Nonlinear standing waves on planar branched systems: Shrinking into metric graph

TL;DR

This paper investigates the behavior of nonlinear standing waves on two-dimensional branched systems as they shrink into one-dimensional metric graphs, providing analytical and numerical insights into boundary condition convergence.

Contribution

It offers a detailed analysis of the convergence of boundary conditions from fat graphs to metric graphs for the nonlinear Schrödinger equation, including numerical validation and practical applications.

Findings

Convergence of boundary conditions from fat to metric graphs demonstrated.

Numerical solutions confirm analytical estimates.

Applicable to various physical systems like Bose-Einstein condensates and optical media.

Abstract

We treat the stationary nonlinear Schroodinger equation on two-dimensional branched domains, so-called fat graphs. The shrinking limit when the domain becomes one-dimensional metric graph is studied by using analytical estimate of the convergence of fat graph boundary conditions into those for metric graph. Detailed analysis of such convergence on the basis of numerical solution of stationary nonlinear Schrodinger equation on a fat graph is provided. Possibility for reproducing different metric graph boundary conditions studied in earlier works is shown. Practical applications of the proposed model for such problems as Bose-Einstein condensation in networks, branched optical media, DNA, conducting polymers and wave dynamics in branched capillary networks are discussed.

Equations

$$-\Delta \psi +(V_{\alpha ,ϵ}-\mu )\psi -|\psi {|}^2\psi =0,$$

$$V_{\alpha ,ϵ}(x)=-\frac{\alpha }{2\pi ϵ}\exp (-|x{|}^2/2{ϵ}^2)$$

$$V_{\alpha ,ϵ}\sim -\alpha ϵ\delta (x)\text{for ϵ→0}.$$

$$-{\varphi }^{\prime \prime }-\mu \varphi -|\varphi {|}^2\varphi =0.$$

$${\varphi }_1={\varphi }_2={\varphi }_3,$$

$${\varphi }_1^{\prime }+{\varphi }_2^{\prime }+{\varphi }_3^{\prime }=\alpha {\varphi }_1.$$

$$\frac{\partial \psi }{\partial n}=0\text{on}\partial {\Omega }_{\epsilon }/({\Gamma }_{1,\epsilon }\cup {\Gamma }_{2,\epsilon }\cup {\Gamma }_{3,\epsilon }).$$

$$\frac{\partial \psi }{\partial n}{|}_{{\Gamma }_{k,\epsilon }}\to {\phi }_k^{\prime },\psi {|}_{{\Gamma }_{k,\epsilon }}\to {\phi }_k\text{at}ϵ\to 0,k=1,2,3,$$

$$-{\Delta }_{y}v(y)-{ϵ}^{2}f(v(y))+\alpha ϵV_{1,1}v(y)=0,$$

$$\underset{{\Omega }_1}{\iint }|{\nabla }_y{v}_m(y){|}^{2}dy=\underset{\partial {\Omega }_{{ϵ}_m}}{\int }{\psi }_m\frac{\partial {\psi }_m}{\partial n}d{s}_x-$$

$$-\iint {\Omega }_{{ϵ}_m}\left[V_{\alpha ,{ϵ}_m}(x){\psi }_m(x)+u_m(x)f({\psi }_m(x))\right]dx⩽$$

$$⩽c_1{c}_2{ϵ}_m\sum _{k=1}^3|{\Gamma }_k|+c_1^4|{\Omega }_1|{ϵ}_m^2+c_1{ϵ}_m\to 0.$$

$$\sum _{k=1}^3\frac{1}{ϵ}\underset{{\Gamma }_{k,ϵ}}{\int }\frac{\partial \psi }{\partial n}d{\Gamma }_{k,ϵ}\to \sum _k|{\Gamma }_1|{\phi }_k^{\prime }.$$

