Globalwell-posedness and I method for the fifth order Korteweg-de Vries equation

The Kawahara equation has fewer symmetries than the KdV equation; in particular, it has no invariant scaling transform and is not completely integrable. Thus its analysis requires different methods. We prove that the Kawahara equation is locally well posed in H ^−7/4, using the ideas of an [`(F)] ^s{\overline F ^s}-type space [8]. Then we show that the equation is globally well posed in H ^s for s ≥ −7/4, using the ideas of the “I-method” [7].     

Quasi exponential decay of a finite difference space discretization of the 1-d wave equation by pointwise interior stabilization

We consider the wave equation on an interval of length 1 with an interior damping at ξ. It is well-known that this system is well-posed in the energy space and that its natural energy is dissipative. Moreover, as it was proved in Ammari et al. (Asymptot Anal 28(3–4):215–240, 2001), the exponential decay property of its solution is equivalent to an observability estimate for the corresponding conservative system. In this case, the observability estimate holds if and only if ξ is a rational number with an irreducible fraction x = \fracpq,\xi=\frac{p}{q}, where p is odd, and therefore under this condition, this system is exponentially stable in the energy space. In this work, we are interested in the finite difference space semi-discretization of the above system. As for other problems (Zuazua, SIAM Rev 47(2):197–243, 2005; Tcheugoué Tébou and Zuazua, Adv Comput Math 26:337–365, 2007), we can expect that the exponential decay of this scheme does not hold in general due to high frequency spurious modes. We first show that this is indeed the case. Secondly we show that a filtering of high frequency modes allows to restore a quasi exponential decay of the discrete energy. This last result is based on a uniform interior observability estimate for filtered solutions of the corresponding conservative semi-discrete system.     

Control and Stabilization of the Korteweg-de Vries Equation on a Periodic Domain

In [38 Russell , D.L. , Zhang , B.-Y. ( 1996 ). Exact controllability and stabilizability of the Korteweg-de Vries equation . Trans. Amer. Math. Soc. 348 : 3643 – 3672 . [Google Scholar]], Russell and Zhang showed that the Korteweg-de Vries equation posed on a periodic domain  with an internal control is locally exactly controllable and locally exponentially stabilizable when the control acts on an arbitrary nonempty subdomain of . In this paper, we show that the system is in fact globally exactly controllable and globally exponentially stabilizable. The global exponential stabilizability is established with the aid of certain properties of propagation of compactness and regularity in Bourgain spaces for the solutions of the associated linear system. With Slemrod's feedback law, the resulting closed-loop system is shown to be locally exponentially stable with an arbitrarily large decay rate. A time-varying feedback law is further designed to ensure a global exponential stability with an arbitrarily large decay rate.     

Explicit Complex-Valued Solutions of the Korteweg–de Vries Equation on the Half-Line and on the Whole-Line

The paper deals with the initial-value problems for the Korteweg–de Vries (KdV) equations on the half-line and on the whole-line for complex-valued measurable and exponentially decreasing potentials. The time evolution equation for the reflection coefficient is derived and then a one-to-one correspondence between the scattering data and the solution of the KdV equation is shown. Families of exact solutions of the KdV equation are represented for the class of reflection-free potentials, in which the inverse scattering problem associated with the KdV equation can be solved exactly. Some helpful examples of soliton solutions of the KdV equation are provided.     

A qualitative analysis of solutions to microelectromechanical systems with curvature and nonlinear permittivity profile

An investigation of qualitative properties of solutions to microelectromechanical systems with nonlinear permittivity profile is established. The considered problem is described by a quasilinear evolution equation for the displacement of a membrane, coupled with an elliptic moving boundary problem for the electrostatic potential between the moving membrane and a rigid ground plate. The system is shown to be well-posed locally in time for arbitrary values λ of the applied voltage. For small values of λ the solution exists even globally in time such that the membrane never touches down on the ground plate. In addition to those fundamental results concerning existence and uniqueness of solutions, sufficient conditions are specified which guarantee non-positivity of the membrane's displacement on the one hand and the occurrence of a singularity after a finite time T on the other hand.     

Exponential Stability for the Generalized Korteweg-de Vries Equation in a Finite Interval with Weak Damping

The aim of this paper is to consider the generalized Korteweg-de Vries equation in a finite interval with a very weak localized dissipation. We obtain the globally uniformly exponentially stability of this equation. The main difficulty in this context comes from the structure of nonlinear term and the lack of regularity.     

