Research Article | Open Access
Chiara Zanini, Fabio Zanolin, "Complex Dynamics in One-Dimensional Nonlinear Schrödinger Equations with Stepwise Potential", Complexity, vol. 2018, Article ID 2101482, 17 pages, 2018. https://doi.org/10.1155/2018/2101482
Complex Dynamics in One-Dimensional Nonlinear Schrödinger Equations with Stepwise Potential
We prove the existence and multiplicity of periodic solutions as well as solutions presenting a complex behavior for the one-dimensional nonlinear Schrödinger equation where the potential approximates a two-step function. The term generalizes the typical -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 although small, can be explicitly estimated.
In a recent paper  we have proved the existence of chaotic dynamics associated with a class of second order nonlinear ODEs of Schrödinger type of the form for a positive periodic coefficient. The study of such equation was motivated by previous works on some models of Bose-Einstein condensates considered in [2–4].
A more classical form of Nonlinear Schrödinger Equation (NLSE from now on) which has been studied by many authors is given by where denotes the Planck constant, is the imaginary unit, is a positive constant, is the potential, and The search of stationary waves, namely, solutions of the form , where and is a real valued function, leads to the study of This latter equation, which is usually written as or equivalently (by a standard rescaling) as has motivated a great deal of research from different points of view (see, for instance, [5–11] just to quote a few classical contributions among a very large and constantly increasing literature on the subject). The case of a periodic potential has been considered as well (see [12, 13]).
In various articles, the hypothesis has been assumed. Setting , this is equivalent to consider the equationwith a positive weight function.
Looking at (6) in one-dimension, we can interpret it as a slowly varying perturbation of the planar Hamiltonian systemwhich presents a hyperbolic equilibrium point (the origin which is a saddle point) with a homoclinic orbit enclosing a region that contains another equilibrium point which is a (local) center (see Figure 1). Similar phase-portraits are common in many different situations and it is known that their perturbations can produce chaotic-like dynamics (see, for instance, [14–18]).
Analogous equations appear in some mathematical models of nonlinear optics derived from Maxwell’s equations . For instance, in  the study of the propagation of electromagnetic waves in layered media leads to the scalar equationwhere the dielectric function takes into account the presence of layers with different refractive indexes. A possible choice of for three layers (one “internal” and two “external”) is given by For some other typical forms of , see  and the references therein. Also this class of equations has been widely investigated in the last decades [20–23].
Finally, we briefly mention another area of research where similar equations arise, that is in the context of wave propagation or stationary solutions for bistable reaction diffusion equations in excitable media (see, for instance, [24–27] and the references therein). In the above quoted papers, a typical one-dimensional model equation takes the form where the function is defined piecewise as follows: (see [26, 27] for different choices of ). The kinetic term is usually taken to be a Nagumo type cubic function (with ) or a McKean’s piecewise linear reaction term . Other possible variants concern the equation with a stepwise function . The case of periodicity in the -variable has been also considered. With this respect we mention a paper of Keener , dealing with the equation where is the so-called -periodic “sawtooth function”. Analogous investigations have been addressed also in [30, 31].
In the present paper we restrict ourselves to the one-dimensional case and we study a second order nonlinear equation which is related to the models considered above. More in detail, we deal with a class of equations of the formwhere, for notational convenience, we consider the independent variable (which usually refers in the above quoted models as a space variable) as a time variable. Such a convention is also motivated by the dynamical systems approach which is followed in the present paper. The choice of introducing two weight functions (that is, for the linear part of the equation and for the nonlinear part) is useful in view of dealing with the more general Schrödinger equation (previously considered in ).