$$\sum _{k=1}^3\frac{1}{ϵ}\underset{{\Gamma }_{k,ϵ}}{\int }\frac{\partial \psi }{\partial n}d{\Gamma }_{k,ϵ}+\alpha \underset{{\Omega }_{ϵ}}{\iint }\frac{1}{{ϵ}^2}{V}_{1,1}\left(\frac{x}{ϵ}\right)$$

$$\frac{1}{ϵ}\underset{{\Omega }_{ϵ}}{\iint }|\psi {|}^2\psi dx\sim O(ϵ),\frac{\mu }{ϵ}\underset{{\Omega }_{ϵ}}{\iint }\psi dx\sim O(ϵ),$$

$$\underset{ϵ\to 0}{\lim }\frac{1}{ϵ}\underset{{\Gamma }_{k,ϵ}}{\int }\frac{\partial \psi }{\partial n}d{\Gamma }_{j,ϵ}=\alpha \psi (0).$$

$$\sum _k|{\Gamma }_k|{\phi }_k^{\prime }=\alpha \phi .$$

$${\varphi }_j(x,a)=\sqrt{2\mu }sech(\sqrt{\mu }(x-a^j)),$$

$$a^j=\frac{1}{\mu }arctanh\left(\frac{\alpha }{(2j-3)\sqrt{\mu }}\right)j=1,2,3.$$

$$-\Delta \psi +(V_{\alpha ,ϵ}-\mu )\psi +|\psi {|}^2\psi =0.$$

Full text

I Introduction

Branched structures and networks appear in many physical systems and in complex systems from biology, ecology, sociology, economy and finance Barabasi1 ; Havlin . Particle and wave dynamics in such systems can be effectively modeled by nonlinear evolution equations on metric graphs. The latter are one dimensional system of bonds which are connected at one or more vertices (branching points). The connection rule is called the topology of the graph. When the bonds can be assigned a length, the graph is called a metric graph. Recently nonlinear evolution equations on metric graph attracted much attention in the literature (see the Refs. zar2010 -Noja2015 ). To solve nonlinear evolution equations on networks one need to impose boundary conditions on graph vertices. Soliton solutions providing reflectionless transmission at the graph vertex together with integrable boundary conditions were derived in zar2010 . Different aspects of nonlinear Schrodinger equation including soliton solutions are discussed in the Refs.adami-eur ; adami-jpa . Solutions of stationary nonlinear Schrödinger equation on graphs for different vertex conditions are obtained in adami2011 ; adami-eur ; Karim2013 ; noja ; Noja2015 . Despite the growing interest to nonlinear evolution equations and soliton dynamics in networks, most of the studies are still restricted by considering metric graphs, i.e. by one-dimensional motion wave motion in branched structures. However, in many cases particle and wave motion in branched structures has certain transverse component, so that the system is two-dimensional. Such systems should be described within two-dimensional evolution equations on planar networks. Such systems can be modeled by so-called fat graphs. Earlier, the linear Schrodinger equation on fat graphs was addressed in number of works expo09 -expo13 by considering metric graph limit as transition to from planar to linear wave motions. Extension of such a study to the case of nonlinear Schrodinger equation based on the numerical treatment of the problem was done in recent work Our2015 . Fat graph treatment of sine-Gordon equation on branched Y-junctions was numerically treated in caputo14 . Pioneering treatment of nonlinear evolution equations on fat graphs date back to Kosugi, who presented in the Ref.kosugi2002 strict mathematical treatment of the nonlinear elliptic differential equations on branched domains and estimates for the shrinking limit. The study of the nonlinear differential equation on branched domains is complicated due to the nonlinearity of the boundary conditions imposed at the branching area or vertex. In case of nonlinear evolution equations related to physics such boundary conditions should be derived from conservation laws or other physical conditions that greatly simplifies these boundary conditions zar2010 ; Our2015 .