Boundary stabilization of a microbeam model

In this paper, we study the boundary stabilization of the deflection of a clamped-free microbeam, which is modeled by a sixth-order hyperbolic equation. We design a boundary feedback control, simpler than the one designed in Vatankhah et al,² that forces the energy associated to the deflection to decay exponentially to zero as the time goes to infinity. The rate in which the energy exponentially decays is explicitly given.     

Stationary Solutions of the KdV Equation

In this article, as a first step to study the large time behavior of solutions to the KdV equation on a half-line, we prove the existence of the stationary solutions to the corresponding problem. The aim is to concentrate on understanding the influence of boundary conditions.     

Exact soliton solutions for the general fifth Korteweg-de Vries equation

With the aid of computer symbolic computation system such as Maple, the extended hyperbolic function method and the Hirota’s bilinear formalism combined with the simplified Hereman form are applied to determine the soliton solutions for the general fifth-order KdV equation. Several new soliton solutions can be obtained if we taking parameters properly in these solutions. The employed methods are straightforward and concise, and they can also be applied to other nonlinear evolution equations in mathematical physics. The article is published in the original.     

Stabilisation faible interne locale de système élastique de Bresse

In [1], Alabau-Boussouira et al. (2011) studied the exponential and polynomial stability of the Bresse system with one globally distributed dissipation law. In this Note, our goal is to extend the results from Alabau-Boussouira et al. (2011) [1], by taking into consideration the important case when the dissipation law is locally distributed and to improve the polynomial energy decay rate. We then study the energy decay rate of the Bresse system with one locally internal distributed dissipation law acting on the equation about the shear angle displacement. Under the equal speed wave propagation condition, we show that the system is exponentially stable. On the contrary, we establish a new polynomial energy decay rate.     

Quasiperiodic Wave Solutions of Supersymmetric KdV Equation in Superspace

A direct and unifying scheme for explicitly constructing quasiperiodic wave solutions (multiperiodic wave solutions) of supersymmetric KdV equation in a superspace is proposed. The scheme is based on the concept of super Hirota forms and on the use of super Riemann theta functions. In contrast to ordinary KdV equation with purely bosonic field, some new phenomena on super quasiperiodic waves occur in the supersymmetric KdV equation with the fermionic field. For instance, it is shown that the supersymmetric KdV equation does not possess an N ‐periodic wave solution for N≥ 2 for arbitrary parameters. It is further observed that there is an influencing band occurred among the quasiperiodic waves under the presence of the Grassmann variable. The quasiperiodic waves are symmetric about the band but collapse along with the band. In addition, the relations between the quasiperiodic wave solutions and soliton solutions are rigorously established. It is shown that quasiperiodic wave solution convergence to the soliton solutions under certain conditions and small amplitude limit.     

Blowup in the Nonlinear Schrödinger Equation Near Critical Dimension

This article contains an analysis of the cubic nonlinear Schrödinger equation and solutions that become singular in finite time. Numerical simulations show that in three dimensions the blowup is self-similar and symmetric. In two dimensions, the blowup still appears to be symmetric but is no longer self-similar. In the case that the dimension, d, is greater than and exponentially close to 2 in terms of a small parameter associated to the norm of the blow-up solution, a locally unique, monotonically decreasing in modulus, self-similar solution that satisfies the boundary and global conditions associated with the blow-up solution is constructed in Kopell and Landman [1995, SIAM J. Appl., Math.55, 1297-1323]. In this article, it is shown that this locally unique solution also exists for d > 2 and algebraically close to 2 in the same small parameter. The central idea of the proof involves constructing a pair of manifolds of solutions (to the nonautonomous ordinary differential equation satisfied by the self-similar solutions) that satisfy the conditions at r = 0 and the asymptotic conditions respectively and then showing that these intersect transversally. A key step involves tracking one of the manifolds over a midrange in which the ordinary differential equation has a turning point and hence obtaining good control over the solutions on the manifold.     

Wave energy decay under fractional derivative controls

^** Present address: Division of Mathematics and Sciences, Rust College, 150 Rust Avenue, Holly Springs, MS 38635, USA In this article, we investigate the asymptotic behaviour ofsolutions of the 1D wave equation with a boundary viscoelasticdamper of the fractional derivative type. We show that the systemis well-posed in the sense of semigroup. We also prove thatthe associated semigroup is not exponentially stable, but onlystrongly asymptotically so. Finally, we establish the followingresult. Provided that the initial states of the system are chosensufficiently smooth and the relaxation function of the viscoelasticdamper is exponentially decreasing, then solutions of the systemwill decay, as time goes to infinity, as [graphic: see PDF] A > 0.     

