Abstract
Keywords:
homoclinic solutions; heteroclinic solutions; multiplicity; nonlinear Schrödinger equation; stepwise potential; topological methodsDedication
Dedicated to Professor Jean Mawhin
1 Introduction
This paper deals with the study of homoclinic and heteroclinic solutions for a class of nonlinear secondorder differential equations of the form
where is a fixed (positive) coefficient and is a bounded weight function. For the nonlinear term, which we split as
we suppose that
(∗) is a locally Lipschitz function which is even, strictly increasing onsuch thatand.
Examples of functions f which are suitable for our considerations are, for instance, as in [1], or () as in [2]. In both these cases, the function f is even. In Section 3 we briefly describe how this restriction could be avoided.
Equations of the form (1.1) naturally arise in the search of particular solutions for some classes of nonlinear Schrödinger equations (NLSE) with inhomogeneous nonlinearities. Typical examples are related to the NLSE
where ħ is the Planck constant, m is the particle’s mass, i is the imaginary unit, Δ is the Laplace operator and . In literature, solutions of the form
where and is a realvalued function, are called stationary waves. Their search leads to the equation
(see [3]). Usually, the Planck constant and the mass are omitted in (1.3) after rescaling.
In many significant models of NLSE, onedimensional waves are considered. They are studied, for example, in nonlinear optics, in the theory of ocean rogue waves and for BoseEinstein condensates (just to mention a few cases). For instance, in [1], BelmonteBeitia and Torres analyzed the onedimensional equation
which is a particular case of (1.3) with , and . In equation (1.4) the nonlinearity is called inhomogeneous due to the presence of a nonconstant weight , which in [1] is assumed to be positive. The sign condition on implies that nontrivial bounded solutions of (1.4) can exist only for (see [[1], Theorem 1]). For this reason, we prefer to set with .
In order to study equation (1.4) or its variants, we are going to follow a dynamical system approach, hence we choose to treat the independent variable (which in the applications has a spatial connotation) as a time variable, via the substitution . Similarly, for the dependent variable, we make the substitution . In this way, equation (1.4) reads as
which belongs to the same class of (1.1).
In spite of the apparent simplicity of equation (1.5), a throughout study of its solutions may be a rather difficult task for a general nonconstant weight function . Looking for homoclinic and heteroclinic solutions of (1.5), it will be natural to focus on the behavior of at ±∞. In similar situations, various authors have confined their study to the case in which is asymptotically constant [46], or eventually constant [7,8]. These assumptions are also justified by the analysis of some physical underlying models, in which a layered structure is present. With this respect, see the introduction in [8], where different eventually autonomous cases are listed for related NLSEs arising in nonlinear optics. Examples in which the nonlinear term presents a piecewise constant weight function have been studied also in biological and chemical models. In particular, these situations occur in the theory of wave propagation for reactiondiffusion systems; see, for instance, [911]. In the context of equation (1.5), examples in which is a piecewise constant function have been considered as well (see [12]).
The aim of the present paper is to provide multiplicity results regarding homoclinic and heteroclinic solutions for equation (1.1) under particular assumptions on the weight function . Actually, we suppose that is an eventually constant piecewise function with only two steps.
Our approach combines phaseplane analysis with timemapping estimates. As in [5,7,8,10,11,13], the solutions are obtained by connecting the unstable and stable manifolds of the equilibrium points of the asymptotically autonomous equations. Such connections are performed by means of orbit paths of an intermediate equation, which represents the behavior of the system during a suitable interval of transition between the asymptotic states. Multiple connecting solutions arise when such interval length is sufficiently large. Lower estimates for will be provided in terms of time mappings, which can be expressed by Abelian integrals. This kind of approach is also reminiscent of some topological methods for the study of SturmLiouville boundary value problems. Indeed, a solution that satisfies the SturmLiouville boundary conditions can be interpreted as a trajectory in the phaseplane that connects two lines (see [1417]). Generalized SturmLiouville solutions, which connect the graphs of two functions or given planar continua, have been considered as well (see [1820]). Often these problems can be settled in the framework of the theory of ODEs with nonlinear boundary conditions (see [21,22]).