In this work we address the problem of stationary (time-independent) nonlinear Schrodinger equation on fat graphs by focusing on metric graph limit in the shrinking of a fat graph. In particular, we obtain estimate for the vertex boundary condition in the shrinking limit and show that they reproduce those for metric graph considered in adami-eur ; Karim2013 . Also, we present a treatment of the problem on the basis of numerical solution of stationary NLSE on fat graph. For small enough bond widths the numerical solution reproduce metric graph solutions from the Refs.adami-eur ; Karim2013 .

Motivation for the study of stationary NLSE comes from such problems as Bose-Einstein condensation on networks Barabasi -Cast , standing waves in branched optical waveguides Assanto , branched liquid crystals and standing wave-polarons in polymers Heeger ; Heeger1 . All these systems require taking into account of transverse component of the wave motion which can be done by considering stationary NLSE on fat graphs. Estimating of the ”boundary” between the planar and linear motions is of importance for experiments on the study of the wave motion in nonlinear networks, as such networks are always somehow planar or tubular.

This paper is organized as follows. In the next section we give formulation of stationary NLSE on fat and metric graphs. Section III presents the results of numerical solution of stationary NLSE on fat graph at different values of the graphs width by considering shrinking limit. In section IV we present detailed discussion of potential application of fat graph problem to some physical systems having planar branched structure. Finally, section V gives some concluding remarks.

II Stationary NLSE on a fat graph

Wave equation on planar branched domains which are often called ”fat graphs” has attracted much attention in the context of wave dynamics during last decade. In particular, stationary linear wave equations on fat graphs have been studied in the Refs.expo05 -expo13 (see Grieser ; pobook and references therein for detailed reviews). Corresponding nonlinear problem is mainly studied for one-dimensional case by considering metric graph approach Different aspects of the nonlinear Schrodinger equation on branched one dimensional branched domains called metric graphs were studied earlier adami2011 -Noja2015 . The metric graph is a set of bonds connected to each other at the vertices according to a rule which called topology of a graph with metrics defined in each bond. Recently the problem of soliton transport in planar branched domains was studied based on numerical solution of NLSE on fat graph Barabasi . Transition from fat to metric graph problem was shown in this study.

Our purpose is to explore solutions of the stationary nonlinear Schrodinger equation on two-dimensional branched domain in the limit when the domain shrinks into a metric graph. Therefore we introduce two problems, ”fat graph” and ”metric graph” problems. Both fat and metric graphs are presented in Fig. 1. Fat graph is a branched domain having two dimensional bonds and vertex as presented (See Fig.1.). In the following we denote bond-lengths of such graph by $l_{1},l_{2}$ and $l_{3}$, bond-widths by $w_{1},w_{2}$ and $w_{3}$. The diameter of the vertex region $\Omega_{\epsilon}=\epsilon\Omega_{1}$ is $r\epsilon$. The stationary nonlinear Schrodinger equation (NLSE) on a ”fat graph” is given as

[TABLE]

where the potential

[TABLE]

is localized at the vertex domain, $\alpha\in R$ is a parameter. The potential is chosen as to reproduce delta-function in the shrinking limit, i.e.

[TABLE]

In the metric graph case we determine coordinate on the $k$th bond from [math] to $l_{k}$, $k=1,2,3$. The metric graph problem is determined by the stationary NLSE which is given on each bond of the graph as

[TABLE]

The equations are related via the vertex conditions given by

[TABLE]

[TABLE]

In the following the problems given by Eqs.(4) -(6) will be called ”metric graph problem”. The aim of this paper is to explore both analytically and numerically the shrinking limit of the fat graph problem given by Eqs.(1) and (2) and determining the conditions providing in the shrinking limit the transition of the fat graph problem with that of metric graph. Such an analysis will consist of two parts, analytical estimate for the convergence of the fat graph problem to that of metric graph and numerical analysis for such convergence. The latter implies the analysis of numerical solution solution of NLSE for fat graph for different small values of the bond and vertex widths. Here we focus on the vertex conditions considered in the Refs.adami-eur ; adami2013 ; noja which are often called ”delta” type boundary conditions.

