An Averaging Theorem for Perturbed KdV Equation

We consider a perturbed KdV equation: [\dot{u}+u_{xxx} - 6uu_x = \epsilon f(x,u(\cdot)), \quad x\in \mathbb{T}, \quad\int_\mathbb{T} u dx=0.] For any periodic function $u(x)$, let $I(u)=(I_1(u),I_2(u),...)\in\mathbb{R}_+^{\infty}$ be the vector, formed by the KdV integrals of motion, calculated for the potential $u(x)$. Assuming that the perturbation $\epsilon f(x,u(\cdot))$ is a smoothing mapping (e.g. it is a smooth function $\epsilon f(x)$, independent from $u$), and that solutions of the perturbed equation satisfy some mild a-priori assumptions, we prove that for solutions $u(t,x)$ with typical initial data and for $0\leqslant t\lesssim \epsilon^{-1}$, the vector $I(u(t))$ may be well approximated by a solution of the averaged equation.Comment: 25 page

On long time dynamics of perturbed KdV equations

Consider perturbed KdV equations: $$u_t+u_{xxx}-6uu_x=\epsilon f(u(\cdot)),\quad x\in\mathbb{T}=\mathbb{R}/\mathbb{Z},\;\int_{\mathbb{T}}u(x,t)dx=0,$$ where the nonlinearity defines analytic operators $u(\cdot)\mapsto f(u(\cdot))$ in sufficiently smooth Sobolev spaces. Assume that the equation has an $\epsilon$-quasi-invariant measure $\mu$ and satisfies some additional mild assumptions. Let $u^{\epsilon}(t)$ be a solution. Then on time intervals of order $\epsilon^{-1}$, as $\epsilon\to0$, its actions $I(u^{\epsilon}(t,\cdot))$ can be approximated by solutions of a certain well-posed averaged equation, provided that the initial datum is $\mu$-typical

Long-time dynamics of resonant weakly nonlinear CGL equations

Consider a weakly nonlinear CGL equation on the torus~$\mathbb{T}^d$: u_t+i\Delta u=\epsilon [\mu(-1)^{m-1}\Delta^{m} u+b|u|^{2p}u+ ic|u|^{2q}u].\eqno{(*)} Here $u=u(t,x)$, $x\in\mathbb{T}^d$, $0<\epsilon<<1$, $\mu\geqslant0$, $b,c\in\mathbb{R}$ and $m,p,q\in\mathbb{N}$. Define \mbox{I(u)=(I_{\dk},\dk\in\mathbb{Z}^d)}, where I_{\dk}=v_{\dk}\bar{v}_{\dk}/2 and v_{\dk}, \dk\in\mathbb{Z}^d, are the Fourier coefficients of the function~$u$ we give. Assume that the equation $(*)$ is well posed on time intervals of order $\epsilon^{-1}$ and its solutions have there a-priori bounds, independent of the small parameter. Let $u(t,x)$ solve the equation $(*)$. If $\epsilon$ is small enough, then for $t\lesssim\epsilon^{-1}$, the quantity $I(u(t,x))$ can be well described by solutions of an {\it effective equation}: $$u_t=\epsilon[\mu(-1)^{m-1}\Delta^m u+ F(u)],$$ where the term $F(u)$ can be constructed through a kind of resonant averaging of the nonlinearity $b|u|^{2p}+ ic|u|^{2q}u$

KdV equation under periodic boundary conditions and its perturbations

In this paper we discuss properties of the KdV equation under periodic boundary conditions, especially those which are important to study perturbations of the equation. Next we review what is known now about long-time behaviour of solutions for perturbed KdV equations

Nearly circular domains which are integrable close to the boundary are ellipses

The Birkhoff conjecture says that the boundary of a strictly convex integrable billiard table is necessarily an ellipse. In this article, we consider a stronger notion of integrability, namely integrability close to the boundary, and prove a local version of this conjecture: a small perturbation of an ellipse of small eccentricity which preserves integrability near the boundary, is itself an ellipse. This extends the result in [1], where integrability was assumed on a larger set. In particular, it shows that (local) integrability near the boundary implies global integrability. One of the crucial ideas in the proof consists in analyzing Taylor expansion of the corresponding action-angle coordinates with respect to the eccentricity parameter, deriving and studying higher order conditions for the preservation of integrable rational caustics.Comment: 64 pages, 3 figures. Final revised version, to appear on Geometric and Functional Analysis (GAFA

A Three-Pole Substrate Integrated Waveguide Bandpass Filter Using New Coupling Scheme

A novel three-pole substrate integrated waveguide (SIW) bandpass filter (BPF) using new coupling scheme is proposed in this paper. Two high order degenerate modes (TE102 and TE201) of a square SIW cavity and a dominant mode (TE101) of a rectangular SIW cavity are coupled to form a three-pole SIW BPF. The coupling scheme of the structure is given and analyzed. Due to the coupling between two cavities, as well as the coupling between source and load, three transmission zeros are created in the stopband of the filter. The proposed three-pole SIW BPF is designed and fabricated. Good agreement between simulated and measured results verifies the validity of the design methodology well