Semiclassical Limit for Generalized KdV Equations Before the Gradient Catastrophe

@article{Masoero2011SemiclassicalLF,
  title={Semiclassical Limit for Generalized KdV Equations Before the Gradient Catastrophe},
  author={Davide Masoero and Andrea Raimondo},
  journal={Letters in Mathematical Physics},
  year={2011},
  volume={103},
  pages={559-583}
}
We study the semiclassical limit of the (generalized) KdV equation, for initial data with Sobolev regularity, before the time of the gradient catastrophe of the limit conservation law. In particular, we show that in the semiclassical limit the solution of the KdV equation: i) converges in Hs to the solution of the Hopf equation, provided the initial data belongs to Hs, ii) admits an asymptotic expansion in powers of the semiclassical parameter, if the initial data belongs to the Schwartz class… 

A Deformation of the Method of Characteristics and the Cauchy Problem for Hamiltonian PDEs in the Small Dispersion Limit

We introduce a deformation of the method of characteristics valid for Hamiltonian perturbations of a scalar conservation law in the small dispersion limit. Our method of analysis is based on the

Integrable viscous conservation laws

We propose an extension of the Dubrovin–Zhang perturbative approach to the study of normal forms for non-Hamiltonian integrable scalar conservation laws. The explicit computation of the first few

Critical asymptotic behavior for the Korteweg-de Vries equation and in random matrix theory

We discuss universality in random matrix theory and in the study of Hamiltonian partial differential equations. We focus on universality of critical behavior and we compare results in unitary random

Asymptotic analysis of noisy fitness maximization, applied to metabolism & growth

A population dynamics model coupling cell growth to a diffusion in the space of metabolic phenotypes as it can be obtained from realistic constraints-based modeling is considered suggesting that suboptimal populations can have a faster response to perturbations.

On Critical Behaviour in Systems of Hamiltonian Partial Differential Equations

It is argued 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$$_I$$I) equation or its fourth-order analogue P$$-I^2$$I2.

References

SHOWING 1-10 OF 24 REFERENCES

Universality of the Break-up Profile for the KdV Equation in the Small Dispersion Limit Using the Riemann-Hilbert Approach

AbstractWe obtain an asymptotic expansion for the solution of the Cauchy problem for the Korteweg-de Vries (KdV) equation $$u_t+6uu_x+\epsilon^{2}u_{xxx}=0,\quad u(x,t=0,\epsilon)=u_0(x),$$for

Semiclassical limit of the nonlinear Schrödinger equation in small time

We study the semi-classical limit of the nonlinear Schr6dinger equation for initial data with Sobolev regularity, before shocks appear in the limit system, and in particular the validity of the WKB

A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the MKdV equation

In this article we present a new and general approach to analyzing the asymptotics of oscillatory Riemann-Hilbert problems. Such problems arise, in particular, when evaluating the long-time behavior

On Hamiltonian Perturbations of Hyperbolic Systems of Conservation Laws, II: Universality of Critical Behaviour

Hamiltonian perturbations of the simplest hyperbolic equation ut + a(u) ux = 0 are studied. We argue that the behaviour of solutions to the perturbed equation near the point of gradient catastrophe

An extension of the steepest descent method for Riemann-Hilbert problems: the small dispersion limit of the Korteweg-de Vries (KdV) equation.

This paper extends the steepest descent method for Riemann-Hilbert problems introduced by Deift and Zhou in a critical new way to small dispersion KdV (Korteweg-de Vries) equation and derives the hyperelliptic asymptotic solution of S. Venakides that describes the oscillations.

The zero dispersion limit of the korteweg‐de vries equation for initial potentials with non‐trivial reflection coefficient

The inverse scattering method is used to determine the distribution limit as ϵ 0 of the solution u(x, t, ϵ) of the initial value problem. Ut − 6uux + ϵ2uxxx = 0, u(x, 0) = v(x), where v(x) is

On the Cauchy Problem for the Generalized Korteweg-de Vries Equation

Abstract We consider the local and global Cauchy problem for the generalized Korteweg-de Vries equation , with initial data in homogeneous and nonhomogeneous Besov spaces. This allows us to slightly

The small dispersion limit of the Korteweg‐de Vries equation. III

In Parts I and II we have derived explicit formulas for the distribution limit u of the solution of the KdV equation as the coefficient of uxxx tends to zero. This formula contains n parameters β1,

The Small Dispersion Limit of the Korteweg-De Vries Equation

There are many physical systems which display shocks i.e. regions in space where the solution develops extremely large slopes. In general, such systems are too complicated to be treated by exact

The initial-value problem for the Korteweg-de Vries equation

  • J. BonaR. Smith
  • Mathematics
    Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences
  • 1975
For the Korteweg-de Vries equation ut+ux+uux+uxxx=0, existence, uniqueness, regularity and continuous dependence results are established for both the pure initial-value problem (posed on -∞<x<∞) and