riemann-Hilbert method. We use this connection with the sigma-Painleve V equation to derive explicit formulae for the joint moments and to show that in the large-matrix limit the joint moments are related to a solution of the sigma-Painleve III equation. Using the conformal block expansion of the tau-functions associated with the sigma-Painleve V and the sigma-Painleve III equations leads to general conjectures for the joint moments.

math-ph↗
arXiv2018

Painlevé IV Critical Asymptotics for Orthogonal Polynomials in the Complex Plane

We study the asymptotic behaviour of orthogonal polynomials in the complex plane that are associated to a certain normal matrix model. The model depends on a parameter and the asymptotic distribution of the eigenvalues undergoes a transition for a special value of the parameter, where it develops a corner-type singularity. In the double scaling limit near the transition we determine the asymptotic behaviour of the orthogonal polynomials in terms of a solution of the Painlevé IV equation. We determine the Fredholm determinant associated to such solution and we compute it numerically on the real line, showing also that the corresponding Painlevé transcendent is pole-free on a semiaxis.

math-ph↗
arXiv2016

On the Tracy-Widom$_β$ Distribution for $β=6$

We study the Tracy-Widom distribution function for Dyson's $β$-ensemble with $β= 6$. The starting point of our analysis is the recent work of I. Rumanov where he produces a Lax-pair representation for the Bloemendal-Virág equation. The latter is a linear PDE which describes the Tracy-Widom functions corresponding to general values of $β$. Using his Lax pair, Rumanov derives an explicit formula for the Tracy-Widom $β=6$ function in terms of the second Painlevé transcendent and the solution of an auxiliary ODE. Rumanov also shows that this formula allows him to derive formally the asymptotic expansion of the Tracy-Widom function. Our goal is to make Rumanov's approach and hence the asymptotic analysis it provides rigorous. In this paper, the first one in a sequel, we show that Rumanov's Lax-pair can be interpreted as a certain gauge transformation of the standard Lax pair for the second Painlevé equation. This gauge transformation though contains functional parameters which are defined via some auxiliary nonlinear ODE which is equivalent to the auxiliary ODE of Rumanov's formula. The gauge-interpretation of Rumanov's Lax-pair allows us to highlight the steps of the original Rumanov's method which needs rigorous justifications in order to make the method complete. We provide a rigorous justification of one of these steps. Namely, we prove that the Painlevé function involved in Rumanov's formula is indeed, as it has been suggested by Rumanov, the Hastings-McLeod solution of the second Painlevé equation. The key issue which we also discuss and which is still open is the question of integrability of the auxiliary ODE in Rumanov's formula. We note that this question is crucial for the rigorous asymptotic analysis of the Tracy-Widom function.

math-ph↗
arXiv2016

Orthogonal polynomials for a class of measures with discrete rotational symmetries in the complex plane

We obtain the strong asymptotics of polynomials $p_n(λ)$, $λ\in\mathbb{C}$, orthogonal with respect to measures in the complex plane of the form $$ e^{-N(|λ|^{2s}-tλ^s-\overline{tλ}^s)}dA(λ), $$ where $s$ is a positive integer, $t$ is a complex parameter and $dA$ stands for the area measure in the plane. Such problem has its origin from normal matrix models. We study the asymptotic behaviour of $p_n(λ)$ in the limit $n,N\to\infty$ in such a way that $n/N\to T$ constant. Such asymptotic behaviour has two distinguished regimes according to the topology of the limiting support of the eigenvalue distribution of the normal matrix model. If $0<|t|^2 T/s$ the eigenvalue distribution support consists of $s$ connected components. Correspondingly the support of the limiting zero distribution of the orthogonal polynomials consists of a closed contour contained in each connected component. Our asymptotic analysis is obtained by reducing the planar orthogonality conditions of the polynomials to an equivalent system of contour integral orthogonality conditions. The strong asymptotics for the orthogonal polynomials is obtained from the corresponding Riemann--Hilbert problem by the Deift--Zhou nonlinear steepest descent method.

math-ph↗
arXiv2015

On critical behaviour in generalized Kadomtsev--Petviashvili equations

An asymptotic description of the formation of dispersive shock waves in solutions to the generalized Kadomtsev--Petviashvili (KP) equation is conjectured. The asymptotic description based on a multiscales expansion is given in terms of a special solution to an ordinary differential equation of the Painleve' I hierarchy. Several examples are discussed numerically to provide strong evidence for the validity of the conjecture.

math-ph↗
arXiv2015

Asymptotics for the partition function in two-cut random matrix models

We obtain large N asymptotics for the Hermitian random matrix partition function \[Z_N(V)=\int_{\mathbb R^N}\prod_{i<j}(x_i-x_j)^2 \prod_{j=1}^N e^{-N V(x_j)}dx_j,\] in the case where the external potential $V$ is a polynomials such that the random matrix eigenvalues accumulate on two disjoint intervals (the two-cut case). We compute leading and sub-leading terms in the asymptotic expansion for $\log Z_N(V)$, up to terms that are small as $N$ goes to infinity. Our approach is based on the explicit computation of the first terms in the asymptotic expansion for a quartic symmetric potential $V$. Afterwards, we use deformation theory of the partition function and of the associated equilibrium measure to generalize our results to general two-cut potentials $V$. The asymptotic expansion of $\log Z_N(V)$ as $N$ goes to infinity contains terms that depend analytically on the potential $V$ and that have already appeared in the literature. In addition our method allows to compute the $V$-independent terms of the asymptotic expansion of $\log Z_N(V)$ which, to the best of our knowledge, had not appeared before in the literature. We use rigorous orthogonal polynomial and Riemann-Hilbert techniques which had so far been successful to compute asymptotics for the partition function only in the one-cut case.

math-ph↗
arXiv2008

Initial Value Problem of the Whitham Equations for the Camassa-Holm Equation

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.

nlin.SI↗
arXiv2008

Universality of the break-up profile for the KdV equation in the small dispersion limit using the Riemann-Hilbert approach

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. The 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.

math-ph↗
arXiv2006

Thomae type formulae for singular Z_N curves

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.

math-ph↗
arXiv2005

Modulation of Camassa--Holm equation and reciprocal transformations

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.

math-ph↗
arXiv2005

Numerical solution of the small dispersion limit of Korteweg de Vries and Whitham equations

The Cauchy problem for the Korteweg de Vries (KdV) equation with small dispersion of order $ε^2$, is characterized by the appearance of a zone of rapid modulated oscillations of wave-length of order $ε$. 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 $ε$ between $10^{-1}$ and $10^{-3}$. The numerical results are compatible with a difference of order $ε$ within the `interior' of the Whitham oscillatory zone, of order $ε^{1/3}$ at the left boundary outside the Whitham zone and of order $ε^{1/2}$ at the right boundary outside the Whitham zone.

math-ph↗
arXiv2005

Large Parameter Behavior of Equilibrium Measures

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).

math-ph↗