In order to make our approach more transparent, we are going to perform our analysis for equation (1.5). This choice is motivated by the sake of avoiding unnecessary technicalities. Our arguments can be modified in a straightforward manner in the case of more general equation (1.1), with f satisfying (∗) (see Section 3). Homoclinic and heteroclinic orbits can be interpreted as solutions for some boundary value problems on unbounded intervals. In the last section we also outline possible applications of our approach to boundary value problems on a compact interval (like the SturmLiouville one).
Besides his manifold achievements in different areas of mathematics, Professor Jean Mawhin is one of the pioneers in the study of topological methods for nonlinear boundary value problems. It is a pleasure and an honor to have the possibility to dedicate our work to his important contributions in this area.
2 Homoclinic and heteroclinic solutions: multiplicity results
2.1 General setting
We consider the secondorder nonlinear differential equation
and its equivalent system in the phaseplane
where is a fixed coefficient and is a bounded measurable function. Solutions of (2.1) are meant in the generalized (i.e., Carathéodory) sense (see [23]). Actually, in our results a step function with only two jumps is considered and the solutions are piecewise smooth.
In the particular case of a constant coefficient , an elementary analysis of the system
shows that the associated phase portrait presents a center and two saddle points . These saddles are connected by two heteroclinic solutions (one connecting to in the upper halfplane and a symmetric one from to in the lower halfplane). The heteroclinic orbits are described by
In [1] BelmonteBeitia and Torres supposed that the weight function is even andTperiodic and proved the existence of a heteroclinic solution connecting two periodic stationary states. Such a result generalizes to the case of periodic coefficients the situation described above for the autonomous system.
As already observed in the introduction, equation (2.1) has been already studied by various authors for its relevance in many applicative models. The aim of the present paper is to provide multiplicity results for heteroclinic and also homoclinic solutions of equation (2.1). A possible way to obtain this goal is to assume that the weight function has a different behavior at infinity and in some intermediate time interval. In this setting, a first step consists of analyzing the situation in which is a stepwise function assuming only two values. We believe that the study of such a simplified case may lead to more general considerations, in which, for instance, is eventually periodic. In this way, multiplicity results for BelmonteBeitia and Torres’s model can be obtained (see Section 3 for a brief overview concerning possible applications of our approach). Equation (2.1) with a periodic stepwise coefficient has recently been studied in [24] in the context of chaoticlike dynamics.
As a tool, in our paper, we use the Poincaré map associated to system (2.2). Given a fixed time interval and a point , we indicate by the solution of (2.2) satisfying the initial condition . The Poincaré map on is defined as
From the fundamental theory of differential equations it follows that the domain of Φ is an open subset of the plane and Φ is an orientation preserving homeomorphism of onto its image .
2.2 Analysis of the equation
Let us consider system (2.2) for the stepwise weight function , defined as
where
System (2.2) can be seen as the superposition of the autonomous systems
and
The first one describes the asymptotic behavior of the equation. We are going to use the orbits of the second system for connecting unstable/stable manifolds of (2.6) during the time interval . Figure 1 shows the superposition of the phase portraits of the two systems.
Figure 1. In the present figure we have considered the superposition of the phase portraits of systems (2.6) (in darker color) and (2.7) (in lighter color) with,,. For graphical reasons, a slightly different x and yscaling has been used.
First of all, we briefly analyze the structure of system (2.6). Its equilibrium points are the origin (which is a center) and two saddle points , where
The stable and unstable manifolds of are illustrated in Figure 2. The sets and are, respectively, the stable and the unstable manifolds of . Symmetrically, and are the stable and the unstable manifolds of .
