Remarks on the Global Attractor of the Weakly Dissipative Benjamin－Ono Equation

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Improved Estimate on Dimension of Inertial Manifold for Kuramoto-Sivashinsky Equation

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Global Attractor for a Generalized Discrete Nonlinear Schrödinger Equation

A generalized discrete nonlinear Schrödinger equation $$i\dot{u}_n(t)+\sum_{m=-\infty}^{+\infty} J(n-m)u_m(t)+g\bigl(u_n(t)\bigr)+i\gamma u_n(t)=f_n,\quad n\in\mathbb{Z}, $$ with long-range interactions in weighted spaces \(\ell_{\mathbf{{q}}}^{2}\) is considered. Under suitable assumptions on the coupling constants J(m), the damping γ and the weight \(\mathbf{{q}}=(q_{n})_{n\in \mathbb{Z}}\) , the existence of a global attractor is proved.     

Regularity of the Attractor for the Dissipative Hamiltonian Amplitude Equation Governing Modulated Wave Instabilities

Abstract In the present paper, the existence of global attractor for dissipative Hamiltonian amplitude equationgoverning the modulated wave instabilities in E_0 is considered.By a decomposition of solution operator,it isshown that the global attractor in E_0 is actually equal to a global attractor in E_1.     

The Global Existence of One Type of Nonlinear Kirchhoff String Equation

In this paper, the global well-posedness of initial-boundary value problem to the nonlinear Kirchhoff equation with source and damping term is established by energy method.     

The Global Attractors for the Dissipative Generalized Hasegawa-Mima Equation

The long time behavior of solutions of the generalized Hasegawa-Mima equation with dissipation term is considered. The existence of global attractors of the periodic initial value problem is proved, and the estimate of the upper bound of the Hausdorff and fractal dimensions for the global attractors is obtained by means of uniform a priori estimates method.     

Global Dissipative Multipeakon Solutions of the Camassa–Holm Equation

We show how to construct globally defined dissipative multipeakon solutions of the Camassa–Holm equation. The construction includes in particular the case with peakon-antipeakon collisions. The solutions are dissipative in the sense that the associated energy is decreasing in time.     

On the Higher-Order Method for the Solution of a Nonlinear Scalar Equation

In this paper, a higher-order method for the solution of a nonlinear scalar equation is presented. It is proved that the new method is locally convergent with an order of (m+2), where m is the highest order derivative used in the iterative formula. Some numerical examples are used to demonstrate the new method.     

Notes on Global Existence for the Nonlinear Schrdinger Equation Involves Derivative

In this paper, we consider the scattering for the nonlinear Schr¨odinger equation with small,smooth, and localized data. In particular, we prove that the solution of the quadratic nonlinear Schr¨odinger equation with nonlinear term |u|2involving some derivatives in two dimension exists globally and scatters. It is worth to note that there exist blow-up solutions of these equations without derivatives. Moreover, for radial data, we prove that for the equation with p-order nonlinearity with derivatives, the similar results hold for p ≥2d+32d-1and d ≥ 2, which is lower than the Strauss exponents.     

Dressing for a Novel Integrable Generalization of the Nonlinear Schrödinger Equation

We implement the dressing method for a novel integrable generalization of the nonlinear Schrödinger equation. As an application, explicit formulas for the N-soliton solutions are derived. As a by-product of the analysis, we find a simplification of the formulas for the N-solitons of the derivative nonlinear Schrödinger equation given by Huang and Chen.     

Weak Continuity of the Flow Map for the Benjamin-Ono Equation on the Line

In this paper we show that the flow map of the Benjamin-Ono equation on the line is weakly continuous in L ²(?), using “local smoothing” estimates. L ²(?) is believed to be a borderline space for the local well-posedness theory of this equation. In the periodic case, Molinet (Math. Ann. 337, 353–383, 2007) has recently proved that the flow map of the Benjamin-Ono equation is not weakly continuous in $L^{2}(\mathbb{T})In this paper we show that the flow map of the Benjamin-Ono equation on the line is weakly continuous in L ²(ℝ), using “local smoothing” estimates. L ²(ℝ) is believed to be a borderline space for the local well-posedness theory of this equation. In the periodic case, Molinet (Math. Ann. 337, 353–383, 2007) has recently proved that the flow map of the Benjamin-Ono equation is not weakly continuous in L²(\mathbbT)L^{2}(\mathbb{T}). Our results are in line with previous work on the cubic nonlinear Schr?dinger equation, where Goubet and Molinet (Nonlinear Anal. 71, 317–320, 2009) showed weak continuity in L ²(ℝ) and Molinet (Am. J. Math. 130, 635–683, 2008) showed lack of weak continuity in L²(\mathbbT)L^{2}(\mathbb{T}).     

Homoclinic and Heteroclinic Flows in Global Attractor for Davey-stewartson Ⅱ Equation

By the variable transformation and generalized Hirota method,exact homoclinic and heteroclinic solutions for Davey-StewartsonⅡ(DSⅡ)equation are obtained.For perturbed DSⅡequation,the existence of a global attractor is proved.The persistence of homoclinic and heteroclinic flows is investigated,and the special homoclinic and heteroclinic structure in attractors is shown.     

On Global Attractor for 2D Kirchhoff–Boussinesq Model with Supercritical Nonlinearity

Dynamics for a class of nonlinear 2D Kirchhoff–Boussinesq models is studied. These nonlinear plate models are characterized by the presence of a nonlinear source that alone leads to finite-time blow up of solutions. In order to counteract, restorative forces are introduced, which however are of a supercritical nature. This raises natural questions such as: (i) wellposedness of finite energy (weak) solutions, (ii) their regularity, and (iii) long time behavior of both weak and strong solutions. It is shown that finite energy solutions do exist globally, are unique and satisfy Hadamard wellposedness criterium. In addition, weak solutions corresponding to “strong” initial data (i.e., strong solutions) enjoy, likewise, the full Hadamard wellposedness. The proof is based on logarithmic control of the lack of Sobolev's embedding. In addition to wellposedness, long time behavior is analyzed. Viscous damping added to the model controls long time behaviour of solutions. It is shown that both weak and (resp. strong) solutions admit compact global attractors in the finite energy norm, (resp. strong topology of strong solutions). The proof of long time behavior is based on Ball's method [2 Ball , J. ( 2004 ). Global attractors for semilinear wave equations . Discr. Cont. Dyn. Sys. 10 : 31 – 52 .[Crossref], [Web of Science ®] , [Google Scholar]] and on recent asymptotic quasi-stability inequalities established in [11 Chueshov , I. , Lasiecka , I. ( 2008 ). Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping . Memoirs of the American Mathematical Society, Vol. 912 . Providence , RI : American Mathematical Society .[Crossref] , [Google Scholar]]. These inequalities enable to prove that strong attractors are finite-dimensional and the corresponding trajectories can exhibit C ^∞ smoothness.     

The Global Existence of One Type of Nonlinear Kirchhoff String Eauation

In this paper, the global well-posedness of initial-boundary value problem to the nonlinear Kirchhoff equation with source and damping term is established by energy method.     

The Hausdorff Dimension of the Double Points on the Brownian Frontier

The frontier of a planar Brownian motion is the boundary of the unbounded component of the complement of its range. In this paper, we find the Hausdorff dimension of the set of double points on the frontier.     

Absence of Global Periodic Solutions for a Schrödinger-Type Nonlinear Evolution Equation

Doklady Mathematics - The problem of the absence of global periodic solutions for a Schrödinger-type nonlinear evolution equation with a linear damping term is investigated. It is proved that...