Equations of Langmuir turbulence and nonlinear Schrödinger equation: Smoothness and approximation
We consider the following family of systems parametrized by e > 0, which describe Langmuir's turbulence. In the case where initial data are sufficiently small, we study the asymptotic behaviour of the solutions (Ee, ne) when e goes to zero, for d = 1, 2, 3. Namely, we state convergence results of (Ee, ne) to the couple (E, − ¦E¦2) where E is the solution of the nonlinear Schrodinger equation: Moreover, in three dimensions we show that the smooth solution (Ee, ne) is defined on [0, Tmax(e)[with Tmax(e) going to infinity as e goes to 0.
We apply the $\delta$-expansion perturbation scheme to the $\lambda \phi^{4}$ self-interacting scalar field theory in 3+1 D at finite temperature. In the $\delta$-expansion the interaction term is written as $\lambda (\phi^{2})^{ 1 + \delta}$ and $\delta$ is considered as the perturbation parameter. We compute within this perturbative approach the renormalized mass at finite temperature at a finite order in $\delta$. The results are compared with the usual loop-expansion at finite temperature.