1,205 research outputs found

### 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 Painlev\'e 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

### Emergence of a singularity for Toeplitz determinants and Painleve V

We obtain asymptotic expansions for Toeplitz determinants corresponding to a
family of symbols depending on a parameter $t$. For $t$ positive, the symbols
are regular so that the determinants obey Szeg\H{o}'s strong limit theorem. If
$t=0$, the symbol possesses a Fisher-Hartwig singularity. Letting $t\to 0$ we
analyze the emergence of a Fisher-Hartwig singularity and a transition between
the two different types of asymptotic behavior for Toeplitz determinants. This
transition is described by a special Painlev\'e V transcendent. A particular
case of our result complements the classical description of Wu, McCoy, Tracy,
and Barouch of the behavior of a 2-spin correlation function for a large
distance between spins in the two-dimensional Ising model as the phase
transition occurs.Comment: 46 page

### 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
$u_t+6uu_x+\epsilon^{2}u_{xxx}=0$ 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.Comment: 25 pages, 4 figure

### Numerical study of the small dispersion limit of the Korteweg-de Vries equation and asymptotic solutions

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

### The existence of a real pole-free solution of the fourth order analogue of the Painleve I equation

We establish the existence of a real solution y(x,T) with no poles on the
real line of the following fourth order analogue of the Painleve I equation,
x=Ty-({1/6}y^3+{1/24}(y_x^2+2yy_{xx})+{1/240}y_{xxxx}). This proves the
existence part of a conjecture posed by Dubrovin. We obtain our result by
proving the solvability of an associated Riemann-Hilbert problem through the
approach of a vanishing lemma. In addition, by applying the Deift/Zhou
steepest-descent method to this Riemann-Hilbert problem, we obtain the
asymptotics for y(x,T) as x\to\pm\infty.Comment: 27 pages, 5 figure

### Asymptotics for a special solution to the second member of the Painleve I hierarchy

We study the asymptotic behavior of a special smooth solution y(x,t) to the
second member of the Painleve I hierarchy. This solution arises in random
matrix theory and in the study of Hamiltonian perturbations of hyperbolic
equations. The asymptotic behavior of y(x,t) if x\to \pm\infty (for fixed t) is
known and relatively simple, but it turns out to be more subtle when x and t
tend to infinity simultaneously. We distinguish a region of algebraic
asymptotic behavior and a region of elliptic asymptotic behavior, and we obtain
rigorous asymptotics in both regions. We also discuss two critical transitional
asymptotic regimes.Comment: 19 page

### Critical asymptotic behavior for the Korteweg\u2013de 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 matrix ensembles with their coun- terparts for the Korteweg\u2013de Vries equation, emphasizing the similarities between both subjects

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