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Zhou Xiaofang 《偏微分方程(英文版)》1994,7(3)
In this paper, we study the hypoellipticity problems for fully nonlinear panial differential equations of order m. For a solution u ∈ C^p_{loc}(Ω), if the linearized operator for the nonlinear equation on u satisfies some subelliptic conditions, we can deduce u ∈ C^∞(Ω) by using the paradifferential operator theory of J. -M. Bony. 相似文献
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这是一篇介绍当前方程界十分重要的课题——非线性微局部分析——的综述文章。作为这一研究领域的开拓者,我们在不太长的篇幅里,从相当的理论高度简洁地介绍该领域近十年来一些最引人注目的工作。本文首先阐述了一般微局部分析的基本思想,然后介绍近十年来对非线性偏微分方程起重要推动作用的仿微分计算(如仿乘积,仿微分算子,仿复合等),以及有着更深刻内容的高次微局部的思想。同时,也大量介绍这些思想在非线性偏微分方程弱奇性分析中的应用,如奇性的传播,反射与绕射,余法型奇性的相互作用,非线性亚椭圆性,以及三个奇性波的相互作用等,这些均是当前方程界的热门课题。 相似文献
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Luc Vrancken 《Mathematische Nachrichten》2002,237(1):129-146
It is well‐known that locally strongly convex affine hyperspheres can be determinedas solutions of differential equations of Monge‐Ampère type. In this paper we study in particular the 3‐dimensional case and we assume that the hypersphere admits a Killing vector field (with respect to the affine metric) whose integral curves are geodesics with respect to both the induced affine connection and the Levi‐Civita connection of the affine metric. We show that besides the already known examples, such hyperspheres can be constructed starting from the 2‐dimensional Poisson equation, the 2‐dimensional sine‐Gordon equation or the 2‐dimensional cosh‐Gordon equation. 相似文献
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By using Xu’s stable-range method,families of explicit exact solutions with multiple parameter functions for the(2+1)-dimensional breaking soliton and KadomtsevPetviashvili equations.These parameter functions make our solutions more applicable to related practical models and boundary value problems. 相似文献
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We consider light propagation in a Kerr-nonlinear 2D waveguide with a Bragg grating in the propagation direction and homogeneous in the transverse direction. Using Newton's iteration method we construct both stationary and travelling solitary wave solutions of the corresponding mathematical model, the 2D nonlinear coupled mode equations (2D CME). We call these solutions 2D gap solitons due to their similarity with the gap solitons of 1D CME (fiber grating). Long-time stable evolution preserving the solitary fashion is demonstrated numerically despite the fact that, as we show, for the 2D CME no local constrained minima of the Hamiltonian functional exist. Building on the 1D study of [ 1 ], we demonstrate trapping of slow enough 2D gap solitons at localized defects. We explain the mechanism of trapping as resonant transfer of energy from the soliton to one or more nonlinear defect modes. For a special class of defects, we construct a family of nonlinear defect modes by numerically following a bifurcation curve starting at analytically or numerically known linear defect modes. Compared to 1D the dynamics of trapping are harder to fully analyze and the existence of many defect modes for a given defect potential causes that slow solitons store a part of their energy for virtually all of the studied attractive defects. 相似文献
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Now at Mathemarics Department, Assiut University Egypt A method is presented to transform parabolic equations to asystem of ordinary differential equations for the solution atthe Chebyshev points. The system may be solved analyticallyor by numerical methods and the Chebyshev coefficients are computed.We have the exact solution of a perturbed problem. 相似文献
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基于延拓结构和Hirota双线性方法研究了广义的变系数耦合非线性Schr(o)dinger方程.首先导出了3组新的变系数可积耦合非线性Schr(o)dinger方程及其线性谱问题(Lax对),然后利用Hirota双线性方法给出了它们的单、双向量孤子解.这些向量孤子解在光孤子通讯中有重要的应用. 相似文献
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基于延拓结构和Hirota双线性方法研究了广义的变系数耦合非线性Schrdinger方程.首先导出了3组新的变系数可积耦合非线性Schrdinger方程及其线性谱问题(Lax对),然后利用Hirota双线性方法给出了它们的单、双向量孤子解.这些向量孤子解在光孤子通讯中有重要的应用. 相似文献
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We consider Dirichlet boundary value problems for a class of nonlinear ordinary differential equations motivated by the study of radial solutions of equations which are perturbations of the p-Laplacian. 相似文献
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The conditional law of an unobservable component x(t) of a diffusion (x(t),y(t)) given the observations {y(s):s[0,t]} is investigated when x(t) lives on a submanifold
of
. The existence of the conditional density with respect to a given measure on
is shown under fairly general conditions, and the analytical properties of this density are characterized in terms of the Sobolev spaces used in the first part of this series. 相似文献
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Yuesheng Xu & Taishan Zeng 《高等学校计算数学学报(英文版)》2023,16(1):58-78
More competent learning models are demanded for data processing due
to increasingly greater amounts of data available in applications. Data that we encounter often have certain embedded sparsity structures. That is, if they are represented in an appropriate basis, their energies can concentrate on a small number
of basis functions. This paper is devoted to a numerical study of adaptive approximation of solutions of nonlinear partial differential equations whose solutions may
have singularities, by deep neural networks (DNNs) with a sparse regularization
with multiple parameters. Noting that DNNs have an intrinsic multi-scale structure
which is favorable for adaptive representation of functions, by employing a penalty
with multiple parameters, we develop DNNs with a multi-scale sparse regularization
(SDNN) for effectively representing functions having certain singularities. We then
apply the proposed SDNN to numerical solutions of the Burgers equation and the
Schrödinger equation. Numerical examples confirm that solutions generated by the
proposed SDNN are sparse and accurate. 相似文献