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

Zanini, Chiara; Zanolin, Fabio

doi:https://doi.org/10.1155/2018/2101482

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Research Article | Open Access

Volume 2018 | Article ID 2101482 | https://doi.org/10.1155/2018/2101482

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

Chiara Zanini¹and Fabio Zanolin²

Academic Editor: Eulalia Martínez

Received31 Jul 2018

Accepted30 Oct 2018

Published02 Dec 2018

Abstract

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.

1. Introduction

In a recent paper [1] 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 [19]. For instance, in [20] 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 [20] 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 [28]. Other possible variants concern the equation with a stepwise function [24]. The case of periodicity in the -variable has been also considered. With this respect we mention a paper of Keener [29], 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 [32]).

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 [33], electric or mechanical systems [34, 35], periodically forced Nagumo equations [36], and biological models subject to seasonal dynamics [37–39], as well as in the context of NLSEs arising in nonlinear optics [23] and in mathematical modelling of structures like crystals or switches in optical fibers [40]. 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 [1] and [41], 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 [48] (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 [57], 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 [58] for the details). As usual, in , the set of two-sided sequence of symbols, we take its standard metric [18] 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 [48] 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 [67]). Our approach is closely related to the theory of topological horseshoes of Kennedy and Yorke [50] as well as to the concept of covering relations introduced by Zgliczyński in [68]. The general theory concerning the “SAP method” has been already exposed in some previous papers (see, for instance, [58] 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

The next result, taken from [69, Theorem 2.1], provides the existence of periodic points and chaotic-like dynamics according to Definition 1, when admits a splitting as in (21).

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 [75], 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 [75] 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]).

Figure 7

The present figure shows some possible rectangular regions which can be considered for the application of Theorem 4. The sets and are the same as in Figure 3 and (as we have already proved in Section 3) they can be used to provide a complex dynamics on positive solutions. If we choose, for instance, the sets and we can prove the presence of a complex dynamics generated by solutions which are negative on the time interval and oscillate in the phase-plane around the point , and then, in the time interval oscillate a certain number of times around the origin. More in detail, given any positive integer , we can produce solutions of (17) which have precisely simple zeros in the interval , provided that is sufficiently large. A lower estimate for can be easily determined by the knowledge of the period of the closed trajectory which bounds “externally” and Similar remarks can be made by selecting other pairs of topological rectangles among those put in evidence with a color. As in the preceding figures, we have considered and , For graphical reasons a slightly different - and -scaling has been used.

Figure 8

The present figure suggests a possible argument to prove multiplicity of sign changing solutions to the Neumann problem (94). We start with system by considering initial points with (recall that is the positive center of , while is the intersection point of the homoclinic orbit with the positive -axis). We parameterize such initial points as an arc If is sufficiently large (with a lower bound which can be easily estimated from the equation) we find that is a spiral-like curve winding a certain number of times around and with an end point on in the fourth quadrant near the origin. If we fix a (small) positive constant and denote by the region between the homoclinic trajectories of and the level line , we observe that the points of remain on under the action of , while the points of run around the origin along the periodic orbit and will perform a certain number of revolutions if is sufficiently large. More precisely, if we denote by the period of and suppose that , we can find solutions of (94) having precisely -zeros in the interval , for every As in the preceding figures, we have considered and , For graphical reasons a slightly different - and -scaling has been used.

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 to any solution for any initial point and

In such a situation, we find that Since is a solution of and, moreover, with , for all , we have that for all In fact, by (18) and , we obtain We have thus proved that the angle is a strictly increasing function of the time variable. Hence, for any positive integer we conclude that(of course, the above relation holds provided that and ).

After these preliminary observations, we are now in position to prove the validity of the first condition of Theorem 4 provided that is large enough.

Let us fix an integer and set

Given as above, we also fix and define The position (61) and the choice of imply that and hence

Now we consider the motion associated with for

First of all, we note that On the other hand, since , we know that Similarly, since , we know that We thus conclude that the range of the angular function covers the interval This interval contains closed disjoint intervals of the form , for Hence, if for each nonnegative integer , we define we obtain nonempty and pairwise disjoint compact subsets of

Let be a (continuous) path such that and and consider the path Passing to the polar coordinates we have that and, moreover, Hence, for every there exists an interval such that for all , and, moreover, Using the fact that for every (by the invariance of the annular region bounded by and with respect to system ), we conclude that the set crosses the region Hence, by the continuity of the map , we can find a subinterval such that and, moreover, We also know that According to Definition 3 we conclude that and thus the first condition in Theorem 4 is fulfilled (see Figure 4 for a graphical description of this step in the proof).

