arXiv subjects
Publications and source records attributed to Joseph Thirouin.
We give a complete classification of the traveling waves of the following quadratic Szeg{ö} equation : $i \partial\_t u = 2JΠ(|u|^2)+\bar{J}u^2, \quad u(0, \cdot)=u\_0$, and we show that they are given by two families of rational functions, one of which is generated by a stable ground state. We prove that the other branch is orbitally unstable.
The purpose of this paper is to go further into the study of the quadratic Szeg{ö} equation, which is the following Hamiltonian PDE : $i \partial\_t u = 2JΠ(|u|^2)+\bar{J}u^2$, $u(0, \cdot)=u\_0$, where $Π$ is the Szeg{ö} projector onto nonnegative modes, and $J = J(u)$ is the complex number given by $J=\int\_\mathbb{T}|u|^2u$. We exhibit an infinite set of new conservation laws $\{\ell\_k \}$ which are in involution. These laws give us a better understanding of the "turbulent" behavior of certain rational solutions of the equation : we show that if the orbit of a rational solution is unbounded in some $H^s$, $s > 1/2$, then one of the $\ell\_k$'s must be zero. As a consequence, we characterize growing solutions which can be written as the sum of two solitons.
In this paper, we study a quadratic equation on the one-dimensional torus : $$i \partial_t u = 2JΠ(|u|^2)+\bar{J}u^2, \quad u(0, \cdot)=u_0,$$ where $J=\int_\mathbb{T}|u|^2u \in\mathbb{C}$ has constant modulus, and $Π$ is the Szegő projector onto functions with nonnegative frequencies. Thanks to a Lax pair structure, we construct a flow on BMO$(\mathbb{T})\cap \mathrm{Im}Π$ which propagates $H^s$ regularity for any $s>0$, whereas the energy level corresponds to $s=1/2$. Then, for each $s>1/2$, we exhibit solutions whose $H^s$ norm goes to $+\infty$ exponentially fast, and we show that this growth is optimal.
This paper is devoted to the study of large time bounds for the Sobolev norms of the solutions of the following fractional cubic Schr{ö}dinger equation on the torus :$$i \partial\_t u = |D|^αu+|u|^2 u, \quad u(0, \cdot)=u\_0,$$where $α$ is a real parameter. We show that, apart from the case $α= 1$, which corresponds to a half-wave equation with no dispersive property at all, solutions of this equation grow at a polynomial rate at most. We also address the case of the cubic and quadratic half-wave equations.