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<ul><li><p>ISSN 1054660X, Laser Physics, 2011, Vol. 21, No. 8, pp. 14911502. Pleiades Publishing, Ltd., 2011.Original Text Astro, Ltd., 2011.</p><p>1491</p><p>1 1. INTRODUCTION</p><p>We propose and compare some numerical schemesto solve the general Schrdinger equations inunbounded media</p><p>(1)</p><p>The wave function is defined in the unboundeddomain N (N 1). In view of a numerical computation, different solutions may be used. For example,one usual scheme consists in splitting the Laplacianand potentialnonlinear parts of the equation and nextsolving the first linear equation, e.g., by FFT methodsand exactly integrating the second nonlinear one (see,e.g., [1]). This kind of scheme is efficient and accurateif the solution remains confined within the computational domain (for instance for solving the GrossPitaevskii equations). Indeed, then periodic boundaryconditions may be applied for the Fourier solutionsince the wave vanishes on the boundary of a largeenough computational domain. In the same situation,other schemes may be used as for example CrankNicolson schemes, RungeKutta methods in time</p><p>1 The article is published in the original.</p><p>it</p><p>x t,( ) x t,( ) V x t,( ) x t,( )+ +</p><p>+ f ( ) x t,( ) 0,=</p><p>x t,( ) N</p><p>]0 T[,,</p><p> x 0,( ) 0 x( ), x N</p><p>.=</p><p>and finite difference or finite element in space. Spectral techniques may be applied too (see [2] for moredetails about some of these techniques). Independently of the numerical discretization, one commonproblem arises when the solution does not remaininside the computational domain. This is for examplethe case for the defocusing nonlinear cubicSchrdinger equation, for linear Schrdinger equation with laser ionization of a onedimensional heliumatom [3], for strong field laser atom interaction [4]Then, it is wellknown that Dirichlet or periodicboundary conditions on the boundary of the computational domain are not adapted. Our goal here is not tofocus on all the physical situations which can arise butrather to propose some different ways of truncatingaccurately the computational domain for (1) andcompare them numerically.</p><p>Concerning truncation methods, the case of thefree Schrdinger equation is now mastered and manypossible solutions can be developed. We refer to [5] foran overview of the techniques. Considering now thelinear Schrdinger equation with potential requiresmore developments. For example, time dependent butspace independent potentials can be considered easilyby gauge change and be treated as the freespace case.The situation of a space variable potential is muchmore complex. In some exceptional cases, explicitexact boundary conditions may be written at the fictitious boundary. However, in most situations, approximate boundary conditions must be derived. Theseboundary conditions are usually called AbsorbingBoundary Conditions (ABCs) since they try to absorb</p><p>Numerical Solution of TimeDependent Nonlinear Schrdinger Equations Using Domain Truncation Techniques Coupled </p><p>with Relaxation Scheme1</p><p>X. Antoinea, *, C. Besseb, **, and P. Kleina, ***a Institut Elie Cartan Nancy, NancyUniversit, CNRS UMR 7502, INRIA CORIDA Team, </p><p>Boulevard des Aiguillettes B.P. 239, 54506 VandoeuvrelsNancy, Franceb Laboratoire Paul Painlev, Univ Lille Nord de France, CNRS UMR 8524, INRIA SIMPAF Team, Universit Lille 1 Sciences et Technologies, Cit Scientifique, 59655 Villeneuve dAscq Cedex, France</p><p>*email: Xavier.Antoine@iecn.unancy.fr**email: Christophe.Besse@math.univlillel.fr</p><p>***email: Pauline.Klein@iecn.unancy.frReceived October 29, 2010; in final form November 16, 2010; published online July 4, 2011</p><p>AbstractThe aim of this paper is to compare different ways for truncating unbounded domains for solvinggeneral nonlinear one and twodimensional Schrdinger equations. We propose to analyze ComplexAbsorbing Potentials, Perfectly Matched Layers and Absorbing Boundary Conditions. The time discretization is made by using a semiimplicit relaxation scheme which avoids any fixed point procedure. The spatialdiscretization involves finite element methods. We propose some numerical experiments to compare theapproaches.</p><p>DOI: 10.1134/S1054660X11150011</p><p>PHYSICS OF COLDTRAPPED ATOMS</p></li><li><p>1492</p><p>LASER PHYSICS Vol. 21 No. 8 2011</p><p>ANTOINE et al.</p><p>waves striking the nonphysical boundary. We refer to[5] for such examples in the onedimensional case. Inthe twodimensional case, only a few solutions can befound for the free and potential cases [5]. In the nonlinear case which is much more complicate, the ABCsat hand are often formally built from the linear casewith potential. To the best of our knowledge, only afew papers propose some ABCs [611]. For the twodimensional case, the only strategies for simulatingABCs have been proposed in [11, 12]. We propose andnumerically test in this paper some new ABCs for (1)for the one and twodimensional cases which arerelated to the ABCs for potentials developed in [13,14]. A very common method in physics is related toComplex Absorbing Potentials (CAPs) [15]. The ideawhich is physically