We divided the fat graph to the following parts: first is vertex region denoted by $\Omega_{\epsilon}=\epsilon\Omega_{1}$, where we assumed $\Omega_{1}$ is convex region with smooth enough boundary ($0\in\Omega_{1}$) and tabular (rectangular) parts. In the limit $\epsilon\to 0$ vertex region give as a vertex point [math], while tabular parts tends to the bonds of metric graph. Convergence problem in the case of tabular region is well studied. Here we refer to kosugi2002 , jimbo . We focus only vertex region. We denote by $\Gamma_{k,\epsilon},(k=1,2,3)$ the those parts(arcs) of the vertex which are connected to the bonds ($\Gamma_{k,\epsilon}=\epsilon\Gamma_{k}$). We impose the following boundary conditions for NLSE given by Eq.(1) in ${\Omega_{\epsilon}}$ :

[TABLE]

Assume that

[TABLE]

where $\varphi_{k}$ and $\varphi^{\prime}_{k}$ ($k=1,2,3$)are constants.

First we show that $\varphi_{1}=\varphi_{2}=\varphi_{3}$.

According to jimbo , kosugi2002 and maximum principle Gilbarg we have $|\psi(x)|<c_{1}$, $|\nabla\psi(x)|\leqslant c_{2}$ for $x\in\Omega_{\epsilon}$

Denote $v(y)=\psi(y/\epsilon)$. The function $v(y)$ satisfies the following equation

[TABLE]

with $f(v)=|v|^{2}v+\mu v$.

The function $v(y)$ satisfies the estimate (see, e.g. kosugi2002 , Gilbarg ) $\|v\|_{C^{2}(\Omega_{1})}\leqslant C_{3}$. From the Ascoli-Arzel theorem, there exist a sequence $\epsilon_{m}\to 0$ and the function $v_{\infty}\in C^{1}(\overline{\Omega_{1}})$, such that $\lim_{k\to\infty}\|v_{m}-v_{\infty}\|_{C^{1}(\overline{\Omega_{1}})}=0.$ Where $v_{m}(y)$ is the function $v(y)$ with $\epsilon=\epsilon_{m}$.

Then we have

[TABLE]

[TABLE]

[TABLE]

From Eq.((9)) we have $v_{\infty}=const$, which proves $\varphi_{1}=\varphi_{2}=\varphi_{3}=\varphi.$

Now we verify the second vertex condition in the limiting problem. For the mean values of the normal derivatives on the boundary we have ($\epsilon=\epsilon_{m}$)

[TABLE]

On the other hand, integrating Eq.(10) over $\Omega_{\epsilon}$, we get

[TABLE]

According to maximum principle Gilbarg for small $\epsilon$ we have $\sup|u|\leqslant C$. Therefore for the integrals we can write

[TABLE]

for small $\epsilon$. Taking into account the above relations and properties of the potential $V_{\alpha,\epsilon}$, we obtain

[TABLE]

Eqs. (10) and (12) lead to

[TABLE]

Thus we showed convergence of the fat graph problem given by Eqs.(1), (2) and (10) to that for metric graph with delta type boundary conditions given by Eqs.(4)-(6). In the next section we will show such convergence on the basis of the numerical solution of stationary NLSE on fat graph.

III Numerical treatment of the shrinking limit

Our purpose is showing convergence of the stationary NLSE on fat graph into that for metric graph in the shrinking limit using both analytically and numerical analysis of the shrinking limit. Such convergence is of practical importance for various problems dealing the wave dynamics in branched structures where one needs to neglect by transverse motion of the waves. Here we will focus on the analysis of behavior wave function itself as well as $|\psi(x,y)|^{2}$. The latter has important physical meaning in the practical applications of the NLSE, e.g., density of particles in BEC, beam intensity in optics, etc.

