A BRIEF SURVEY ON CONSTRUCTING HOMOCLINIC STRUCTURES OF SOLITON EQUATIONS

2006 ◽  
Vol 16 (10) ◽  
pp. 2799-2813 ◽  
Author(s):  
RANCHAO WU ◽  
JIANHUA SUN
Keyword(s):  
Nonlinear Partial Differential Equations ◽  
Plane Waves ◽  
Homoclinic Orbits ◽  
Lax Pair ◽  
Dressing Method ◽  
Soliton Equations ◽  
Coupled Equations ◽  
Mathematical Proofs ◽  
Extended Application ◽  
Uniform Plane

To give rigorous mathematical proofs of chaotic behaviors in a given system, it is necessary to identify the homoclinic structures in the system. In this tutorial review, methods for constructing explicit solutions for nonlinear partial differential equations are presented, with more emphasis placed on those utilizing complete integrability associated with soliton equations. As an extended application, homoclinic orbits to spatial uniform plane waves of coupled modified nonlinear Schrödinger equations are obtained via the dressing method. During the procedure, it is necessary to introduce the Lax pair for these coupled equations, as well as its Floquet spectral analysis and corresponding Bloch functions.

Integrable variable-coefficient derivative Spin-1 Gross–Pitaevskii equations and their explicit solutions

2021 ◽  
Author(s):  
Ting Su ◽  
Junhong Yao ◽  
Yanan Huang
Keyword(s):  
Variable Coefficient ◽  
Separation Of Variables ◽  
Lax Pair ◽  
Explicit Solutions ◽  
Dressing Method

Based on the generalized dressing method, we propose a new integrable variable coefficient Spin-1 Gross–Pitaevskii equations and derive their Lax pair. Using separation of variables, we have derived explicit solutions of the equations. In order to analyze the characteristic of derived solution, the graphical wave of the solutions is plotted with the aid of Matlab.

Convection heat transfer in a Maxwell fluid at a non-isothermal surface

Open Physics ◽  
2011 ◽  
Vol 9 (3) ◽  
Author(s):  
Kuppalapalle Vajravelu ◽  
Kerehalli Prasad ◽  
Ashwatha Sujatha
Keyword(s):  
Heat Transfer ◽  
Differential Equations ◽  
Variable Temperature ◽  
Nonlinear Partial Differential Equations ◽  
Convection Heat Transfer ◽  
Maxwell Fluid ◽  
Convection Heat ◽  
Stretching Surface ◽  
Coupled Equations ◽  
Wide Range

AbstractAnalysis is carried out to study the convection heat transfer in an upper convected Maxwell fluid at a non-isothermal stretching surface. This is a generalization of the paper by Sadeghy et al. [21] to study the effects of free convection currents, variable thermal conductivity and the variable temperature at the stretching surface. Unlike in Sadeghy et al., here the governing nonlinear partial differential equations are coupled. These coupled equations are transformed in to a system of nonlinear ordinary differential equations and are solved numerically by a finite difference scheme (known as the Keller-Box method) and the numerical results are presented through graphs and tables for a wide range of governing parameters. The results obtained for the flow and heat transfer characteristics reveal many interesting behaviors that warrant further study of nonlinear convection heat transfer.

scholarly journals NONCOMMUTATIVE EXTENDED WAVES AND SOLITON-LIKE CONFIGURATIONS IN N = 2 STRING THEORY

2003 ◽  
Vol 18 (26) ◽  
pp. 4889-4931 ◽  
Author(s):  
MATTHIAS IHL ◽  
SEBASTIAN UHLMANN
Keyword(s):  
Field Theory ◽  
String Theory ◽  
String Field Theory ◽  
Plane Waves ◽  
Equation Of Motion ◽  
Lax Pair ◽  
The Self ◽  
Yang Mills ◽  
Self Duality ◽  
Wave Solutions

The Seiberg–Witten limit of fermionic N = 2 string theory with nonvanishing B-field is governed by noncommutative self-dual Yang–Mills theory (ncSDYM) in 2+2 dimensions. Conversely, the self-duality equations are contained in the equation of motion of N = 2 string field theory in a B-field background. Therefore finding solutions to noncommutative self-dual Yang–Mills theory on ℝ2,2 might help to improve our understanding of nonperturbative properties of string (field) theory. In this paper, we construct nonlinear soliton-like and multi-plane wave solutions of the ncSDYM equations corresponding to certain D-brane configurations by employing a solution generating technique, an extension of the so-called dressing approach. The underlying Lax pair is discussed in two different gauges, the unitary and the Hermitian gauge. Several examples and applications for both situations are considered, including Abelian solutions constructed from GMS-like projectors, noncommutative U(2) soliton-like configurations and interacting plane waves. We display a correspondence to earlier work on string field theory and argue that the solutions found here can serve as a guideline in the search for nonperturbative solutions of nonpolynomial string field theory.

scholarly journals Numerical simulation on convection-diffusion in an overlapping stenosed artery

2014 ◽  
Vol 9 (3) ◽  
Author(s):  
Ilyani Abdullah ◽  
Amira Husni Talib ◽  
Siti Noor Izzati Che Mohd Sabri
Keyword(s):  
Mass Transport ◽  
Low Density Lipoprotein ◽  
Nonlinear Partial Differential Equations ◽  
Density Lipoprotein ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Coupled Equations ◽  
Stenosed Artery ◽  
Flowing Blood

Consider an unsteady Newtonian blood flow coupled with mass transport in which flowing through an artery with the presence of an overlapping stenosis. The flowing blood is governed by nonlinear partial differential equations while the convection-diffusion equation to blood is employed to couple with the Newtonian equation in order to characterize the mass transport of blood-borne components such as low-density lipoprotein (LDL). This mass transport refers to the movement of blood-borne molecules from flowing blood into the artery wall, or vice versa. These coupled equations are solved numerically using finite-difference method with an appropriate prescribed initial and boundary conditions. The graphical results of velocity profiles and mass concentration of the solute are presented along the distributions over the entire considered arterial segment. These results show the important role of mass transport in stenosed artery.

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