scholarly journals Complex Dynamics in One-Dimensional Nonlinear Schrödinger Equations with Stepwise Potential

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-17
Author(s):  
Chiara Zanini ◽  
Fabio Zanolin
Keyword(s):  
Complex Dynamics ◽  
Neumann Boundary ◽  
Step Function ◽  
One Dimensional ◽  
Nonlinear Schrödinger ◽  
Existence And Multiplicity ◽  
Twist Maps ◽  
Topological Horseshoes ◽  
Recent Developments ◽  
The One

We prove the existence and multiplicity of periodic solutions as well as solutions presenting a complex behavior for the one-dimensional nonlinear Schrödinger equation -ε2u′′+V(x)u=f(u), where the potential V(x) approximates a two-step function. The term f(u) generalizes the typical p-power nonlinearity considered by several authors in this context. Our approach is based on some recent developments of the theory of topological horseshoes, in connection with a linked twist maps geometry, which are applied to the discrete dynamics of the Poincaré map. We discuss the periodic and the Neumann boundary conditions. The value of the term ε>0, although small, can be explicitly estimated.

scholarly journals Order parameter profiles in a system with Neumann – Neumann boundary conditions

2018 ◽  
Vol 145 ◽  
pp. 01009 ◽  
Author(s):  
Vassil M. Vassilev ◽  
Daniel M. Dantchev ◽  
Peter A. Djondjorov
Keyword(s):  
Boundary Conditions ◽  
Order Parameter ◽  
Neumann Boundary ◽  
Elliptic Functions ◽  
Thermodynamic System ◽  
Neumann Boundary Conditions ◽  
Ginzburg Landau ◽  
One Dimensional ◽  
The One ◽  
Field Plane

In this article we consider a critical thermodynamic system with the shape of a thin film confined between two parallel planes. It is assumed that the state of the system at a given temperature and external ordering field is described by order-parameter profiles, which minimize the one-dimensional counterpart of the standard ϕ4 Ginzburg–Landau Hamiltonian and meet the so-called Neumann – Neumann boundary conditions. We give analytic representation of the extremals of this variational problem in terms ofWeierstrass elliptic functions. Then, depending on the temperature and ordering field we determine the minimizers and obtain the phase diagram in the temperature-field plane.

REACTION-FRONT DYNAMICS IN A+B→C WITH INITIALLY-SEPARATED REACTANTS

Fractals ◽  
1993 ◽  
Vol 01 (03) ◽  
pp. 405-415 ◽  
Author(s):  
S. HAVLIN ◽  
M. ARAUJO ◽  
H. LARRALDE ◽  
A. SHEHTER ◽  
H.E. STANLEY
Keyword(s):  
Reaction Front ◽  
Reaction Rate ◽  
Reaction System ◽  
Diffusion Reaction ◽  
Dimensional System ◽  
Scaling Exponent ◽  
One Dimensional ◽  
Recent Developments ◽  
The One ◽  
Front Dynamics

We review recent developments in the study of the diffusion reaction system of the type A+B→C in which the reactants are initially separated. We consider the case where the A and B particles are initially placed uniformly in Euclidean space at x>0 and x<0 respectively. We find that whereas for d≥2 a single scaling exponent characterizes the width of the reaction zone, a multiscaling approach is needed to describe the one-dimensional system. We also present analytical and numerical results for the reaction rate on fractals and percolation systems.

scholarly journals Positive Solutions of the One-Dimensional p-Laplacian with Nonlinearity Defined on a Finite Interval

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Ruyun Ma ◽  
Chunjie Xie ◽  
Abubaker Ahmed
Keyword(s):  
Boundary Value Problems ◽  
Positive Solutions ◽  
Finite Interval ◽  
Boundary Value ◽  
Quadrature Method ◽  
One Dimensional ◽  
Existence And Multiplicity ◽  
The One ◽  
Multiplicity Of Positive Solutions

We use the quadrature method to show the existence and multiplicity of positive solutions of the boundary value problems involving one-dimensional p-Laplacian u′t|p−2u′t′+λfut=0, t∈0,1, u(0)=u(1)=0, where p∈(1,2], λ∈(0,∞) is a parameter, f∈C1([0,r),[0,∞)) for some constant r>0, f(s)>0 in (0,r), and lims→r-(r-s)p-1f(s)=+∞.

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