The nonlinear term in (14) is assumed to be a continuously differentiable function and is chosen in order to include, as a particular case, the polynomial nonlinearities which usually appear in the context of the NLSEs (see Section 2 for the precise assumptions on ). The main hypothesis on the coefficients and , which are supposed to be nonnegative, is that they are close in the -norm to stepwise functions. Such a particular choice for the shape of the coefficients is mainly motivated by mathematical convenience, as it permits to develop the proofs in a simpler and more transparent way and thus to avoid more complicated technicalities. However, it is interesting to observe that second order equations or, more generally, first order planar systems with piecewise constant coefficients naturally appear in several applications, such as the theory of switching control , electric or mechanical systems [34, 35], periodically forced Nagumo equations , and biological models subject to seasonal dynamics [37–39], as well as in the context of NLSEs arising in nonlinear optics  and in mathematical modelling of structures like crystals or switches in optical fibers . In this connection and as already observed at the beginning of the Introduction, stepwise periodic coefficients have been recently considered also in some (Gross-Pitaevskii) equations describing the phenomenon of Bose-Einstein condensation where the existence of complex dynamics for and was proved in  and , respectively. Periodically forced second order nonlinear equations with stepwise coefficients are widely analyzed also in connection with Littlewood’s example of unbounded solutions to Duffing equations and its generalizations [42–45]. Finally, we observe that variants of these equations with a stepwise weight function have been considered with respect to the search of multiple “large” solutions, namely, solutions presenting a blow-up phenomenon at the boundary of a given interval (see [46, 47]).
The main part of the paper is devoted to the study of the periodic boundary value problem associated with (14). In doing so, we prove the presence of infinitely many subharmonic solutions and also the existence of solutions with certain oscillatory properties which can reproduce any prescribed sequence of coin-tossing type  (see Definition 1 in Section 2). Such chaotic-like solutions are obtained by an application of the theory of topological horseshoes [49, 50], in a variant developed in [51, 52]. We will also discuss how the arguments of the proofs can be modified in order to deal with the Neumann and the Dirichlet boundary value problems. In this latter context, the recent years have witnessed a growing interest toward the search of existence and multiplicity results of solutions of on a bounded domain with interior and/or boundary peaks [53–56]. With this respect, we stress the fact that our multiplicity results appear to be of completely different nature; they are typically one-dimensional, even if they could be applied to PDEs on thin annular domains of
For simplicity in the exposition, we will focus our presentation mainly to the study of positive solutions. We shall explain how to obtain sign changing solutions with prescribed nodal properties with some illustrative remarks at the end of the article.
2. Setting of the Problem and Main Results
We consider the second order nonlinear equation (of Schrödinger type)where is a continuously differentiable function of the formwith satisfying As a consequence of it follows that , for and Since we are looking for positive solutions, the actual behavior of for will not affect our result. For simplicity, we suppose that is odd (that is, is even). We assume such a symmetry condition also in order to cover the classical example with All the results of the present paper could be proved for a locally Lipschitz continuous function satisfying and with strictly increasing on (and strictly decreasing on ). We prefer to consider the smooth case for simplicity in the presentation.
For the potential we suppose that is a -periodic stepwise function (for some ) of the formwith Writing (17) as the equivalent first order system in the phase-plane, we can describe the presence of a piecewise constant -periodic coefficient as follows: the trajectories are governed by the autonomous system in the time interval At the time we have a switching to system which, in turns, rules the motions for a time interval of length All this switching behavior is then repeated in a -periodic fashion.
Recall that, given a first order differential system , its Poincaré map, on a time interval , is the function which maps any initial point to , where , is the solution of the differential system satisfying the initial condition In our setting, it is straightforward to check that the Poincaré map on for system , that we denote by , can be decomposed aswhere is the Poincaré map associated with the autonomous system along the time interval Notice that, due to the autonomous nature of the subsystem, the map coincides with the Poincaré map of on The assumptions on guarantee the global existence of the solutions for all the Cauchy problems and therefore is a global homeomorphism of the plane.
Our goal is to prove the existence of periodic solutions (harmonic and subharmonic) for (17). Following a classical procedure , this will be achieved by looking for the fixed points of and its iterates. In our approach we apply some recent results on planar maps which provide not only the existence of fixed points and periodic points, but also the fact that the associated discrete dynamical system is “chaotic”. In the literature one can find several different methods which guarantee the presence of chaos for planar maps or, more generally, for homeomorphisms (or diffeomorphisms) in finite dimensional spaces. Moreover, different definitions of chaotic dynamics have been proposed by various authors. For the reader convenience, we recall now the concept of chaos that we are going to consider. Although the main definitions and the abstract setting can be presented in the framework of metric spaces, we confine ourselves to the case of homeomorphisms of the plane, which is the situation encountered by dealing with the Poincaré map associated with a planar system.
