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

The Modulation of Multiple Phases Leading to the Modified KdV Equation

This paper seeks to derive the modified KdV (mKdV) equation using a novel approach from systems generated from abstract Lagrangians that possess a two-parameter symmetry group. The method to do uses a modified modulation approach, which results in the mKdV emerging with coefficients related to the conservation laws possessed by the original Lagrangian system. Alongside this, an adaptation of the method of Kuramoto is developed, providing a simpler mechanism to determine the coefficients of the nonlinear term. The theory is illustrated using two examples of physical interest, one in stratified hydrodynamics and another using a coupled Nonlinear Schr\"odinger model, to illustrate how the criterion for the mKdV equation to emerge may be assessed and its coefficients generated.Comment: 35 pages, 5 figure

Resonant nonlinearity management for nonlinear-Schr\"{o}dinger solitons

We consider effects of a periodic modulation of the nonlinearity coefficient on fundamental and higher-order solitons in the one-dimensional NLS equation, which is an issue of direct interest to Bose-Einstein condensates in the context of the Feshbach-resonance control, and fiber-optic telecommunications as concerns periodic compensation of the nonlinearity. We find from simulations, and explain by means of a straightforward analysis, that the response of a fundamental soliton to the weak perturbation is resonant, if the modulation frequency $\omega$ is close to the intrinsic frequency of the soliton. For higher-order $n$-solitons with $n=2$ and 3, the response to an extremely weak perturbation is also resonant, if $\omega$ is close to the corresponding intrinsic frequency. More importantly, a slightly stronger drive splits the 2- or 3-soliton, respectively, into a set of two or three moving fundamental solitons. The dependence of the threshold perturbation amplitude, necessary for the splitting, on $\omega$ has a resonant character too. Amplitudes and velocities of the emerging fundamental solitons are accurately predicted, using exact and approximate conservation laws of the perturbed NLS equation.Comment: 14 pages, 6 figure

The nonlinear electromigration of analytes into confined spaces

We consider the problem of electromigration of a sample ion (analyte) within a uniform background electrolyte when the confining channel undergoes a sudden contraction. One example of such a situation arises in microfluidics in the electrokinetic injection of the analyte into a micro-capillary from a reservoir of much larger size. Here the sample concentration propagates as a wave driven by the electric field. The dynamics is governed by the Nerst-Planck-Poisson system of equations for ionic transport.A reduced one dimensional nonlinear equation describing the evolution of the sample concentration is derived.We integrate this equation numerically to obtain the evolution of the wave shape and determine how the the injected mass depends on the sample concentration in the reservoir.It is shown that due to the nonlinear coupling of the ionic concentrations and the electric field, the concentration of the injected sample could be substantially less than the concentration of the sample in the reservoir.Comment: 14 pages, 5 Figures, 1 Appendi

Modeling M-Theory Vacua via Gauged S-Duality

We construct a model of M-theory vacua using gauged S-duality and the Chan-Paton symmetries by introducing an infinite number of open string charges. In the Bechi-Rouet-Stora-Tyutin formalism, the local description of the gauged S-duality on its moduli space of vacua is fully determined by one physical state condition on the vacua. We introduce the string probe of the spatial degrees of freedom and define the increment of the cosmic time. The dimensionality of space-time and the gauge group of the low energy effective theory originate in the symmetries (with or without their breakdown) in our model. This modeling leads to the derived category formulation of the quantum mechanical world including gravity and to the concept of a non-linear potential of gauged and affinized S-duality which specifies the morphism structure of this derived category.Comment: 31 pages, version reflecting the erratum. arXiv admin note: substantial text overlap with arXiv:1102.460

Anomalous Scaling and Solitary Waves in Systems with Non-Linear Diffusion

We study a non-linear convective-diffusive equation, local in space and time, which has its background in the dynamics of the thickness of a wetting film. The presence of a non-linear diffusion predicts the existence of fronts as well as shock fronts. Despite the absence of memory effects, solutions in the case of pure non-linear diffusion exhibit an anomalous sub-diffusive scaling. Due to a balance between non-linear diffusion and convection we, in particular, show that solitary waves appear. For large times they merge into a single solitary wave exhibiting a topological stability. Even though our results concern a specific equation, numerical simulations supports the view that anomalous diffusion and the solitary waves disclosed will be general features in such non-linear convective-diffusive dynamics.Comment: Corrected typos, added 3 references and 2 figure

On the relationship between nonlinear equations integrable by the method of characteristics and equations associated with commuting vector fields

It was shown recently that Frobenius reduction of the matrix fields reveals interesting relations among the nonlinear Partial Differential Equations (PDEs) integrable by the Inverse Spectral Transform Method ($S$-integrable PDEs), linearizable by the Hoph-Cole substitution ($C$-integrable PDEs) and integrable by the method of characteristics ($Ch$-integrable PDEs). However, only two classes of $S$-integrable PDEs have been involved: soliton equations like Korteweg-de Vries, Nonlinear Shr\"odinger, Kadomtsev-Petviashvili and Davey-Stewartson equations, and GL(N,\CC) Self-dual type PDEs, like Yang-Mills equation. In this paper we consider the simple five-dimensional nonlinear PDE from another class of $S$-integrable PDEs, namely, scalar nonlinear PDE which is commutativity condition of the pair of vector fields. We show its origin from the (1+1)-dimensional hierarchy of $Ch$-integrable PDEs after certain composition of Frobenius type and differential reductions imposed on the matrix fields. Matrix generalization of the above scalar nonlinear PDE will be derived as well.Comment: 14 pages, 1 figur

Higher-order splitting algorithms for solving the nonlinear Schr\"odinger equation and their instabilities

Since the kinetic and the potential energy term of the real time nonlinear Schr\"odinger equation can each be solved exactly, the entire equation can be solved to any order via splitting algorithms. We verified the fourth-order convergence of some well known algorithms by solving the Gross-Pitaevskii equation numerically. All such splitting algorithms suffer from a latent numerical instability even when the total energy is very well conserved. A detail error analysis reveals that the noise, or elementary excitations of the nonlinear Schr\"odinger, obeys the Bogoliubov spectrum and the instability is due to the exponential growth of high wave number noises caused by the splitting process. For a continuum wave function, this instability is unavoidable no matter how small the time step. For a discrete wave function, the instability can be avoided only for \dt k_{max}^2{<\atop\sim}2 \pi, where $k_{max}=\pi/\Delta x$.Comment: 10 pages, 8 figures, submitted to Phys. Rev.