The estimate presented by Eqs.(12) and (13) shows convergence of the vertex boundary conditions of a fat graph into those for metric graph. Such a convergence can be shown on the basis of numerical the solution of the stationary NLSE on fat graph by considering the shrinking limit, $\epsilon\to 0$.

Here we will explore numerical solutions of Eq.(1) with the fat graph boundary conditions given by Eq.(10) at different values of the fat graph bond length by considering both attractive and repulsive nonlinearities. They describe bright and dark (static) solitons, respectively. Assuming $w_{1}=\epsilon w_{2}=\epsilon w_{3}$, we show that these numerical solutions reproduce the solutions of metric graph problem analytically obtained in the Refs.adami-eur . Solution of the metric graph problem given by Eqs.(4) -(6) was obtained in adami-eur and can be written as

[TABLE]

where

[TABLE]

In Fig. 2 solutions of stationary NLSE on fat graph obtained by numerical solutions of two-dimensional stationary NLSE (1) with the boundary conditions (10) are for different values of the bond width $\epsilon=3,{\rm and}1$ ($\;\mu=-1,\;\alpha=0.5$). As it can be seen, the wave function is localized at the vertex in all cases.

Fig. 3 compares solution of Eq.(1) for the bond width, $\epsilon=1$ with that of for metric graph given by Eqs.(4) -(6). It can be observed the convergence of the solution of fat graph problem to that of metric graph in the shrinking limit.

Besides NLSE with attractive (focusing) nonlinearity given by Eq.(1), one can consider repulsive case when the wave equation given as

[TABLE]

In Fig.4 solutions for fat and metric graphs are compared for the case of repulsive nonlinearity and $\epsilon=0.3$. Unlike to attractive case, the wave function localized on the bonds for this repulsive nonlinearity. Fig. 5 shows how the solution NLSE on fat graph depends on chemical potential $\mu$.

IV Standing nonlinear waves in branched planar structures

Two-dimensional branched structures where the wave dynamics is described by nonlinear Schrodinger equation appear in many areas of physics. Here we will briefly discuss physical systems where the above model can be applied.

BEC in planar networks. Bose-Einstein condensation is a remarkable phenomenon that can be described by a version of NLSE which is called Gross-Pitaevskii equation Barabasi -Burioni3 . It can be realized in trapped cold atoms and depending on the type of a trap NLSE based model has different potentials and boundary conditions. The above model describes BEC in planar branched structures/traps which can be experimentally created in several physical systems. We note that the Bose-Einstein condensation in networks attracted much attention recently (see, e.g. Refs.Barabasi -Cast ). Despite the fact that nonlinear Schrodinger equation is the powerful tool for description of BEC dynamics, all the studies of BEC in networks used so far tight binding and statistical mechanics based approaches Barabasi -Cast . Moreover, most of the studies of this problem does not discuss experimental realization of BEC in networks. Planar BEC can be experimentally realized in surface optical traps Butov , superconductive BEC for exitons in planar systems Ben and atom chip films Oberthaler . All these systems can be constructed in in branched form in which the BEC standing wave can be described by Eq. (1). In this case, parameter $\mu$ in Eq. (1) corresponds to chemical potential, while $|\psi|^{2}$ describes number of atoms in condensate. Transition from planar to one-dimensional BEC dynamics corresponding to the shrinking limit of the above fat graph can be then treated similarly to that in the Ref.Carr2000 . Other types of networks and branched systems where planar and one-dimensional BEC can occur are the different types of Josephson junctions Burger -Oberthaler . Networks or branched structures of Josephson junctions can be realized in different versions Hasegawa -Luca1 . Planar Josephson junctions can be fabricated in using different techniques Kras ; Nus . The standing wave states of the condensate in planar Josephson network can be described by our model. Conducting polymers are also branched systems where BEC can be experimentally realized Cast .

