We solve the Cauchy problem for the modified Korteweg–de Vries equation with steplike quasi-periodic, finite-gap initial conditions under the assumption that the perturbations have a given number of derivatives and moments finite.

A limit theorem for the total current in the asymmetric simple exclusion process (ASEP) with step initial condition is proved. This extends the result of Johansson on TASEP to ASEP.

Let Ω be a an open bounded domain in R , N ≥ 2, with a smooth boundary ∂Ω. Let QT Ω × 0, T and ΓT ∂Ω × 0, T where T is a positive real number. Then ΓT is smooth and any point on ΓT satisfies the inside and outside strong sphere property see 1 . For u : QT → R we denote its partial derivatives in the distributional sense when they exist byDtu ∂u/∂t, Diu ∂u/∂xi,DiDju ∂u/∂xi∂xj , i, j 1, . . . ,N....

We study the asymptotic behavior for solutions to nonlocal diffusion models of the form ut = J ∗ u − u in the whole R with an initial condition u(x, 0) = u0(x). Under suitable hypotheses on J (involving its Fourier transform) and u0, it is proved an expansion of the form ∥∥u(u)− ∑ |α|≤k (−1)|α| α! ( ∫ u0(x)x dx ) ∂Kt ∥∥ Lq(Rd) ≤ Ct−A, where Kt is the regular part of the fundamental solution and...

In this note, we propose two uniform sliding mode controllers (SMCs) for a second-order uncertain system. They provide convergence to an arbitrarily small vicinity centred at the origin in finite time, which can be bounded by some constant independent of the initial conditions and uncertainties. Towards this aim, a non-linear sliding surface is designed ensuring convergence, during the sliding ...

We derive a formula for computing explicitly the optimal non-linear filter for a class of problems which admit finite dimensional filters. The result includes all known results of this kind as special cases.

We illustrate general techniques for assessing dynamic stability in games of incomplete information by re-analyzing two models of preference evolution, the Arce (2007) PrincipalAgent game and the Friedman and Singh (2009) Noisy Trust game. The techniques include extensions of replicator and gradient dynamics, and for both models they confirm local stability of the key static equilibria. That is...

We describe a new, spectrally accurate method for solving matrixvalued Riemann–Hilbert problems numerically. The effectiveness of this approach is demonstrated by computing solutions to the homogeneous Painlevé II equation. This can be used to relate initial conditions with asymptotic behaviour.

We consider the wave equation in an unbounded conical domain, with initial conditions and boundary conditions of Dirichlet or Neumann type. We give a uniform decay estimate of the solution in terms of weighted Sobolev norms of the initial data. The decay rate is the same as in the full space case.

Abstract—Laplace transformations have wide applications in engineering and sciences. All previous studies of modified Laplace transformations depend on differential equation with initial conditions. The purpose of our paper is to solve the linear differential equations (not initial value problem) and then find the general solution (not particular) via the Laplace transformations without needed ...