scholarly journals On the Marchenko System and the Long-time Behavior of Multi-soliton Solutions of the One-dimensional Gross-Pitaevskii Equation

2015 ◽  
Vol 28 (2) ◽  
pp. 167-196
Author(s):  
Mohamad Haidar
Keyword(s):  
Soliton Solutions ◽  
Pitaevskii Equation ◽  
Time Behavior ◽  
Long Time Behavior ◽  
One Dimensional ◽  
Long Time ◽  
The One

Long-time behavior for the full one-dimensional Frémond model for shape memory alloys

2000 ◽  
Vol 12 (6) ◽  
pp. 423-433 ◽  
Author(s):  
Pierluigi Colli ◽  
Philippe Laurençot ◽  
Ulisse Stefanelli
Keyword(s):  
Shape Memory Alloys ◽  
Shape Memory ◽  
Time Behavior ◽  
Long Time Behavior ◽  
One Dimensional ◽  
Long Time

scholarly journals Asymptotic behavior of solutions to a tumor angiogenesis model with chemotaxis–haptotaxis

2019 ◽  
Vol 29 (07) ◽  
pp. 1387-1412 ◽  
Author(s):  
Peter Y. H. Pang ◽  
Yifu Wang
Keyword(s):  
Tumor Angiogenesis ◽  
Asymptotic Behavior Of Solutions ◽  
Time Behavior ◽  
System Of Differential Equations ◽  
One Dimensional ◽  
Long Time ◽  
Bounded Smooth ◽  
The One ◽  
Global Boundedness ◽  
Angiogenesis Model

This paper studies the following system of differential equations modeling tumor angiogenesis in a bounded smooth domain [Formula: see text] ([Formula: see text]): [Formula: see text] where [Formula: see text] and [Formula: see text] are positive parameters. For any reasonably regular initial data [Formula: see text], we prove the global boundedness ([Formula: see text]-norm) of [Formula: see text] via an iterative method. Furthermore, we investigate the long-time behavior of solutions to the above system under an additional mild condition, and improve previously known results. In particular, in the one-dimensional case, we show that the solution [Formula: see text] converges to [Formula: see text] with an explicit exponential rate as time tends to infinity.

scholarly journals DISPERSIVE EFFECTIVE MODELS FOR WAVES IN HETEROGENEOUS MEDIA

2011 ◽  
Vol 21 (09) ◽  
pp. 1871-1899 ◽  
Author(s):  
AGNES LAMACZ
Keyword(s):  
Heterogeneous Media ◽  
Variable Coefficients ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Smooth Functions ◽  
Long Time ◽  
Constant Coefficients ◽  
The One ◽  
Effective Models ◽  
Dispersive Models

We study the long-time behavior of waves in a strongly heterogeneous medium, starting from the one-dimensional scalar wave equation with variable coefficients. We assume that the coefficients are periodic with period ε and ε > 0 is a small length parameter. Our main result concerns homogenization and consists in the rigorous derivation of two different dispersive models. The first is a fourth-order equation with constant coefficients including powers of ε. In the second model, the ε-dependence is completely avoided by considering a third-order linearized Korteweg–de Vries equation. Our result is that both simplified models describe the long-time behavior well. An essential tool in our analysis is an adaption operator which modifies smooth functions according to the periodic structure of the medium.

FROM BOUNDARY VALUE PROBLEMS TO DIFFERENCE EQUATIONS: A METHOD OF INVESTIGATION OF CHAOTIC VIBRATIONS

1999 ◽  
Vol 09 (07) ◽  
pp. 1285-1306 ◽  
Author(s):  
E. YU. ROMANENKO ◽  
A. N. SHARKOVSKY
Keyword(s):  
Difference Equation ◽  
Boundary Value Problems ◽  
Difference Equations ◽  
Boundary Value ◽  
Dynamical Behavior ◽  
Time Behavior ◽  
One Dimensional ◽  
Long Time ◽  
The Difference ◽  
The One

Among evolutionary boundary value problems for partial differential equations, there is a wide class of problems reducible to difference, differential-difference and other relevant equations. Of especial promise for investigation are problems that reduce to difference equations with continuous argument. Such problems, even in their simplest form, may exhibit solutions with extremely complicated long-time behavior to the extent of possessing evolutions that are indistinguishable from random ones when time is large. It is owing to the reduction to a difference equation followed by the employment of the properties of the one-dimensional map associated with the difference equation, that, it is in many cases possible to establish mathematical mechanisms for one or other type of dynamical behavior of solutions. The paper presents the overall picture in the study of boundary value problems reducible to difference equations (on which the authors have a direct bearing over the last ten years) and demonstrates with several simplest examples the potentialities that such a reduction opens up.

Exact Chirped Soliton Solutions for the One-Dimensional Gross– Pitaevskii Equation with Time-Dependent Parameters

2012 ◽  
Vol 67 (3-4) ◽  
pp. 141-146 ◽  
Author(s):  
Zhenyun Qina ◽  
Gui Mu
Keyword(s):  
Soliton Solution ◽  
Soliton Solutions ◽  
Einstein Condensate ◽  
Time Dependent ◽  
Pitaevskii Equation ◽  
Zero Temperature ◽  
One Dimensional ◽  
Bose Einstein Condensate ◽  
The One ◽  
Bose Einstein

The Gross-Pitaevskii equation (GPE) describing the dynamics of a Bose-Einstein condensate at absolute zero temperature, is a generalized form of the nonlinear Schr¨odinger equation. In this work, the exact bright one-soliton solution of the one-dimensional GPE with time-dependent parameters is directly obtained by using the well-known Hirota method under the same conditions as in S. Rajendran et al., Physica D 239, 366 (2010). In addition, the two-soliton solution is also constructed effectively

scholarly journals Long-time behavior of the one-phase Stefan problem in periodic and random media

2018 ◽  
Vol 11 (5) ◽  
pp. 991-1010
Author(s):  
Norbert Požár ◽  
◽  
Giang Thi Thu Vu ◽  
Keyword(s):  
Stefan Problem ◽  
Random Media ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Long Time ◽  
The One
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