Figure 2. The stable and unstable manifolds ofand. The arrows show the direction of the flow along the orbits.
In the case of the cubic nonlinearity, we find
The solutions of system (2.2) with defined as in (2.5) will be obtained by connecting suitably chosen parts of stable and unstable manifolds of (2.6) with trajectories of (2.7). More in detail, we proceed as follows.
Due to the Hamiltonian nature of the equation under consideration, we can consider the ‘energy’ level lines of system (2.7), given by
We also set
where
The curve is the part of the line at level in the strip . For every c such that
the set is a periodic orbit of (2.7) run in the clockwise sense, which intersects the yaxis at the points . The (minimal) period of can be expressed by the following timemapping formula:
where is the potential associated to the equation and , are such that
In our case,
and by symmetry . More in detail, we can express the period by means of an elliptic integral in the following way:
for , and , . Notice that in (2.10). Therefore
where is the Jacobi elliptic sine function of modulus (see [25,26]).
The level line contains the saddle points of system (2.7) and their heteroclinic connections in the upper halfplane and in the lower halfplane. The energy level
corresponds to the closed curve , which is tangent to and . Moreover, for
the energy level line contains the saddle points of system (2.6).
Using the parameters , and , we define the regions
Both ℰ and ℱ are invariant sets for (2.7); they are filled by periodic orbits. In the sequel, the orbits in the region ℰ will be called internal, while those in ℱ will be called external. The choice of these names is made in order to distinguish the trajectories with respect to , which contains the saddle points of system (2.6). (See Figure 3.)
Figure 3. The energy level lines,andfor system (2.7) superimposed over the stable and unstable manifolds offor system (2.7). The colored regions are the invariant sets ℰ (the darker one) and ℱ (lighter). The figure has been plotted with the parameters , , . For graphical reasons, a slightly different x and yscaling has been used.
In order to obtain homoclinic or heteroclinic solutions for system (2.2) with defined as in (2.5), we connect the unstable manifold of the point to the stable manifold of via an orbit path of for . Actually, this is the only way to get the desired solutions. If , such connection will follow a trajectory with , so it will lie in the invariant annular region ℰ bounded by and . On the contrary, for the connection lies in the set ℱ between and .
To provide more explicit details, we consider the case . A solution of (2.2), which is homoclinic to , can be produced in two ways. One consists in connecting to , using an orbit path of for . Another possibility is given by connecting a point of to a point of via an orbit path of for . For , we can obtain our solution if and only if is an integer multiple of the fundamental period of the orbit . Indeed, such a solution will be constantly equal to for , and for it coincides with the periodic solution of (2.7) such that and . This solution makes turns around the origin for .
Similar considerations can be developed with respect to heteroclinic solutions. For instance, a heteroclinic orbit from to can be obtained as follows: by connecting a point of with one of using an orbit path of for , or by a connection of two different points of via an orbit path of for . For , we can obtain our solution if and only if is an odd multiple of , where is the period of the orbit . Indeed, such a solution will be constantly equal to for and constantly equal to for . Moreover, it coincides for with the solution of (2.7) such that and . This solution makes halfturns around the origin for .
Figure 4, although not exhaustive of all the conceivable cases, summarizes several different possibilities. Indeed, a trajectory homoclinic to can be obtained as follows: move by system (2.6) along from to the intersection point of with a closed curve external to , namely δ (for ). This point is put in evidence with a small black circle. Next, follow δ (by system (2.7)) in order to reach the intersection of δ with (for ). Such an intersection point is indicated by a grey square. Finally, switch to system (2.6) and move toward along (for ). With a similar procedure, we can obtain a heteroclinic trajectory from to . In fact, we can move by system (2.6) along from to the intersection point of with a closed curve external to , namely γ (for ). Next, follow γ (by system (2.7)) to the intersection with (for ). Finally, switch to system (2.6) and move toward along (for ).