Definition 1. Let be a homeomorphism and let be a nonempty set. Assume also that is an integer. We say that induces chaotic dynamics on symbols in the set if there exist nonempty pairwise disjoint compact sets such that, for each two-sided sequence of symbols there exists a corresponding sequence withand, whenever is a periodic sequence (that is, ) for some , there exists a periodic sequence satisfying (24).
Note that, as a particular consequence of this definition, we have that for each there is at least one fixed point of in Since is a homeomorphism from onto its image, it follows also that there exists a nonempty compact set which is invariant for (i.e., ) and such that is semiconjugate to the two-sided Bernoulli shift on symbols according to the commutative diagram where is a continuous and surjective function. Moreover, as a consequence of Definition 1 we can take such that it contains as a dense subset the periodic points of and such that the counterimage (by the semiconjugacy ) of any periodic sequence in contains a periodic point of (see  for the details). As usual, in , the set of two-sided sequence of symbols, we take its standard metric  for which turns out to be a compact set with the product topology.
We observe that Definition 1 is related to the concept of chaos in the sense of coin-tossing  and it also implies the presence of chaotic dynamics according to Block and Coppel [59, 60], as well as a positive topological entropy for the map Similar examples of complex dynamics for the Poincaré map associated with differential systems have been discussed, e.g., in [61–65], using different methods. See also [1, 31, 66] for recent contributions in this direction.
Now we are in position to state our main result for (17).
Theorem 2. Let be a -function of the form (18), with satisfying Let be a -periodic stepwise function as in (19), such that Then, there exist a compact set and, for every integer , two positive constants and such that, if and , the Poincaré map for system on (with ) induces chaotic dynamics on symbols in the set Moreover, for the corresponding solutions of we have for every
The constants and can be explicitly determined in terms of and some Abelian integrals depending by , and as in formula (41). The set is explicitly exhibited in the course of the proof. Indeed, we have with defined in (54).
The proof is based on a topological technique, named stretching along the paths (SAP), which is a variant of the classical Smale’s horseshoe geometry (see ). Our approach is closely related to the theory of topological horseshoes of Kennedy and Yorke  as well as to the concept of covering relations introduced by Zgliczyński in . The general theory concerning the “SAP method” has been already exposed in some previous papers (see, for instance,  and the references therein). In order to make our paper self-contained, we recall the main notation and the results which are needed for the proof of Theorem 2.
By path we mean a continuous mapping and we set Without loss of generality we will usually take By a sub-path of we mean the restriction of to a compact subinterval of its domain. An arc is the homeomorphic image of the compact interval We define an oriented rectangle in as a pair where is homeomorphic to the unit square (we usually refer to as a topological rectangle) and is the disjoint union of two disjoint compact arcs (which are called the components or sides of ). We also denote by the closure of which is the union of two compact arcs and The subscripts stand, conventionally, for left, right, up, and down.
Suppose that is a planar homeomorphism of onto its image. Let and be oriented rectangles.
Definition 3. Let be a compact set. We say that stretches to along the paths and writeif for every path such that and (or and ), there exists a subinterval such that and, moreover, and belong to different components of In the special case in which , we simply write
Theorem 4. Let and be continuous maps and let , be oriented rectangles. Suppose that the following conditions are satisfied: (i)there exist pairwise disjoint compact sets such that (ii)there is a compact set such that Then the map induces chaotic dynamics on symbols in the set Moreover, for each sequence of symbols , there exists a compact connected set with and , such that, for every , there exists a sequence with and
A dual version of Theorem 4 holds if we interchange the hypotheses on and , namely, if we suppose that(i)there is a compact set such that (ii)there exist pairwise disjoint compact sets such that The corresponding conclusion has to be modified accordingly.
The application of Theorem 4 to Theorem 2 is possible thanks to a linked twist maps geometry which appears from the phase-plane analysis of the systems and . The theory of “linked twist maps” regards the case in which a map can be expressed as a composition of two twist maps acting on two annuli crossing each other (see [70–74] for an introduction of the topic and for interesting applications to chaotic mixing). The main argument in the proof of Theorem 2 relies on the construction of two annular regions which cross each other in a suitable manner (see Figures 2 and 3) and such that acts on them as a linked twist map.