After the time we switch to equation . As previously observed, due to the autonomous nature of the system, the study of the solutions in the time interval is equivalent to the study for Now we setand fix

Arguing in a similar manner as before, we introduce a system of polar coordinates with center at the point 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 to any solution with initial point and Repeating the same argument as above, one can see that the angle is a strictly increasing function of the time variable and, moreover, for any positive integer (of course, the above relation holds provided that and for ).

Let be a (continuous) path such that and and consider the path Passing to the polar coordinates we have that and, moreover, Indeed, the point lies on the homoclinic trajectory and therefore remains in the fourth quadrant. On the other hand, the point lies on the periodic orbit and, since (which is the period of ), it follows that makes at least one turn around the center during the time interval

Hence, there exists an interval such that Using the fact that for every (by the invariance of the region bounded by and with respect to system ), we conclude that the set crosses the region Hence, by the continuity of the map , we can find a subinterval such that and, moreover, According to Definition 3 we conclude that and thus the second condition in Theorem 4 is fulfilled for (see Figure 5 for a graphical description of this last step in the proof).

Then, Theorem 4 applies and the proof of Theorem 2 is complete (with respect to the “chaotic part”), providing the existence of complex dynamics on symbols for the Poincaré map (of system ) on the compact set , according to Definition 1.

Regarding the fact that all the solutions we find via Theorem 4 are always positive in the -variable, we need only to observe that, in the first step of the proof concerning the property we have also found that for all when Moreover, by construction, it follows that for all when This concludes the proof.

Remark 6. From the proof of Theorem 2 (and the geometric construction involving and ) it follows that our result is stable with respect to small perturbations of the coefficients. Indeed, from the fundamental theory of ODEs we know that, for any fixed time interval , a “small” perturbation of the coefficients in the -norm on produces a “small” perturbation on the Poincaré map (in the sense of the continuous dependence of the solutions from the data). With this respect, the following result holds.

Theorem 7. Let and be as in Theorem 2, with Given and and according to Theorem 2, let us fix and and let Then, there is such that equationhas infinitely many periodic (subharmonic) solutions as well as solutions presenting a complex dynamics, with , provided that is a (measurable) -periodic function satisfying

In Theorem 7 the solutions are considered in the Carathéodory sense [78] (when is only measurable). On the other hand, we can also take an arbitrarily smooth function which approximates the step function , provided that (85) is satisfied. The stability of Theorem 2 with respect to small perturbations of the Poincaré map is not confined to the coefficient of the nonlinearity. For instance, we can obtain the same result for a perturbed equation of the form provided that , and are sufficiently small and is a -periodic function with small, too. Finally, we stress that and can be easily computed and are not necessarily “large” (see, for instance, the examples considered in Figures 4-5).

In the previous sections we have discussed the presence of chaotic-like dynamics (including the existence of infinitely many subharmonic solutions) for (17), by assuming that the period is large enough. From this point of view, our results can be interpreted in line with analogous theorem on Hamiltonian systems with slowly varying coefficients (see [14, 16]). On the other hand, via a simple change of variable, we can apply our results to a typical Schrödinger equation of the formfor a periodic stepwise function of fixed period , of the formwith and As before, we also set Writing (87) as an equivalent first order systemin the phase-plane, we obtain the following result.

Theorem 8. Let be a -function of the form (18), with satisfying . Let be a -periodic stepwise function as in (88). Then, there exist a compact set and, for every integer , a constant such that, for every the Poincaré map associated with (89) on induces chaotic dynamics on symbols in the set Moreover, for the corresponding solutions of (89) we have for every

The constant can be explicitly determined in terms of and

Proof. As in the proof of Theorem 2, we suppose (the treatment of the other situation is completely similar and thus is omitted).
The change of variables and , , transforms system (89) to the equivalent first order systemfor a stepwise periodic function of period By (88) and setting and , it follows that on and on with Notice that if we denote by the Poincaré map associated with system (91) on , then it follows that (one can easily check this fact, because ).
Now, for (91) we can apply Theorem 2. In particular, through the proof of that result in Section 3, we find a compact region and two constants and such that the chaotic dynamics for (according to Definition 1) is ensured provided that and The lower estimates on and transfer to an upper bound for , so that if then the chaotic dynamics for on the same set is guaranteed. In particular, recalling the definition of in (61) and in (75), we derive a precise estimate to For instance, if we take , and (like in the example of Figure 2) then we know that and therefore, if we choose , the conclusion of Theorem 8 holds.

We observe that, similarly to Remark 6, the stability of the result with respect to small perturbations of the coefficients is guaranteed, too.

