Multi-component generalizations of the CH equation: geometrical aspects, peakons and numerical examples

Multi-component generalizations of the Hirota-Satsuma coupled KdV equation

In this paper, we consider multi-component generalizations of the Hirota–Satsuma coupled Korteweg–de Vries (KdV) equation. By introducing a Lax pair, we present a matrix generalization of the Hirota–Satsuma coupled KdV equation, which is shown to be reduced to a vector Hirota–Satsuma coupled KdV equation. By using Hirota's bilinear method, we find a few soliton solutions to the vector Hirota–Sa...

Geometrical Aspects of Exact Boundary Controllability for the Wave Equation a Numerical Study

This essentially numerical study sets out to investigate vari ous geometrical properties of exact boundary controllability of the wave equation when the control is applied on a part of the boundary Re lationships between the geometry of the domain the geometry of the controlled boundary the time needed to control and the energy of the control are dealt with A new norm of the control and an ener...

Bifurcation of Peakons and Cuspons of the Integrable Novikov Equation

By applying the bifurcation theory of dynamical systems to the Novikov equation, a new feature of non-smooth traveling wave solutions, two peakons or two cuspons that coexist for the same wave speed, is put forward. It is shown that 0 = g is the peakon bifurcation value in the process of obtaining the bifurcation of phase portraits, where g is a certain integration constant. In particular, we o...

Numerical Aspects of the Deterministic Solution of the Wigner Equation

Generalizations of the Wave Equation

The main result of this paper is a generalization of the property that, for smooth u, uxy = 0 implies (*) u(x, y) = a(x) + b(y). Any function having generalized unsymmetric mixed partial derivative identically zero is of the form (*). There is a function with generalized symmetric mixed partial derivative identically zero not of the form (*), but (*) does follow here with the additional assumpt...