3. Technical Estimates and Proofs
As already observed in Section 2, the motion associated with system is given by a switching in a -periodic fashion between the orbits of the two autonomous systems and . Such systems have the same qualitative structure and differ only for the value of the -coefficient. For this reason, we first perform a phase-plane analysis of the planar systemfor a given parameter. System (36) is a conservative one with associated energy where As a consequence of , there is a unique solution of the equation The corresponding equilibrium point is a center surrounded by a trajectory which represents the homoclinic solution at zero.
The origin and the homoclinic trajectory determine the part of the level line at energy zero contained in the half-plane We denote by the intersection point of the homoclinic orbit with the (positive) -axis. Notice that is the unique (positive) solution of the equation As a consequence of , both and , thought as functions of the parameter , are strictly monotone increasing. Observe also that, for every constant with the level line is a closed curve which is a (positive) periodic orbit of (36). The period of can be computed by the quadrature formulawhere and are the solutions of the equation , with Moreover, we have that Without further assumptions on (or, equivalently, on ) we cannot guarantee the monotonicity of the time-mapping function Sufficient conditions ensuring that is strictly increasing can be found in literature. For instance, according to , the convexity of the auxiliary function guarantees that is increasing.
Example 5. Consider the typical nonlinear term , with In this case, condition holds for and Moreover, we find that In order to prove the monotonicity of the time-map, via the Chicone theorem in  we have to study the sign of auxiliary function on the open interval After performing the required computations and using the change of variable , one can see that the sign of for is the same of the expression for
For instance, if , it is easy to check that the above expression is strictly positive for (which corresponds to ) and therefore the time-mapping function is strictly increasing on The case is the model situation that we have chosen in all our illustrative examples of Figures 1–8.
The monotonicity of the period map still holds for an arbitrary The proof in this case is a more complicated task (see [76, 77]).
Until now we have considered some general properties of the solutions of system (36). As a next step, in order to investigate the dynamics associate to system for a -periodic potential defined as in (19), we need to make a comparison between the phase–portraits associated with the autonomous systems and . Keeping the notation just introduced, we set and denote by the associated energy, for Accordingly, we indicate by and the corresponding equilibrium points and the intersection points of the homoclinic orbits with the positive -axis. Moreover, denotes the fundamental period of the closed orbits defined in (41) for the potential functions
Just to fix a case of study, we suppose thatThe case when can be treated in a similar manner. Observe that from (47) it follows that the homoclinic orbit of system is contained in the part of the right half-plane bounded by the homoclinic orbit of system .
Now we are in position to introduce the rectangular regions (topological rectangles) and in order to apply Theorem 4. As a first step, we chose a closed trajectory of system which intersects in two distinct points the homoclinic trajectory of system . From an analytic point of view, this corresponds to solving the system for a suitable , with It is clear that this system has a pair of solutions with if and only if The latter condition holds if and only if We conclude that the desired geometry can be produced if and only if we choose an energy level for such thatFrom now on, we suppose to have fixed a constant satisfying (50). Let us call such a constant and denote by the intersection of the closed orbit with the positive -axis which is closer to the origin. Next, we choose with and consider the level line of system passing through This is the closed orbit , for For we further require thatWe notice that it is always possible to find an open interval such that for each the condition (51) holds. This follows from the fact that as In the special case in which the time-mapping is strictly monotone increasing, one can take an arbitrary Observe that, by construction, we also haveAs a last step, we fix a constant such thatBy this latter choice, the corresponding (periodic) trajectory of system intersects both and in the region bounded by the homoclinic orbit Figure 2 illustrates the geometric construction performed above.
Next we defineand its specular image with respect to the -axis, namely,For these regions we select an orientation as follows: (see Figure 3).
Now we describe the behavior of the points in the regions and under the action of the Poincaré maps and , respectively.
Suppose that For each point the solution of is indeed a solution of the autonomous conservative system and therefore, , for all The orbit for is closed trajectory surrounding the equilibrium point of . Consistently with the previous notation, the period of the points in is denoted by Since all the points of (for ) move in the clockwise sense along the orbit under the action of the dynamical system associated with , it will be convenient to introduce a system of polar coordinates with center at and take the clockwise orientation as a positive orientation for the angles starting from the positive half-line In this manner, we can associate an angular coordinate