245 research outputs found

### Long-time asymptotics for the Degasperis-Procesi equation on the half-line

We analyze the long-time asymptotics for the Degasperis--Procesi equation on
the half-line. By applying nonlinear steepest descent techniques to an
associated $3 \times 3$-matrix valued Riemann--Hilbert problem, we find an
explicit formula for the leading order asymptotics of the solution in the
similarity region in terms of the initial and boundary values.Comment: 61 pages, 11 figure

### Absence of continuous spectral types for certain nonstationary random models

We consider continuum random Schr\"odinger operators of the type $H_{\omega}
= -\Delta + V_0 + V_{\omega}$ with a deterministic background potential $V_0$.
We establish criteria for the absence of continuous and absolutely continuous
spectrum, respectively, outside the spectrum of $-\Delta +V_0$. The models we
treat include random surface potentials as well as sparse or slowly decaying
random potentials. In particular, we establish absence of absolutely continuous
surface spectrum for random potentials supported near a one-dimensional surface
(``random tube'') in arbitrary dimension.Comment: 14 pages, 2 figure

### Asymptotics of large eigenvalues for a class of band matrices

We investigate the asymptotic behaviour of large eigenvalues for a class of
finite difference self-adjoint operators with compact resolvent in $l^2$

### The Unified Method: II NLS on the Half-Line with $t$-Periodic Boundary Conditions

Boundary value problems for integrable nonlinear evolution PDEs formulated on
the half-line can be analyzed by the unified method introduced by one of the
authors and used extensively in the literature. The implementation of this
general method to this particular class of problems yields the solution in
terms of the unique solution of a matrix Riemann-Hilbert problem formulated in
the complex $k$-plane (the Fourier plane), which has a jump matrix with
explicit $(x,t)$-dependence involving four scalar functions of $k$, called
spectral functions. Two of these functions depend on the initial data, whereas
the other two depend on all boundary values. The most difficult step of the new
method is the characterization of the latter two spectral functions in terms of
the given initial and boundary data, i.e. the elimination of the unknown
boundary values. For certain boundary conditions, called linearizable, this can
be achieved simply using algebraic manipulations. Here, we first present an
effective characterization of the spectral functions in terms of the given
initial and boundary data for the general case of non-linearizable boundary
conditions. This characterization is based on the analysis of the so-called
global relation and on the introduction of the so-called
Gelfand-Levitan-Marchenko representations of the eigenfunctions defining the
spectral functions. We then concentrate on the physically significant case of
$t$-periodic Dirichlet boundary data. After presenting certain heuristic
arguments which suggest that the Neumann boundary values become periodic as
$t\to\infty$, we show that for the case of the NLS with a sine-wave as
Dirichlet data, the asymptotics of the Neumann boundary values can be computed
explicitly at least up to third order in a perturbative expansion and indeed at
least up to this order are asymptotically periodic.Comment: 29 page

### The Generalized Dirichlet to Neumann map for the KdV equation on the half-line

For the two versions of the KdV equation on the positive half-line an
initial-boundary value problem is well posed if one prescribes an initial
condition plus either one boundary condition if $q_{t}$ and $q_{xxx}$ have the
same sign (KdVI) or two boundary conditions if $q_{t}$ and $q_{xxx}$ have
opposite sign (KdVII). Constructing the generalized Dirichlet to Neumann map
for the above problems means characterizing the unknown boundary values in
terms of the given initial and boundary conditions. For example, if
$\{q(x,0),q(0,t) \}$ and $\{q(x,0),q(0,t),q_{x}(0,t) \}$ are given for the KdVI
and KdVII equations, respectively, then one must construct the unknown boundary
values $\{q_{x}(0,t),q_{xx}(0,t) \}$ and $\{q_{xx}(0,t) \}$, respectively. We
show that this can be achieved without solving for $q(x,t)$ by analysing a
certain ``global relation'' which couples the given initial and boundary
conditions with the unknown boundary values, as well as with the function
$\Phi^{(t)}(t,k)$, where $\Phi^{(t)}$ satisifies the $t$-part of the associated
Lax pair evaluated at $x=0$. Indeed, by employing a Gelfand--Levitan--Marchenko
triangular representation for $\Phi^{(t)}$, the global relation can be solved
\emph{explicitly} for the unknown boundary values in terms of the given initial
and boundary conditions and the function $\Phi^{(t)}$. This yields the unknown
boundary values in terms of a nonlinear Volterra integral equation.Comment: 21 pages, 3 figure

### Long-Time Asymptotics for the Camassa-Holm Equation

We apply the method of nonlinear steepest descent to compute the long-time
asymptotics of the Camassa-Holm equation for decaying initial data, completing
previous results by A. Boutet de Monvel and D. Shepelsky.Comment: 30 page

### The random link approximation for the Euclidean traveling salesman problem

The traveling salesman problem (TSP) consists of finding the length of the
shortest closed tour visiting N ``cities''. We consider the Euclidean TSP where
the cities are distributed randomly and independently in a d-dimensional unit
hypercube. Working with periodic boundary conditions and inspired by a
remarkable universality in the kth nearest neighbor distribution, we find for
the average optimum tour length = beta_E(d) N^{1-1/d} [1+O(1/N)] with
beta_E(2) = 0.7120 +- 0.0002 and beta_E(3) = 0.6979 +- 0.0002. We then derive
analytical predictions for these quantities using the random link
approximation, where the lengths between cities are taken as independent random
variables. From the ``cavity'' equations developed by Krauth, Mezard and
Parisi, we calculate the associated random link values beta_RL(d). For d=1,2,3,
numerical results show that the random link approximation is a good one, with a
discrepancy of less than 2.1% between beta_E(d) and beta_RL(d). For large d, we
argue that the approximation is exact up to O(1/d^2) and give a conjecture for
beta_E(d), in terms of a power series in 1/d, specifying both leading and
subleading coefficients.Comment: 29 pages, 6 figures; formatting and typos correcte

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