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 Schrödinger equation in the semiclassical limit.

VL - 474 UR - https://royalsocietypublishing.org/doi/abs/10.1098/rspa.2017.0458 ER - TY - RPRT T1 - On critical behaviour in systems of Hamiltonian partial differential equations Y1 - 2013 A1 - Boris Dubrovin A1 - Tamara Grava A1 - Christian Klein A1 - Antonio Moro AB -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.

PB - SISSA U1 - 7280 U2 - Mathematics U4 - 1 U5 - MAT/07 FISICA MATEMATICA ER - TY - RPRT T1 - On the tritronquée solutions of P$_I^2$ Y1 - 2013 A1 - Tamara Grava A1 - Andrey Kapaev A1 - Christian Klein AB -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.

PB - SISSA U1 - 7282 U2 - Mathematics U4 - 1 U5 - MAT/07 FISICA MATEMATICA ER - TY - JOUR T1 - Numerical study of the small dispersion limit of the Korteweg-de Vries equation and asymptotic solutions JF - Physica D 241, nr. 23-24 (2012): 2246-2264 Y1 - 2012 A1 - Tamara Grava A1 - Christian Klein KW - Korteweg-de Vries equation AB - 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. PB - Elsevier U1 - 7069 U2 - Physics U4 - -1 ER - TY - JOUR T1 - Numerical Study of breakup in generalized Korteweg-de Vries and Kawahara equations JF - SIAM J. Appl. Math. 71 (2011) 983-1008 Y1 - 2011 A1 - Boris Dubrovin A1 - Tamara Grava A1 - Christian Klein AB - This article is concerned with a conjecture in [B. Dubrovin, Comm. Math. Phys., 267 (2006), pp. 117–139] 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 Painlevé-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. PB - SIAM UR - http://hdl.handle.net/1963/4951 U1 - 4732 U2 - Mathematics U3 - Mathematical Physics U4 - -1 ER - TY - RPRT T1 - Numerical Solution of the Small Dispersion Limit of the Camassa-Holm and Whitham Equations and Multiscale Expansions Y1 - 2010 A1 - Simonetta Abenda A1 - Tamara Grava A1 - Christian Klein AB - 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.... UR - http://hdl.handle.net/1963/3840 U1 - 487 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - On universality of critical behaviour in the focusing nonlinear Schrödinger equation, elliptic umbilic catastrophe and the \\\\it tritronquée solution to the Painlevé-I equation JF - J. Nonlinear Sci. 19 (2009) 57-94 Y1 - 2009 A1 - Boris Dubrovin A1 - Tamara Grava A1 - Christian Klein AB - We argue that the critical behaviour near the point of ``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. UR - http://hdl.handle.net/1963/2525 U1 - 1593 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Numerical study of a multiscale expansion of the Korteweg-de Vries equation and Painlevé-II equation JF - Proc. R. Soc. A 464 (2008) 733-757 Y1 - 2008 A1 - Tamara Grava A1 - Christian Klein AB - 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 $\\\\epsilon^{2/3}$. UR - http://hdl.handle.net/1963/2592 U1 - 1530 U2 - Mathematics U3 - Mathematical Physics ER - TY - RPRT T1 - Numerical solution of the small dispersion limit of Korteweg de Vries and Whitham equations Y1 - 2007 A1 - Tamara Grava A1 - Christian Klein AB - 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. JF - Comm. Pure Appl. Math. 60 (2007) 1623-1664 UR - http://hdl.handle.net/1963/1788 U1 - 2756 U2 - Mathematics U3 - Mathematical Physics ER - TY - RPRT T1 - Numerical study of a multiscale expansion of KdV and Camassa-Holm equation Y1 - 2007 A1 - Tamara Grava A1 - Christian Klein AB - 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 UR - http://hdl.handle.net/1963/2527 U1 - 1591 U2 - Mathematics U3 - Mathematical Physics ER -