Moving mesh method for problems with blow-up on unbounded domains

This paper studies the numerical solutions of semilinear parabolic partial differential equations (PDEs) on unbounded spatial domains whose solutions blow up in finite time. There are two major difficulties usually in numerical solutions: the singularity of blow-up and the unboundedness. We propose local absorbing boundary conditions (LABCs) on the selected artificial boundaries by using the idea of unified approach (Brunner et al., SIAM J Sci Comput 31:4478–4496, (2010). Since the uniform fixed spatial meshes may be inefficient, we adopt moving mesh partial differential equation (MMPDE) method to adapt the spatial mesh as the singularity develops. Combining LABCs and MMPDE, we can effectively capture the qualitative behavior of the blow-up singularities in the unbounded domain. Moreover, the implementation of the combination consists of two independent parts. Numerical examples also illustrate the efficiency and the accuracy of the new method.     

Boundary controllability for the nonlinear Korteweg–de Vries equation on any critical domain

It is known that the linear Korteweg–de Vries (KdV) equation with homogeneous Dirichlet boundary conditions and Neumann boundary control is not controllable for some critical spatial domains. In this paper, we prove in these critical cases, that the nonlinear KdV equation is locally controllable around the origin provided that the time of control is large enough. It is done by performing a power series expansion of the solution and studying the cascade system resulting of this expansion.     

On the Derivation of the Modified Kadomtsev-Petviashvili Equation

It is well known that the Kadomtsev-Petviashvili (KP) equation is the two-dimensional analogue of the Korteweg—de Vries (KdV) equation. We reconsider the derivation of the KP equation, modified to include the effects of rotation, in order to determine the nature of the initial conditions. The motivation for this is that if the solutions of the modified KP equation are assumed to be locally confined, then they satisfy a certain constraint, which appears to restrict considerably the class of allowed initial conditions. The outcome of the analysis presented here is that in general it is not permissible to assume that solutions of the modified KP equation are locally confined, and hence the constraint cannot be applied. The reason for this is the radiation of Poincaré waves, which appear behind the main part of the solution described by the modified KP equation.     

Bifurcations in the Generalized Korteweg–de Vries Equation

We study the generalized Korteweg–de Vries (KdV) equation and the Korteweg–de Vries–Burgers (KdVB) equation with periodic in the spatial variable boundary conditions. For various values of parameters, in a sufficiently small neighborhood of the zero equilibrium state we construct asymptotics of periodic solutions and invariant tori. Separately we consider the case when the stability spectrum of the zero solution contains a countable number of roots of the characteristic equation. In this case we state a special nonlinear boundary-value problem which plays the role of a normal form and determines the dynamics of the initial problem.     

Orbital Stability of Solitary Waves of the Nonlinear Schrodinger-KdV Equation

This paper concerns the orbital stability of solitary waves of the system of KdV equation coupling with nonlinear Schrödinger equation. By applying the abstract results of Grillakis et al. [1- 2] and detailed spectral analysis, we obtain the stability of the solitary waves.     

Generalized Hunter–Saxton equation and the geometry of the group of circle diffeomorphisms

We study an equation lying ‘mid-way’ between the periodic Hunter–Saxton and Camassa–Holm equations, and which describes evolution of rotators in liquid crystals with external magnetic field and self-interaction. We prove that it is an Euler equation on the diffeomorphism group of the circle corresponding to a natural right-invariant Sobolev metric. We show that the equation is bihamiltonian and admits both cusped and smooth traveling-wave solutions which are natural candidates for solitons. We also prove that it is locally well-posed and establish results on the lifespan of its solutions. Throughout the paper we argue that despite similarities to the KdV, CH and HS equations, the new equation manifests several distinctive features that set it apart from the other three.     

弹性弦Dirichlet边界反馈控制的镇定与Riesz基生成

本文通过一端固定 ,一端 Dirichlet边界控制的一维波动方程说明系统是 Salamon- W eiss意义下适定和正则的 .由此说明 ,由 J.L.Lions引入的用于研究双曲方程精确可控性的 H ilbert唯一性方法是控制论中著名的对偶原理 .我们讨论了系统的指数镇定及闭环系统的广义本征函数生成 Riesz基和谱确定增长条件 .我们希望通过本文使读者对目前线性偏微分控制理论的一个新动向有一基本的了解 .