Self-Similar Blowup Solutions to the 2-Component Camassa-Holm Equations

In this article, we study the self-similar solutions of the 2-component Camassa-Holm equations% \begin{equation} \left\{ \begin{array} [c]{c}% \rho_{t}+u\rho_{x}+\rho u_{x}=0 m_{t}+2u_{x}m+um_{x}+\sigma\rho\rho_{x}=0 \end{array} \right. \end{equation} with \begin{equation} m=u-\alpha^{2}u_{xx}. \end{equation} By the separation method, we can obtain a class of blowup or global solutions for $\sigma=1$ or $-1$. In particular, for the integrable system with $\sigma=1$, we have the global solutions:% \begin{equation} \left\{ \begin{array} [c]{c}% \rho(t,x)=\left\{ \begin{array} [c]{c}% \frac{f\left( \eta\right) }{a(3t)^{1/3}},\text{ for }\eta^{2}<\frac {\alpha^{2}}{\xi} 0,\text{ for }\eta^{2}\geq\frac{\alpha^{2}}{\xi}% \end{array} \right. ,u(t,x)=\frac{\overset{\cdot}{a}(3t)}{a(3t)}x \overset{\cdot\cdot}{a}(s)-\frac{\xi}{3a(s)^{1/3}}=0,\text{ }a(0)=a_{0}% >0,\text{ }\overset{\cdot}{a}(0)=a_{1} f(\eta)=\xi\sqrt{-\frac{1}{\xi}\eta^{2}+\left( \frac{\alpha}{\xi}\right) ^{2}}% \end{array} \right. \end{equation} where $\eta=\frac{x}{a(s)^{1/3}}$ with $s=3t;$ $\xi>0$ and $\alpha\geq0$ are arbitrary constants.\newline Our analytical solutions could provide concrete examples for testing the validation and stabilities of numerical methods for the systems.Comment: 5 more figures can be found in the corresponding journal paper (J. Math. Phys. 51, 093524 (2010) ). Key Words: 2-Component Camassa-Holm Equations, Shallow Water System, Analytical Solutions, Blowup, Global, Self-Similar, Separation Method, Construction of Solutions, Moving Boundar

Acoustoelectric effect in a finite-length ballistic quantum channel

The dc current induced by a coherent surface acoustic wave (SAW) of wave vector q in a ballistic channel of length L is calculated. The current contains two contributions, even and odd in q. The even current exists only in a asymmetric channel, when the electron reflection coefficients r_1 and r_2 at both channel ends are different. The direction of the even current does not depend on the direction of the SAW propagation, but is reversed upon interchanging r_1 and r_2. The direction of the odd current is correlated with the direction of the SAW propagation, but is insensitive to the interchange of r_1 and r_2. It is shown that both contributions to the current are non zero only when the electron reflection coefficients at the channel ends are energy dependent. The current exhibits geometric oscillations as function of qL. These oscillations are the hallmark of the coherence of the SAW and are completely washed out when the current is induced by a flux of non-coherent phonons. The results are compared with those obtained previously by different methods and under different assumptions.Comment: 7 pages, 2 figure

Gravity travelling waves for two superposed fluid layers, one being of infinite depth: a new type of bifurcation

International audienceIn this paper, we study the travelling gravity waves in a system of two layers of perfect fluids, the bottom one being infinitely deep, the upper one having a finite thickness h. We assume that the flow is potential, and the dimensionless parameters are the ratio between densities ρ = ρ 2 /ρ 1 and λ = gh/c^2. We study special values of the parameters such that λ(1 − ρ) is near 1 − , where a bifurcation of a new type occurs. We formulate the problem as a spatial reversible dynamical system, where U = 0 corresponds to a uniform state (velocity c in a moving reference frame), and we consider the linearized operator around 0. We show that its spectrum contains the entire real axis (essential spectrum), with in addition a double eigenvalue in 0, a pair of simple imaginary eigenvalues ±iλ at a distance O(1) from 0, and for λ(1 − ρ) above 1, another pair of simple imaginary eigenvalues tending towards 0 as λ(1 − ρ) → 1 +. When λ(1 − ρ) ≤ 1 this pair disappears into the essential spectrum. The rest of the spectrum lies at a distance at least O(1) from the imaginary axis. We show in this paper that for λ(1 − ρ) close to 1 − , there is a family of periodic solutions like in the Lyapunov-Devaney theorem (despite the resonance due to the point 0 in the spectrum). Moreover, showing that the full system can be seen as a perturbation of the Benjamin-Ono equation, coupled with a nonlinear oscillation, we also prove the existence of a family of homoclinic connections to these periodic orbits, provided that these ones are not too small