natural consists in adding to thelinear Schrdinger equation a complex absorbingpotential to damp the incoming wave in a surroundinglayer. We try to extend this approach here to (1). As wewill see, this approach fails to work and generates largereflections. In particular, the choice of the absorbingpotential and its parameters is non trivial and extension to a nonlinear problem does not appear as natural. A related technique not analyzed here is themethod of Exterior Complex Scaling (see examples in[16, 17]). A closely powerful method introduced byBrenger [18] for electromagnetic waves is the Perfectly Matched Layer (PML) approach. The methodintroduces dissipation but inside the Laplacian termand not the potential term. We apply this techniquehere [19] to (1) to show its accuracy. It appears that theaccuracy that can be expected from the ABCs andPMLs approaches is about the same, generally showing an error of reflection of the order of 0.1% or less,and is therefore useful for practical computations. Foran easier implementation of all the truncation techniques, we use a semiimplicit relaxation scheme [20]which leads to a flexible implementation of themethod. In particular, it does not require any iterationlike in a fixed point or Newton procedure for the nonlinearity. Since the PML and ABCs approaches areaccurate for the onedimensional case, we next introduce their extension to twodimensional problems. We</p><p>detail the discretization issues and analyze their accuracy in the case of the propagation of a soliton in acubic media.</p><p>The plan of the paper is the following. In Section 2,we introduce the ABCs, CAPs and PMLs techniquesfor the onedimensional Eq. (1). We propose someschemes based on relaxation techniques coupled toFinite Element Methods (FEM). Finally, we numerically test and compare the different approaches. In thethird Section, we extend our methods and discretizations to the twodimensional nonlinear Schrdingerequation. Some numerical simulations confirm theaccuracy of the methods. Finally, Section 4 gives aconclusion. Let us note here that the codes corresponding to Sections 2 and 3 can be, downloadedfreely at http://microwave.math.cnrs.fr/code/index.html if the reader wants to know more about theimplementation issues of all these techniques.</p><p>2. ONEDIMENSIONAL NONLINEAR PROBLEMS</p><p>2.1. Absorbing Boundary Conditions (ABCs)</p><p>The first approach that we investigate concernsabsorption at the boundary. We consider the timedependent onedimensional nonlinear Schrdingerequation with a variable potential and a nonlinearterm</p><p>(2)</p><p>We assume here that := ]xl, xr[ represents a boundedcomputational domain of boundary := := {xl, xr}and set T := ]0; T[, t := ]0; T[ (see Fig. 1).</p><p>Furthermore, 0 is supposed to be a compactlysupported initial data inside . If the potential V andnonlinear interaction f are constant outside , werespectively call them localized potential and interaction. Then exact absorption at the boundary can beobtained. To write this boundary condition (also calledtransparent), let us assume that the potentials areequal to zero outside . Then, it is now standard thatthe boundary condition is given by</p><p>(3)</p><p>where n is the outwardly directed unit normal vector</p><p>to . The operator is the halforder derivativeoperator</p><p>(4)</p><p>it x t,( ) x2 x t,( ) V x t,( ) x t,( )+ +</p><p>+ f ( ) x t,( ) 0,=</p><p>x t,( ) T,</p><p> x 0,( ) 0 x( ), x .=</p><p>n x t,( ) ei/4 t</p><p>1/2 x t,( )+ 0, on T,=</p><p>t1/2</p><p>t1/2 x t,( ) := t</p><p>1</p><p> x,( )</p><p>t .d</p><p>0</p><p>t</p><p>n</p><p>xl xr</p><p>T</p><p>M(xr, t)</p><p>Tt</p><p>n</p><p>Fig. 1. Computational domain T and fictitious boundaryT = ]0; T[.</p></li><li><p>LASER PHYSICS Vol. 21 No. 8 2011</p><p>NUMERICAL SOLUTION OF TIMEDEPENDENT 1493</p><p>It can be proved that system (2)(3) is wellposed in amathematical setting and that we have the energybound</p><p>(5)</p><p>for any time t > 0, where ||u(, t)||0, designates the 2norm of over </p><p>which can also be interpreted as the probability offinding in and translates the absorbing property ofthe boundary condition. The boundary condition (3)is exact in the sense that there is no reflection back intothe computational domain. Mathematically, thisimplies that the solution to (2)(3) is strictly equal to</p><p>the restriction , solution to (1).</p><p>In the case of unbounded potential and interaction, then the situation is much more complex. Essentially, the possibility of writing the exact boundarycondition (3) is related to the fact that for localizedinteractions, the Laplace transform can be used in theleft and right exterior domains to write down theboundary condition through the Green function. Inthe case of nonlocal interactions, this no longer possible. Except in some special situations of potentials(e.g., linear potential) and nonlinearity (essentiallyintegrable systems like the cubic case), it is impossibleto get the exact expression of the absorption conditions. Here, we present without any mathematicaldetails which are too cumbersome the boundary conditions that can be set at the boundary. We refer to [13]for the mathematical