Perturbational Blowup Solutions to the 2-Component Camassa-Holm Equations

In this article, we study the perturbational method to construct the non-radially symmetric solutions of the compressible 2-component Camassa-Holm equations. In detail, we first combine the substitutional method and the separation method to construct a new class of analytical solutions for that system. In fact, we perturb the linear velocity: u=c(t)x+b(t), and substitute it into the system. Then, by comparing the coefficients of the polynomial, we can deduce the functional differential equations involving $(c(t),b(t),\rho^{2}(0,t)).$ Additionally, we could apply the Hubble's transformation c(t)={\dot{a}(3t)}/{a(3t)}, to simplify the ordinary differential system involving $(a(3t),b(t),\rho ^{2}(0,t))$. After proving the global or local existences of the corresponding dynamical system, a new class of analytical solutions is shown. And the corresponding solutions in radial symmetry are also given. To determine that the solutions exist globally or blow up, we just use the qualitative properties about the well-known Emden equation: {array} [c]{c} {d^{2}/{dt^{2}}}a(3t)= {\xi}{a^{1/3}(3t)}, a(0)=a_{0}>0 ,\dot{a}(0)=a_{1} {array} . Our solutions obtained by the perturbational method, fully cover the previous known results in "M.W. Yuen, \textit{Self-Similar Blowup Solutions to the 2-Component Camassa-Holm Equations,}J. Math. Phys., \textbf{51} (2010) 093524, 14pp." by the separation method.Comment: 12 page

Some Exact Blowup Solutions to the Pressureless Euler Equations in R^N

The pressureless Euler equations can be used as simple models of cosmology or plasma physics. In this paper, we construct the exact solutions in non-radial symmetry to the pressureless Euler equations in $R^{N}:$% [c]{c}% \rho(t,\vec{x})=\frac{f(\frac{1}{a(t)^{s}}\underset{i=1}{\overset {N}{\sum}}x_{i}^{s})}{a(t)^{N}}\text{,}\vec{u}(t,\vec{x}% )=\frac{\overset{\cdot}{a}(t)}{a(t)}\vec{x}, a(t)=a_{1}+a_{2}t. \label{eq234}% where the arbitrary function $f\geq0$ and $f\in C^{1};$ $s\geq1$, $a_{1}>0$ and $a_{2}$ are constants$.$\newline In particular, for $a_{2}<0$, the solutions blow up on the finite time $T=-a_{1}/a_{2}$. Moreover, the functions (\ref{eq234}) are also the solutions to the pressureless Navier-Stokes equations.Comment: 7 pages Key Words: Pressureless Gas, Euler Equations, Exact Solutions, Non-Radial Symmetry, Navier-Stokes Equations, Blowup, Free Boundar

Self-Similar Blowup Solutions to the 2-Component Degasperis-Procesi Shallow Water System

In this article, we study the self-similar solutions of the 2-component Degasperis-Procesi water system:% [c]{c}% \rho_{t}+k_{2}u\rho_{x}+(k_{1}+k_{2})\rho u_{x}=0 u_{t}-u_{xxt}+4uu_{x}-3u_{x}u_{xx}-uu_{xxx}+k_{3}\rho\rho_{x}=0. By the separation method, we can obtain a class of self-similar solutions,% [c]{c}% \rho(t,x)=\max(\frac{f(\eta)}{a(4t)^{(k_{1}+k_{2})/4}},\text{}0),\text{}u(t,x)=\frac{\overset{\cdot}{a}(4t)}{a(4t)}x \overset{\cdot\cdot}{a}(s)-\frac{\xi}{4a(s)^{\kappa}}=0,\text{}a(0)=a_{0}% \neq0,\text{}\overset{\cdot}{a}(0)=a_{1} f(\eta)=\frac{k_{3}}{\xi}\sqrt{-\frac{\xi}{k_{3}}\eta^{2}+(\frac{\xi}{k_{3}}\alpha) ^{2}}% where $\eta=\frac{x}{a(s)^{1/4}}$ with $s=4t;$ $\kappa=\frac{k_{1}}{2}+k_{2}-1,$ $\alpha\geq0,$ $\xi<0$, $a_{0}$ and $a_{1}$ are constants. which the local or global behavior can be determined by the corresponding Emden equation. The results are very similar to the one obtained for the 2-component Camassa-Holm equations. Our analytical solutions could provide concrete examples for testing the validation and stabilities of numerical methods for the systems. With the characteristic line method, blowup phenomenon for $k_{3}\geq0$ is also studied.Comment: 13 Pages, Key Words: 2-Component Degasperis-Procesi, Shallow Water System, Analytical Solutions, Blowup, Global, Self-Similar, Separation Method, Construction of Solutions, Moving Boundary, 2-Component Camassa-Holm Equation

On the Security of Y-00 under Fast Correlation and Other Attacks on the Key

The potential weakness of the Y-00 direct encryption protocol when the encryption box ENC in Y-00 is not chosen properly is demonstrated in a fast correlation attack by S. Donnet et al in Phys. Lett. A 35, 6 (2006) 406-410. In this paper, we show how this weakness can be eliminated with a proper design of ENC. In particular, we present a Y-00 configuration that is more secure than AES under known-plaintext attack. It is also shown that under any ciphertext-only attack, full information-theoretic security on the Y-00 seed key is obtained for any ENC when proper deliberate signal randomization is employed