Browsing by Author "Klein, Christian"

Now showing 1 - 5 of 5
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    Numerical solution of the small dispersion limit of Korteweg de Vries and Whitham equations
    (2005) Grava, Tamara; Klein, Christian; Mathematics; Mathematical Physics
    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 `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.
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    Numerical Solution of the Small Dispersion Limit of the Camassa-Holm and Whitham Equations and Multiscale Expansions
    (2010-02-05T10:20:57Z) Abenda, Simonetta; Grava, Tamara; Klein, Christian; Mathematics; Mathematical Physics
    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....
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    Numerical study of a multiscale expansion of KdV and Camassa-Holm equation
    (2007-12-12T12:39:24Z) Grava, Tamara; Klein, Christian; Mathematics; Mathematical Physics
    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
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    On critical behaviour in systems of Hamiltonian partial differential equations
    (SISSA, 2014-01-15) Dubrovin, Boris; Grava, Tamara; Klein, Christian; Moro, Antonio; Mathematics
    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.
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    On the tritronquée solutions of P$_I^2$
    (SISSA, 2014-01-15) Grava, Tamara; Kapaev, Andrei; Klein, Christian; Mathematics
    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.