5. Boundary Value Problems on Finite Intervals: Positive Solutions

We start this section by briefly describing how the method applied in the proof of Theorem 2 can be adapted to obtain multiplicity results for the Neumann boundary value problem(see (87)). Our aim is to find multiple positive solutions for (93), where the number of the solutions becomes arbitrarily large as Actually, such kind of result can be obtained by a variant of Theorem 2, via a change of variables as in the proof of Theorem 8. With this respect, we study the equivalent problem(see (17)) and look for multiplicity results, where the number of the solutions increases as the time-interval length grows.

As in Section 2 we suppose that is a stepwise function of the form (19) for (the case in which can be treated in a similar manner). The assumptions on are the same as in Section 2.

Following the argument of the proof of Theorem 2 we consider the Poincaré map associated with the planar system as well as its components associated with systems The difference with respect to the proof of Theorem 2, consists into the fact that this time we look for initial points , with such that the second component of vanishes. In other words, we apply a shooting method, looking for a solution which departs at time from a point on the positive -axis and hits again the (positive) -axis at the time , with for all

We repeat step by step (keeping the same notation) the geometrical construction in the proof of Theorem 2. In particular, as before, we choose the closed orbits and of system and of system in order to produce the regions and in the phase-plane. Recall also that (see (51)). We have already indicated by the intersection point of with the positive -axis which is closer to the origin. We introduce now also the second intersection point of with the positive -axis, which will be denoted by Clearly we have (compared also with Figure 2).

We produce the solutions of (94) by shooting from initial points with Just to fix one case for our discussion, let us assume that the former of the above alternative occurs. More precisely, we shall develop the following argument that we first briefly describe in an heuristic manner for the reader’s convenience. We start from an initial point in the segment of the phase-plane and apply the Poincaré map for a time sufficiently long so that the image of such segment crosses at least times the set Then we switch to the Poincaré map and apply it for a time sufficiently long so that the above arcs crossing will be transformed (by ) to some curves winding around the point and crossing at least times the -axis. Putting all together these facts we conclude that there are at least solutions to (94). We present now the technical justification, by slightly modifying the argument of the proof of Theorem 2.

As before, we represent the solutions of in polar coordinates with respect to the center , using a clockwise orientation for the angular coordinate. In this case, instead of taking , we have and therefore Accordingly, we replace the path with the map which parameterizes the segment , namely we take As a consequence, for we have that For an integer , we define as in (61) and fix Then, repeating the same argument of the proof of Theorem 2, we define the integers and and obtain and (which, in this case, is an obvious choice). As a consequence we find pairwise disjoint intervals for such that and, moreover, After the time we switch to equation . As remarked before, the study of the solutions in the time interval is equivalent to the study for , where, for the moment, is not yet fixed.

We introduce another system of polar coordinates with center at the point 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 to any solution with initial point and In order to have the condition satisfied, we look for solutions (in the phase-plane) such that In terms of these new angular coordinates, this corresponds to the condition for some positive integer

We have already proved that the angle is a strictly increasing function of the time variable. Moreover, as a consequence of (60) and the symmetry of the orbits with respect to the -axis, we find that, for any positive integer and for any , (of course, the above relation holds provided that and for ).

Now we are ready to introduce the constant which will be defined as follows. Let us fix a positive integer and setand fix

The point lies on the level line of the homoclinic trajectory and therefore, On the other hand, lies on the level line of the closed trajectory of period and therefore we have Hence, we can conclude that the arc , which connect in the opposite sides and , is mapped by to an arc which crosses at least times the -axis. In other words, for each integer there exists one point such that The corresponding solution has precisely -zeros of the derivative in the interval

Finally, recalling that the points are images through of corresponding points in the interval , allows us to conclude with the following claim.

Theorem 9. Let be a -function of the form (18), with satisfying . Let be a stepwise function as in (19), and such that Then, for every pair of integers with and , there exist two positive constants and such that, if and , the problem (94) has at least positive solutions on (with ). More precisely, for each there are at least solutions of (94) with and such that has precisely -zeros in

The solutions that we have produced are only those obtained by shooting from , achieving the set at the time and coming back at the -axis at the time With obvious changes in the argument, we could produce solutions which are in at the time Moreover, we could also start from the segment and reach the region (or, respectively, ) at the time Therefore, with the same technique, we can produce four different classes of solutions. Finally, we can consider (by suitably modifying the same approach) also the case

A variant of Theorem 9 for problem (93) with sufficiently small can be also obtained via the same change of variables as in the proof of Theorem 8.