Figure 4. In the present figure we have considered the superposition of (2.6) and (2.7) for,,. The closed curves α, β, γ, δ represent four different level lines of system (2.7): the curves α and β in ℰ correspond to a level , while γ and δ in ℱ refer to . In order to describe a connection from an unstable manifold to a stable one of system (2.6) via an orbit path of (2.7), we have marked with a black circle possible starting points and with a grey square some available end points on the same level lines.
In general, the connections through the trajectories of system (2.7) will either involve only an arc of the closed curve , or they will require to perform a certain number of winds around the origin, depending on the available time. If the length of the time interval is small, only the first case is possible. However, if is large enough, more choices arise. For instance, looking at the orbit δ in Figure 4, we could make a certain number of loops from the black circle before reaching the grey square on the same orbit. Similar considerations apply to the heteroclinic trajectory described above with reference to γ.
Until now we have described the ‘external connections’, namely those which lie in the region ℱ. Further possibilities appear if we consider ‘internal connections’ by means of orbit paths contained in ℰ. For example, if we look at the orbit β in Figure 4, we can obtain trajectories homoclinic to by choosing one of the two black circles as a starting point from and one of the two grey squares as an end point on . Loops along β will be permitted if the time is sufficiently large. The construction of heteroclinic orbits from to via the line α follows a similar procedure.
2.3 Study of the Poincaré map
Another possible point of view to describe the previous construction of homoclinic and heteroclinic solutions consists in considering the Poincaré map associated to system (2.7) for the time interval . Due to the autonomous nature of this system, we have that , denoted simply by Φ when no confusion may occur. Moreover, the region
is a compact invariant set contained in the domain of Φ. Let us denote by the part of the unstable manifold between and , including the extreme points. Hence the external connections can be precisely described by looking at the intersection points of with (for the homoclinics) and with (for the heteroclinics), as illustrated in Figure 5.
Figure 5. In the present figure we show the transformation ofunder the Poincaré mapΦfor. The parameters used are , , . For graphical reasons a slightly different x and yscaling has been used.
With reference to Figure 5, we observe that the intersection point of with (indicated by a small circle) is moved by Φ along to a point very close to . On the other hand, during the time interval , the point makes a little more than a complete turn around the origin; the final position is indicated by a cross. The arc is a spirallike curve with one end near and the other on . For the time length , considered in Figure 5, there are precisely two intersections of with , corresponding to external homoclinic solutions. Meanwhile, we have also an external heteroclinic solution due to the intersection of with . All these three intersection points have been indicated by a grey square (following the same convention used in Figure 4).
In order to describe further these solutions, let us consider their energy at the time that we denote by , , , with . Let us define the points as for . The solution of (2.2) with or has the following behavior: for , with and convex for . Moreover, is increasing on and decreasing on . On the interval , if , the solution is concave, while if , the solution has two maxima and one (negative) minimum separated by two simple zeros. The solution of system (2.2) with is such that for with and for with ; is convex for . Moreover, the solution has one maximum and one minimum separated by one simple zero.
Even if the time interval length is small, we always have at least one intersection of with and thus a homoclinic solution. On the other hand, if grows, the spiral curve will wind more times around the origin, hence more homoclinic/heteroclinic solutions will appear.
If we look for the internal connections (made by orbit paths lying in the region ℰ), we proceed as follows.
Let us set and denote by its symmetric part with respect to the xaxis. Figure 6 illustrates the involved geometry from the point of view of the Poincaré map. We observe that all the points of ℒ are contained in ℰ, hence they are periodic points (of different periods). As before, during the time interval , the point moves clockwise around the origin along ; the final position is indicated by a cross. Symmetrically, the point performs exactly the same angle around the origin, reaching the final position , indicated by a small circle. All the other points of move, under the action of Φ, on the energy level lines with . The arc is a curve contained in ℰ, connecting the two antipodal points and leaning on one point of . This tangent point is the image through Φ of the point . Such a property of implies that intersects both ