Instability and stability properties of traveling waves for the double dispersion equation

In this article we are concerned with the instability and stability properties of traveling wave solutions of the double dispersion equation $~u_{tt} -u_{xx}+a u_{xxxx}-bu_{xxtt} = - (|u|^{p-1}u)_{xx}~$ for $~p>1$, $~a\geq b>0$. The main characteristic of this equation is the existence of two sources of dispersion, characterized by the terms $u_{xxxx}$ and $u_{xxtt}$. We obtain an explicit condition in terms of $a$, $b$ and $p$ on wave velocities ensuring that traveling wave solutions of the double dispersion equation are strongly unstable by blow up. In the special case of the Boussinesq equation ($b=0$), our condition reduces to the one given in the literature. For the double dispersion equation, we also investigate orbital stability of traveling waves by considering the convexity of a scalar function. We provide both analytical and numerical results on the variation of the stability region of wave velocities with $a$, $b$ and $p$ and then state explicitly the conditions under which the traveling waves are orbitally stable.Comment: 16 pages, 4 figure

Traveling waves in one-dimensional nonlinear models of strain-limiting viscoelasticity

In this article we investigate traveling wave solutions of a nonlinear differential equation describing the behaviour of one-dimensional viscoelastic medium with implicit constitutive relations. We focus on a subclass of such models known as the strain-limiting models introduced by Rajagopal. To describe the response of viscoelastic solids we assume a nonlinear relationship among the linearized strain, the strain rate and the Cauchy stress. We then concentrate on traveling wave solutions that correspond to the heteroclinic connections between the two constant states. We establish conditions for the existence of such solutions, and find those solutions, explicitly, implicitly or numerically, for various forms of the nonlinear constitutive relation

A Comparison of Solutions of Two Convolution-Type Unidirectional Wave Equations

In this work, we prove a comparison result for a general class of nonlinear dispersive unidirectional wave equations. The dispersive nature of one-dimensional waves occurs because of a convolution integral in space. For two specific choices of the kernel function, the Benjamin-Bona-Mahony equation and the Rosenau equation that are particularly suitable to model water waves and elastic waves, respectively, are two members of the class. We first prove an energy estimate for the Cauchy problem of the nonlocal unidirectional wave equation. Then, for the same initial data, we consider two distinct solutions corresponding to two different kernel functions. Our main result is that the difference between the solutions remains small in a suitable Sobolev norm if the two kernel functions have similar dispersive characteristics in the long-wave limit. As a sample case of this comparison result, we provide the approximations to the hyperbolic conservation law.Comment: 12 pages, to appear in Applicable Analysi

The Camassa-Holm equation as the long-wave limit of the improved Boussinesq equation and of a class of nonlocal wave equations

In the present study we prove rigorously that in the long-wave limit, the unidirectional solutions of a class of nonlocal wave equations to which the improved Boussinesq equation belongs are well approximated by the solutions of the Camassa-Holm equation over a long time scale. This general class of nonlocal wave equations model bidirectional wave propagation in a nonlocally and nonlinearly elastic medium whose constitutive equation is given by a convolution integral. To justify the Camassa-Holm approximation we show that approximation errors remain small over a long time interval. To be more precise, we obtain error estimates in terms of two independent, small, positive parameters $\epsilon$ and $\delta$ measuring the effect of nonlinearity and dispersion, respectively. We further show that similar conclusions are also valid for the lower order approximations: the Benjamin-Bona-Mahony approximation and the Korteweg-de Vries approximation.Comment: 24 pages, to appear in Discrete and Continuous Dynamical System

The Cauchy problem for a class of two-dimensional nonlocal nonlinear wave equations governing anti-plane shear motions in elastic materials

This paper is concerned with the analysis of the Cauchy problem of a general class of two-dimensional nonlinear nonlocal wave equations governing anti-plane shear motions in nonlocal elasticity. The nonlocal nature of the problem is reflected by a convolution integral in the space variables. The Fourier transform of the convolution kernel is nonnegative and satisfies a certain growth condition at infinity. For initial data in $L^{2}$ Sobolev spaces, conditions for global existence or finite time blow-up of the solutions of the Cauchy problem are established.Comment: 15 pages. "Section 6 The Anisotropic Case" added and minor changes. Accepted for publication in Nonlinearit

Spor Bilimlerinde Hareket Yakalama Teknolojisi: Kinect ile Üç Boyutlu Sanal Spor Platformu

Düzenli spor yapmanın yaşam ve sağlık kalitesi için yararlı olduğu kabul edilmektedir. Fakat düzenli spor yapma olanaklarının özellikle gelişmekte olan ülkelerde yeterince yaygınlaşmadığı gözlenmektedir. İnternet ve Kinect gibi güncel ve gelişmiş teknolojilerin sağladığı olanaklar ile, spor yapamayanlara fiziksel olarak aynı ortamda bulunmaksızın, sanal eğitmen kontrolünde spor yapabilme olanağı sunmanın alternatif bir çözüm yaratabileceği düşünülmektedir. Bu doğrultudaki çalışmalarla, Kinect üç boyutlu sanal spor platformu tasarlanıp geliştirilmiştir. Geliştirilen platformun özellikleri, yeni teknolojilerin avantajlarının kullanıldığı platformlar ile alternatif spor yapma olanakları gözden geçirilmektedirRegular sports activity is considered to be beneficial for life quality and health. On the other hand, regular sporting opportunities are not widespread, especially in developing countries. It may be possible to provide an alternative solution for physical activity in the control of a virtual instructor, without being physically in the same place, thanks to the facilities provided by technologies such as Internet and Kinect. In this respect, the three dimensional virtual sports platform Kinect was designed and developed. The different features of the platform are described, and the advantages of new technology achievement and possibilities of alternative sports practice are give

Macrophage mal1 deficiency suppresses atherosclerosis in low-density lipoprotein receptor-null mice by activating peroxisome proliferator-activated receptor-γ-regulated genes

Objective- The adipocyte/macrophage fatty acid-binding proteins aP2 (FABP4) and Mal1 (FABP5) are intracellular lipid chaperones that modulate systemic glucose metabolism, insulin sensitivity, and atherosclerosis. Combined deficiency of aP2 and Mal1 has been shown to reduce the development of atherosclerosis, but the independent role of macrophage Mal1 expression in atherogenesis remains unclear. Methods and Results- We transplanted wild-type (WT), Mal1, or aP2 bone marrow into low-density lipoprotein receptor-null (LDLR) mice and fed them a Western diet for 8 weeks. Mal1→LDLR mice had significantly reduced (36%) atherosclerosis in the proximal aorta compared with control WT→LDLR mice. Interestingly, peritoneal macrophages isolated from Mal1-deficient mice displayed increased peroxisome proliferator-activated receptor-γ (PPARγ) activity and upregulation of a PPARγ-related cholesterol trafficking gene, CD36. Mal1 macrophages showed suppression of inflammatory genes, such as COX2 and interleukin 6. Mal1→LDLR mice had significantly decreased macrophage numbers in the aortic atherosclerotic lesions compared with WT→LDLR mice, suggesting that monocyte recruitment may be impaired. Indeed, blood monocytes isolated from Mal1→LDLR mice on a high-fat diet had decreased CC chemokine receptor 2 gene and protein expression levels compared with WT monocytes. Conclusion- Taken together, our results demonstrate that Mal1 plays a proatherogenic role by suppressing PPARγ activity, which increases expression of CC chemokine receptor 2 by monocytes, promoting their recruitment to atherosclerotic lesions. © 2011 American Heart Association, Inc