Double diffusion structure of logarithmically damped wave equations with a small parameter

The subject of our consideration is a family of semilinear wave equations with a small parameter and nonlinearities which provide the existence of kink-type solutions (solitons). Using asymptotic analysis and numerical simulation, we demonstrate that solitons of the same type (kinks or antikinks) interact in the same manner as for the sine-Gordon equation. However, solitons of the different type preserve the shape after the interaction only in the case of two or three waves, and, moreover, under some additional conditions.

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Asymptotic integration of systems of two wave equations with small parameter

Lietuvos matematikos rinkinys ◽

10.15388/lmr.b.2016.06 ◽

2016 ◽

Vol 57 ◽

Author(s):

Aleksandras Krylovas

Keyword(s):

Small Parameter ◽

Wave Equations ◽

Space Variable ◽

Initial Conditions ◽

Perturbation Technique ◽

Asymptotic Series ◽

Asymptotic Integration ◽

Weakly Nonlinear ◽

Multi Scale ◽

Scale Perturbation

In this article we consider a hyperbolic system of two weakly nonlinear equations. Coefficients of the equations and initial conditions are periodical functions of the space variable. A multi-scale perturbation technique and integrating along characteristics are used to construct asymptotic series without secular members. The scheme of asymptotic integration is applied to analysis of oscillations of nonlinear non-homogeneous strings.

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UNIFORMLY VALID ASYMPTOTICS FOR CARRIER’S MATHEMATICAL MODEL OF STRING OSCILLATIONS

Mathematical Modelling and Analysis ◽

10.3846/13926292.2017.1309702 ◽

2017 ◽

Vol 22 (3) ◽

pp. 337-351

Author(s):

Paulius Miškinis ◽

Aleksandras Krylovas ◽

Olga Lavcel-Budko

Keyword(s):

Mathematical Model ◽

Small Parameter ◽

Asymptotic Approximation ◽

Wave Equations ◽

Initial Conditions ◽

Resonant Interaction ◽

Nonlinear Wave Equations ◽

Time Interval ◽

Initial Moment ◽

Weakly Nonlinear

In the paper, an asymptotic analysis of G.F. Carrier’s mathematical model of string oscillation is presented. The model consists of a system of two nonlinear second order partial differential equations and periodic initial conditions. The longitudinal and transversal string oscillations are analyzed together when at the initial moment of time the system’s solutions have amplitudes proportional to a small parameter. The problem is reduced to a system of two weakly nonlinear wave equations. The resonant interaction of periodic waves is analyzed. An uniformly valid asymptotic approximation in the long time interval, which is inversely proportional to the small parameter, is constructed. This asymptotic approximation is a solution of averaged along characteristics integro-differential system. Conditions of appearance of combinatoric resonances in the system have been established. The results of numerical experiments are presented.

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A singular perturbed problem for semilinear wave equations with small parameter

Discrete and Continuous Dynamical Systems ◽

10.3934/dcds.1999.5.473 ◽

1999 ◽

Vol 5 (3) ◽

pp. 473-488 ◽

Cited By ~ 2

Author(s):

Han Yang ◽

Keyword(s):

Small Parameter ◽

Wave Equations ◽

Semilinear Wave Equations

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A Perturbation Theory for the Population of Atomic Energy Levels in Non-Lte Plasmas

International Astronomical Union Colloquium ◽

10.1017/s025292110010805x ◽

1988 ◽

Vol 102 ◽

pp. 343-347

Author(s):

M. Klapisch

Keyword(s):

Perturbation Theory ◽

Small Parameter ◽

Classification Scheme ◽

Sum Rules ◽

Energy Levels ◽

Natural Classification ◽

Quantum Numbers ◽

Formal Expansion ◽

Atomic Energy Levels ◽

As Effects

AbstractA formal expansion of the CRM in powers of a small parameter is presented. The terms of the expansion are products of matrices. Inverses are interpreted as effects of cascades.It will be shown that this allows for the separation of the different contributions to the populations, thus providing a natural classification scheme for processes involving atoms in plasmas. Sum rules can be formulated, allowing the population of the levels, in some simple cases, to be related in a transparent way to the quantum numbers.

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Double diffusion

AccessScience ◽

10.1036/1097-8542.757558 ◽

2015 ◽

Keyword(s):

Double Diffusion

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Remarks on nonlinear elastic waves with null conditions

ESAIM Control Optimisation and Calculus of Variations ◽

10.1051/cocv/2020039 ◽

2020 ◽

Vol 26 ◽

pp. 121

Author(s):

Dongbing Zha ◽

Weimin Peng

Keyword(s):

Initial Data ◽

Global Existence ◽

Elastic Waves ◽

Wave Equations ◽

Alternative Proof ◽

Classical Solutions ◽

Small Initial Data ◽

Hyperelastic Materials ◽

Variational Structure ◽

Nonlinear Elastic

For the Cauchy problem of nonlinear elastic wave equations for 3D isotropic, homogeneous and hyperelastic materials with null conditions, global existence of classical solutions with small initial data was proved in R. Agemi (Invent. Math. 142 (2000) 225–250) and T. C. Sideris (Ann. Math. 151 (2000) 849–874) independently. In this paper, we will give some remarks and an alternative proof for it. First, we give the explicit variational structure of nonlinear elastic waves. Thus we can identify whether materials satisfy the null condition by checking the stored energy function directly. Furthermore, by some careful analyses on the nonlinear structure, we show that the Helmholtz projection, which is usually considered to be ill-suited for nonlinear analysis, can be in fact used to show the global existence result. We also improve the amount of Sobolev regularity of initial data, which seems optimal in the framework of classical solutions.

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Immunological Evidence for Prothrombin in Human Platelets

Thrombosis and Haemostasis ◽

10.1055/s-0038-1661534 ◽

1986 ◽

Vol 55 (02) ◽

pp. 268-270

Author(s):

R J Alexander

Keyword(s):

Electrophoretic Mobility ◽

Platelet Activation ◽

Double Diffusion ◽

Human Platelets ◽

Secreted Proteins ◽

Platelet Surface ◽

Activated Platelets ◽

Similar Protein ◽

Diffusion Assay ◽

Supernatant Solution

SummaryAn attempt was made to isolate from plasma the platelet surface substrate for thrombin, glycoprotein V (GPV), because a GPV antigen was reported to be present in plasma (3). Plasma fractionation based on procedures for purification of GPV from platelets revealed a thrombin-sensitive protein with appropriate electrophoretic mobility. The protein was purified; an antiserum against it i) reacted with detergent-solubilized platelet proteins or secreted proteins in a double diffusion assay, ii) adsorbed a protein from the supernatant solution of activated platelets, and iii) inhibited thrombin-induced platelet activation, but the antiserum did not adsorb labeled GPV. The purified protein was immunochemically related to prothrombin rather than to GPV. Other antibodies against prothrombin were also able to adsorb a protein from platelets. It is concluded that 1) plasma does not contain appreciable amounts of GPV, and 2) platelets contain prothrombin or an immunochemically similar protein.

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