Silvana De Lillo

On the Burgers equation with moving boundary

Abstract A general method for solving the Dirichlet problem for the Burgers equation with a moving boundary is introduced. The method reduces the initial value problem to a linear integral equation of Volterra type with mildly singular kernel, which admits a unique solution under rather general assumptions. Two explicit cases are considered: a boundary moving with constant velocity and a rapidly oscillating boundary.

“Follow the leader” learning dynamics on networks

Abstract A learning dynamics on network is introduced, characterized by binary and multiple nonlinear interactions among the individuals distributed in the different nodes. A particular topology of the network is considered by introducing a leader node which influences all the other “follower” nodes without being influenced in turn. Numerical simulations are provided, particularly focusing on the effect of the network structure and of the nonlinear interactions on the emerging behaviour of the system. It turns out that the leader node always exhibits an autonomous evolution, while the follower nodes may have a regression when the interactions with the leader node are switched off. There is instead a remarkable change in the final configurations of the follower nodes, even if only one of them is connected to the leader: indeed, due to the microscopic interactions among them, all the follower nodes feel a strong “leader effect”.

Neumann problem on the semi-line for the Burgers equation

In this article, the Neumann problem on the semi-line for the Burgers equation is considered. The problem is reduced to a nonlinear integral equation in one independent variable, whose unique solution is proven to exist for small time. An explicit solution is discussed as well.

Stochastic Effects on the Eckhaus Equation

The random-force driven Eckhaus equation is studied in the case of a long range correlated noise. The ensemble average of the Kink solution is obtained, and some relevant correlation functions are obtained.

On the Forced Eckhaus Equation

Parametrically forced nonlinear Schroedinger type equations have recently been investigated1 in connection with the existence of nonlinear bound state solutions and of their stability. In this note we review some recent results2, concerning the Eckhaus equation in an external potential V(x) and the properties of its bound state solutions. Some new results, related to the time asymptotic behavior of the solutions, will also be reported.

Note on a Complex Eckhaus Equation

A complex Eckhaus equation which is easily Bose quantized possesses an infinity of local conservation laws. Thus this equation is S-integrable as well as C-integrable.