details. Essentially, the derivation is based on highfrequency asymptotic expansions in the Laplace domain using the extended theoryof pseudodifferential operators. The resulting boundary conditions are no longer exactly nonreflecting.They are then called Absorbing (and not transparent)Boundary Conditions (ABCs), and we need to precisethe order related to their asymptotics with the aim ofmeasuring the a priori accuracy of the boundary condition. The ABC of order two is given by</p><p>(6)</p><p>on T. The squareroot operator of it + V + f(||)means that we consider the spectral squarerootdecomposition of this operator. The resulting operatoris nonlocal but will be localized later for the numericalpurpose through Pad approximants. HigherorderABCs can be derived [13] but will not be tested here.We cannot expect that an ABC works well for anypotential. In practical computations and for f = 0, onephysically admissible assumption is that the potentialis repulsive which means that V : + is smooth andthat we have xxV(x, t) > 0, (x, t) l, r +, where</p><p> t,( ) 0 , 0 ( ) 0 , ,</p><p> t,( ) 0 , x t,( )2</p><p>xd</p><p>1/2</p><p>=</p><p> T</p><p>n i it V f ( )+ + 0,=</p><p>l := ]; xl] and r := [xr; [. An example is V(x, t) =2|x|a, with 0 < a 2. The assumption on the nonlinearity is not clear most particularly when a potential isadded. The only intuitive assumption is that thesolution is outgoing to the bounded domain and thatno nonlinear or potential effect makes it reflectingback into , which is a priori difficult to check becausemathematically hard to write. Finally, we can prove[13] that (5) still holds for the secondorder ABC (6)and for a time independent potential V(x) which translates the absorbing property of the boundary condition.</p><p>2.2. Complex Absorbing Potential (CAP), Exterior Complex Scaling and Perfectly Matched Layers (PMLs)</p><p>Another useful and widely studied approach forcomputing solutions to timedependent Schrdingerequations with a potential term by using an absorbingdomain is first the technique of Complex AbsorbingPotential (CAP). Essentially, the idea consists in introducing a complex potential in the exterior domain toabsorb the travelling wave. Mathematically, this consists in adding a spatial potential iW in some exteriorlayers l = ]xlp, xl[ and r = ]xr, xrp[ (see Fig. 2). Ofcourse, to coincide with the solution to (2), W isrequired to be zero in and with a positive real part inthe layers to damp the incoming wavefield. Anotherway to analyze this approach is called the ExteriorComplex Scaling approach which consists in interpreting the introduction of the complex potential asthe complex scaling: x xei, where is a rotationangle which must be correctly chosen. Extensionincludes the Smooth Exterior Scaling approach [16,17]. From the numerical point of view, the CAPapproach is direct to code since we have to solve</p><p>(7)</p><p>it x t,( ) x2 x t,( ) V x t,( ) x t,( )+ +</p><p> iW x( ) x t,( ) f ( ) x t,( )+ 0,=</p><p>x t,( ) Text</p><p>,</p><p> x 0,( ) 0 x( ), x ext</p><p>,=</p><p>xlp xl xr xrp</p><p>l r</p><p>Tt</p><p>Fig. 2. Computational domain for the CAP and PMLapproaches.</p></li><li><p>1494</p><p>LASER PHYSICS Vol. 21 No. 8 2011</p><p>ANTOINE et al.</p><p>in the extended domain ext := l r = ]xlp, xrp[with boundary ext = {xlp, xrp}. Here, according to [16,21], we consider the quadratic profile</p><p>(8)</p><p>for a real positive value of W0. The thickness of thelayer is = |xrp xr| = |xlp xl|. Other choices includeexponential type absorbing functions [16]. At theboundary points xlp and xrp, a boundary condition mustbe imposed. Here, we consider the classical homogeneous Dirichlet boundary conditions (x, t) = 0 at xlpand xrp. However, a suitable extension does not seemdirect for nonlinear problems as we will see later.</p><p>We concentrate now on another closely relatedapproach called the Perfectly Matched Layers (PMLs)method which was introduced by Brenger [18] forMaxwell equations. The idea consists in introducing acomplexification of the derivative operator throughdamping in the extended domains l and r. In thecase of the nonlinear Schrdinger equation, this canbe written down as</p><p>(9)</p><p>in ext (Fig. 2). Function S is given by S(x) := 1 +R(x). The layer parameters R and must be chosencarefully. Optimization techniques and adaptive discretizations can be developed [5]. Here, we will use theparameter values derived in [19], i.e., R = ei/4 and isthe quadratic function</p><p>(10)</p><p>The distance := |xrp xr| = |xl xlp| (that we takeequal on both sides here for simplicity) corresponds tothe thickness of the left and right absorption regions ofthe computational domain. Again, we fix the homogeneous Dirichlet boundary conditions: (x, t) = 0 at xlpand xrp. It is interesting to note the close form of CAPand PML approaches even if they lead to differentequations to solve.</p><p>Unlike the ABCs, both CAP, ECS, and PMLs mustbe adapted and optimized according to each situation.They have the advantage of being easy to code but atthe price of an extended domain of computation extwhere the potential V must be known. This is not</p><p>W x( )</p><p>W02</p><p>x xl( )2, xlp x xl,</p></li></ul>