Networks of planar optical waveguides and fibers. Optical waveguides are the systems where the wave propagation can be described by linear or nonlinear Schrodinger equation. Stationary nonlinear Schrodinger equation describes standing wave modes in such waveguides. Optical waveguides and fibers can be realized in linear, planar and cylindric forms. Few papers discuss different ways for experimental realizations of planar waveguides Sand ; Eis . Networks of such waveguides is of importance from the viewpoint of practical importance. In particular, modern optical telecommunication technology requires using networks of such waveguides rather than separate fibers. Earlier branching of nonlinear optical waves in Y-junction optical media was discussed in the Refs. Assanto ; Santos . However, these works did not use the above fat or metric graph approaches. In branched optical system, the function $\psi$ in Eq.(1) describes amplitude of the wave, while quantity $\mu$ corresponds to the propagation constant. By exploring the shrinking limit of NLSE on a fat graph one can determine minimal width of the fibers for which wave transport in such networks can be considered as one dimensional. Also, the above model can be useful for fabrication the materials and devices with tunable optical properties.

Standing waves in DNA. Remarkable branched structure where solitons and nonlinear waves appear is the DNA. Depending on the model and approach, such waves can be described either by sine-Gordon or nonlinear Schrodinger equations Yakushevich1 -Sataric . Within so-called Peyrard-Bishop model nonlinear dynamics of DNA base pair is described by NLSE. The base pair of DNA has a branched structure and can be considered as a star graph. Realistic wave motion in DNA is two-dimensional and the transverse component of the oscillations can play important role in DNA dynamics. Therefore by studying shrinking limit allows to determine the boundary between the one- and two-dimensional approaches. Two dimensional model of DNA dynamics was considered earlier in Yakushevich1 -Sataric .

Conducting polymers. Polymers are molecular networks having complicated topology whose structural units can be often reduced to star graph. A type of polymers which are called conjugated polymers can exhibit metallic or semiconductor properties. Such polymers, which are called conducting polymers attracted much attention due their wide range of electronic applications Heeger ; Heeger1 . Charge and spin excitations and their transport in conducting polymers can have solitons and standing nonlinear waves described by NLSE. The motion of such waves is usually two-dimensional that can be effectively described to one dimensional form. Due to finite size of the branching points a polymer chain can be considered as a fat graph. The above NLSE on fat graph and its shrinking limit ca be powerful tool for description of charge and spin carriers dynamics in conducting polymers. Especially this can be effective method for charge transport, recombination and separation in polymer based materials such as light-emitting diodes, organic solar cells etc.

Capillary networks. Solitons and nonlinear standing waves appear in hydrodynamics capillary systems, where fluid dynamics can be described by NLSE Rassm . Planar version of capillary systems as discussed inRassm . Nonlinear wave in branched capillary systems should be described by NLSE on a fat graph. The connection point of such network has finite size. Our model allows to estimate characteristic width of the capillary tube, which which wave motion in such network can be considered as linear.

V Conclusions

In this work we studied the stationary nonlinear Schrodinger equation on fat graphs by focusing on metric graph transition in the shrinking limit. Analytical estimate for the shrinking of boundary conditions are obtained. It is shown that in the shrinking limit fat graph boundary conditions reproduce those for metric graph. Such a convergence is also shown on the basis of numerical treatment of NLSE on a fat graph. Detailed discussion of the potential application of the model to BEC in networks, branched optical materials, DNA double helix, conducting polymers and capillary networks is discussed. The model can be extended for other fat graph topologies, such as tree, ring and complete graphs. The above study allows to determine the border between planar and linear motion in branched systems, where particle and wave transport is effectively considered as one dimensional. The results of this paper can be useful for the problems of engineering materials and devices on the basis of branched optical and electronic structures.

Acknowledgement

This work is supported by a grant of the Volkswagen Foundation. We thank Hannes Uecker for his valuable comments on the formulation of the problem and for the help in numerical part of the work.