Asymptotic Analysis of a Coupled System of Nonlocal Equations with Oscillatory Coefficients

James M. Scott; Tadele Mengesha

doi:10.1137/19m1288085

Generalized Darboux transformation and asymptotic analysis on the degenerate dark-bright solitons for a coupled nonlinear Schrödinger system

Physica Scripta ◽

10.1088/1402-4896/ac38d7 ◽

2021 ◽

Author(s):

He-yuan Tian ◽

Bo Tian ◽

Yan Sun ◽

Su-Su Chen

Keyword(s):

Asymptotic Analysis ◽

Darboux Transformation ◽

Coupled System ◽

Bound State ◽

Bright Soliton ◽

Spectral Parameters ◽

Nonlinear Superposition ◽

Bright Solitons ◽

Generalized Darboux Transformation ◽

The Given

Abstract In this paper, our work is based on a coupled nonlinear Schr ̈odinger system in a two-mode nonlinear fiber. A (N,m)-generalized Darboux transformation is constructed to derive the Nth-order solutions, where the positive integers N and m denote the numbers of iterative times and of distinct spectral parameters, respectively. Based on the Nth-order solutions and the given steps to perform the asymptotic analysis, it is found that a degenerate dark-bright soliton is the nonlinear superposition of several asymptotic dark-bright solitons possessing the same profile. For those asymptotic dark-bright solitons, their velocities are z-dependent except that one of those velocities could become z-independent under the certain condition, where z denotes the evolution dimension. Those asymptotic dark-bright solitons are reflected during the interaction. When a degenerate dark-bright soliton interacts with a nondegenerate/degenerate dark-bright soliton, the interaction is elastic, and the asymptotic bound-state dark-bright soliton with z-dependent or z-independent velocity could take place under certain conditions. Our study extends the investigation on the degenerate solitons from the bright soliton case for the scalar equations to the dark-bright soliton case for a coupled system.

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Asymptotic analysis of a viscous fluid–thin plate interaction: Periodic flow

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202514500079 ◽

2014 ◽

Vol 24 (09) ◽

pp. 1781-1822 ◽

Cited By ~ 10

Author(s):

G. P. Panasenko ◽

R. Stavre

Keyword(s):

Viscous Fluid ◽

Asymptotic Analysis ◽

Coupled System ◽

Thin Elastic Plate ◽

Periodic Flow ◽

Complete Asymptotic Expansion ◽

Thin Channel ◽

Elastic Wall ◽

Leading Term ◽

2D Elasticity

The first goal of this paper is to provide an asymptotic derivation and justification of the model studied in [Asymptotic analysis of a periodic flow in a thin channel with visco-elastic wall, J. Math. Pures Appl.85 (2006) 558–579]. We consider the coupled system "viscous fluid flow–thin elastic plate" when the thickness of the plate, ε, tends to zero, while the density and the Young's modulus of the plate material are of order ε-1and ε-3, respectively. The plate lies on the fluid which occupies a thick domain. The complete asymptotic expansion is constructed when ε tends to zero and it is proved that the leading term of the expansion satisfies the equations of [Asymptotic analysis of a periodic flow in a thin channel with visco-elastic wall, J. Math. Pures Appl.85 (2006) 558–579]. The second goal is the partial asymptotic decomposition formulation of the original problem when a part of the plate is described by a one-dimensional (1D) model while the other part is simulated by the two-dimensional (2D) elasticity equations. The appropriate junction conditions based on the previous asymptotic analysis are proposed at the interface point between the 1D and 2D equations. The error of the method is evaluated.

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The asymptotic analysis of a Darcy-Stokessystem coupled through a curved interface

Revista Integración ◽

10.18273/revint.v37n2-2019004 ◽

2019 ◽

Vol 37 (2) ◽

pp. 261-297

Author(s):

Fernando A. Morales

Keyword(s):

Porous Medium ◽

Asymptotic Analysis ◽

Stokes Flow ◽

Coupled System ◽

Narrow Channel ◽

Limit Problem ◽

Darcy Flow ◽

Cylindrical Domain ◽

Fluid Exchange ◽

Curved Interface

We present the asymptotic analysis of a Darcy-Stokes coupledsystem, modeling the fluid exchange between a narrow channel(Stokes flow)and a porous medium (Darcy flow), coupled through aC2curved interface.The channel is a cylindrical domain between the interface (Γ) and a paralleltranslation of itself (Γ +ǫbeN, ǫ >0). The introduction of a change variable(to fix the domain geometry) and the introduction of two systems of coor-dinates: the Cartesian and a local one (consistent with the geometry of thesurface), permit to find the limiting form of the system when the width of thechannel tends to zero (ǫ→0). The limit problem is a coupled system withDarcy flow in the porous medium and Brinkman flow on the curved interface(Γ).

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On the asymptotic analysis of problems involving fractional Laplacian in cylindrical domains tending to infinity

Communications in Contemporary Mathematics ◽

10.1142/s0219199716500358 ◽

2016 ◽

Vol 19 (05) ◽

pp. 1650035 ◽

Cited By ~ 3

Author(s):

Indranil Chowdhury ◽

Prosenjit Roy

Keyword(s):

Asymptotic Behavior ◽

Asymptotic Analysis ◽

Fractional Laplacian ◽

Elliptic Problems ◽

Weighted Estimate ◽

Dirichlet Problems ◽

Nonlocal Equations ◽

Cylindrical Domains ◽

Non Local ◽

Second Order Elliptic Problems

The paper is an attempt to investigate the issues of asymptotic analysis for problems involving fractional Laplacian where the domains tend to become unbounded in one-direction. Motivated from the pioneering work on second-order elliptic problems by Chipot and Rougirel in [On the asymptotic behaviour of the solution of elliptic problems in cylindrical domains becoming unbounded, Commun. Contemp. Math. 4(1) (2002) 15–44], where the force functions are considered on the cross-section of domains, we prove the non-local counterpart of their result.Recently in [Asymptotic behavior of elliptic nonlocal equations set in cylinders, Asymptot. Anal. 89(1–2) (2014) 21–35] Yeressian established a weighted estimate for solutions of non-local Dirichlet problems which exhibit the asymptotic behavior. The case when [Formula: see text] was also treated as an example to show how the weighted estimate might be used to achieve the asymptotic behavior. In this paper, we extend this result to each order between [Formula: see text] and [Formula: see text].

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