In order to obtain positive solutions for the Dirichlet (two-point) boundary value problemor (after a suitable rescaling)a natural implementation of the shooting approach considered for the periodic and the Neumann problem consists into shooting from the positive -axis of the phase-plane. More precisely, we consider the initial condition for system and look for a suitable value of the parameter such that satisfies In such a situation, for a weight function with only two steps, we do not obtain multiplicity results (in general). However, one could easily provide a mechanism for multiple solutions by taking a weight function with three steps defined as follows:with and positive constants with In fact, under these assumptions, we can find solutions of (108) which are positive on and oscillate a certain (prescribed) number of times in the phase-plane around the point , during the time interval The number of such solutions can be larger than a preassigned number , provided that for a sufficiently large time (depending on ). An analogous conclusion can be derived for problem (107).

As a consequence of the technique we can also prove the stability of the multiplicity results with respect to small perturbations of the coefficients.

6. Final Remarks

6.1. Sign Changing Solutions

In the previous sections we have focused our attention only to the search of positive solutions. We stress the fact that the same approach works well in order to produce nodal solutions, that is solutions with a prescribed number of zeros in a given interval. For the sake of simplicity in the exposition and consistently with the classical case , we confine ourselves to the case of an odd nonlinear term in the equation. If is not odd, we can still adapt the same argument with minor efforts.

The assumption (18), with an even function satisfying , implies that the phase–portrait of (36) is similar to that of Figure 1 with a mirror symmetry with respect to the -axis (see Figure 6).

Now we consider again (17) with a periodic stepwise weight function as in (19), with Actually, in order to enter in the same situation analyzed in Section 3, we assumeAs previously observed, the dynamics associated with system can be described as that of a couple of switching systems of the form and . This time we are interested also in sign changing solutions and therefore we exploit the properties of the trajectories in the negative half-plane, too. If we overlap the energy level lines of the two autonomous systems we have the possibility of applying our abstract result (namely, Theorem 4) to different choices of rectangular regions and which are determined through the intersections of the level lines of the two systems. An illustrative example of possible choices of topological rectangles is given in Figure 7 where we have put in evidence some sets which are suitable for the application of Theorem 4.

As a consequence, we can find different rectangular regions where chaotic dynamics occur. The corresponding solutions can have different qualitative behaviors on the intervals and/or , depending on the regions where we apply Theorem 4. For instance, we can find solutions with prescribed nodal properties on the intervals and , as well as solutions which are positive (or negative) on and have a prescribed number of zeros on

For the search of sign changing solutions to the Neumann problems (93) or (94), we suppose (114) and apply a shooting method by selecting initial points on the -axis of the phase-plane. A possible argument is illustrated in Figure 8.

Finally, for the Dirichlet problem (108), where again we suppose that is step function like in (19), we can prove the existence of multiple sign changing solutions by shooting from the -axis in the phase-plane and suitably adapting the argument described above for the Neumann problem.

6.2. Remarks on the Weight Functions

Throughout the paper we have focused our analysis to an equation of the form (17), namely We point out that the same technique works for the case of equationwith a positive stepwise function. This follows from the elementary observation that the equation , with constant coefficients , is equivalent to , for Hence, if we take as a step function on an interval (with possibly periodic), we find that (115) turns out to be equivalent to (17). For the Dirichlet problem, existence, multiplicity and stability of positive solutions to equations of the form (115) have been obtained in [79] in the more general PDEs case for a sign changing weight function. In our setting, we suppose for every

Finally, we observe that a simple adaptation of our approach (as described in Sections 2 and 3 can be applied to (14) provided that both and are close to stepwise functions.

Data Availability

No data were used to support this study.

Disclosure

Preliminary results from this paper have been presented at the “Workshop on Nonlinear Partial Differential Equations”, June 19-20, 2013, at the Universidad Complutense de Madrid.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work has been performed under the auspices of GNAMPA-INDAM and partially supported by the PRIN-2009 project “Equazioni Differenziali Ordinarie e Applicazioni”. F. Zanolin gratefully thanks Professor Julián López-Gómez for the kind invitation and the hospitality. The authors also thank their colleagues A. Boscaggin, W. Dambrosio, and D. Papini who quoted a preliminary version of their article in some of their own recent works [80, 81].

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Copyright © 2018 Chiara Zanini and Fabio Zanolin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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

Abstract

1. Introduction

2. Setting of the Problem and Main Results

3. Technical Estimates and Proofs

4. Related Results

5. Boundary Value Problems on Finite Intervals: Positive Solutions

6. Final Remarks

6.1. Sign Changing Solutions

6.2. Remarks on the Weight Functions

Data Availability

Disclosure

Conflicts of Interest

Acknowledgments

References

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