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T. Grava

Publications and source records attributed to T. Grava.

At least 19 recordsLinked to original sources
arXiv2021

Adiabatic invariants for the FPUT and Toda chain in the thermodynamic limit

We consider the Fermi-Pasta-Ulam-Tsingou (FPUT) chain composed by $N \gg 1$ particles and periodic boundary conditions, and endow the phase space with the Gibbs measure at small temperature $β^{-1}$. Given a fixed ${1\leq m \ll N}$, we prove that the first $m$ integrals of motion of the periodic Toda chain are adiabatic invariants of FPUT (namely they are approximately constant along the Hamiltonian flow of the FPUT) for times of order $β$, for initial data in a set of large measure. We also prove that special linear combinations of the harmonic energies are adiabatic invariants of the FPUT on the same time scale, whereas they become adiabatic invariants for all times for the Toda dynamics.

math-ph↗
arXiv2017

Numerical study of the Kadomtsev--Petviashvili equation and dispersive shock waves

A detailed numerical study of the long time behaviour of dispersive shock waves in solutions to the Kadomtsev-Petviashvili (KP) I equation is presented. It is shown that modulated lump solutions emerge from the dispersive shock waves. For the description of dispersive shock waves, Whitham modulation equations for KP are obtained. It is shown that the modulation equations near the soliton line are hyperbolic for the KPII equation while they are elliptic for the KPI equation leading to a focusing effect and the formation of lumps. Such a behaviour is similar to the appearance of breathers for the focusing nonlinear Schrodinger equation in the semiclassical limit.

math.NA↗
arXiv2016

Spatial structure of shock formation

The formation of a singularity in a compressible gas, as described by the Euler equation, is characterized by the steepening, and eventual overturning of a wave. Using a self-similar description in two space dimensions, we show that the spatial structure of this process, which starts at a point, is equivalent to the formation of a caustic, i.e. to a cusp catastrophe. The lines along which the profile has infinite slope correspond to the caustic lines, from which we construct the position of the shock. By solving the similarity equation, we obtain a complete local description of wave steepening and of the spreading of the shock from a point.

physics.flu-dyn↗
arXiv2016

Shock formation in the dispersionless Kadomtsev-Petviashvili equation

The dispersionless Kadomtsev-Petviashvili (dKP) equation $(u_t+uu_x)_x=u_{yy}$ is one of the simplest nonlinear wave equations describing two-dimensional shocks. To solve the dKP equation we use a coordinate transformation inspired by the method of characteristics for the one-dimensional Hopf equation $u_t+uu_x=0$. We show numerically that the solutions to the transformed equation do not develop shocks. This permits us to extend the dKP solution as the graph of a multivalued function beyond the critical time when the gradients blow up. This overturned solution is multivalued in a lip shape region in the $(x,y)$ plane, where the solution of the dKP equation exists in a weak sense only, and a shock front develops. A local expansion reveals the universal scaling structure of the shock, which after a suitable change of coordinates corresponds to a generic cusp catastrophe. We provide a heuristic derivation of the shock front position near the critical point for the solution of the dKP equation, and study the solution of the dKP equation when a small amount of dissipation is added. Using multiple-scale analysis, we show that in the limit of small dissipation and near the critical point of the dKP solution, the solution of the dissipative dKP equation converges to a Pearcey integral. We test and illustrate our results by detailed comparisons with numerical simulations of both the regularized equation, the dKP equation, and the asymptotic description given in terms of the Pearcey integral.

math-ph↗
arXiv2015

On the tritronquée solutions of P$_I^2$

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 $u^{(m)}(x,t)$, $\hat u^{(m)}(x,t)$, $m=0,...,6$, called {\em tritronquée}, 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 tritronquée solutions.

math-ph↗
arXiv2013

On critical behaviour in systems of Hamiltonian partial differential equations

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 Painlevé-I (P$_I$) equation or its fourth order analogue P$_I^2$. As concrete examples we discuss nonlinear Schrödinger equations in the semiclassical limit. A numerical study of these cases provides strong evidence in support of the conjecture.

math-ph↗
arXiv2011

The KdV hierarchy: universality and a Painleve transcendent

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 Painlevé 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.

math-ph↗
arXiv2010

Numerical Study of breakup in generalized Korteweg-de Vries and Kawahara equations

This article is concerned with a conjecture by one of the authors on the formation of dispersive shocks in a class of Hamiltonian dispersive regularizations of the quasilinear 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 behaviour of their solution near the point of gradient catastrophe for the transport equation is described locally by a special solution of a Painlevé-type equation. This local description holds also for solutions to equations where blow up 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 $ε^2$ where $ε^2$ is the small dispersion parameter. Corrections up to order $ε^4 $ are obtained and tested numerically.

math-ph↗
arXiv2009

Solitonic asymptotics for the Korteweg-de Vries equation in the small dispersion limit

We study the small dispersion limit for the Korteweg-de Vries (KdV) equation $u_t+6uu_x+ε^{2}u_{xxx}=0$ 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.

math-ph↗
arXiv2009

Numerical Solution of the Small Dispersion Limit of the Camassa-Holm and Whitham Equations and Multiscale Expansions

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. We present a quantitative numerical comparison between the CH and the asymptotic solution. The dependence on the small dispersion parameter $ε$ is studied in the interior and at the boundaries of the Whitham zone. In the interior of the zone, the difference between CH and asymptotic solution is of the order $ε$, at the trailing edge of the order $\sqrtε$ and at the leading edge of the order $ε^{1/3}$. For the latter we present a multiscale expansion which describes the amplitude of the oscillations in terms of the Hastings-McLeod solution of the Painlevé II equation. We show numerically that this multiscale solution provides an enhanced asymptotic description near the leading edge.

math-ph↗
arXiv2008

Painleve II asymptotics near the leading edge of the oscillatory zone for the Korteweg-de Vries equation in the small dispersion limit

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 Painlevé II equation. We prove our results using the Riemann-Hilbert approach.

math-ph↗
arXiv2008

Numerical study of a multiscale expansion of the Korteweg de Vries equation and Painlevé-II equation

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 $ε$ in the interior of the Whitham oscillatory zone, it is known to be only of order $ε^{1/3}$ 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 Painlevé-II equation. We show numerically that the resulting multiscale solution approximates the KdV solution, in the small dispersion limit, to the order $ε^{2/3}$.

math-ph↗
arXiv2007

Reciprocal transformations and flat metrics on Hurwitz spaces

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.

nlin.SI↗
arXiv2007

On universality of critical behaviour in the focusing nonlinear Schrödinger equation, elliptic umbilic catastrophe and the {\it tritronquée} solution to the Painlevé-I equation

We argue that the critical behaviour near the point of ``gradient catastrophe" of the solution to the Cauchy problem for the focusing nonlinear Schrödinger equation $ iεψ_t +\frac{ε^2}2ψ_{xx}+ |ψ|^2 ψ=0$ with analytic initial data of the form $ψ(x,0;ε) =A(x) e^{\frac{i}ε S(x)}$ is approximately described by a particular solution to the Painlevé-I equation.

math.AP↗
arXiv2007

Numerical study of a multiscale expansion of KdV and Camassa-Holm equation

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 Painlevé 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

math-ph↗
arXiv2003

Singular Z_N curves, Riemann-Hilbert problem and modular solutions of the Schlesinger equation

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 Γ_0(3) of the modular group. The inverse function t=t(T) generates a solution of a general Halphen system.

math-ph↗
arXiv2002

Eigenvalue correlations on Hyperelliptic Riemann surfaces

In this note we compute the functional derivative of the induced charge density, on a thin conductor, consisting of the union of g+1 disjoint intervals, $J:=\cup_{j=1}^{g+1}(a_j,b_j),$ with respect to an external potential. In the context of random matrix theory this object gives the eigenvalue fluctuations of Hermitian random matrix ensembles where the eigenvalue density is supported on J.

math-ph↗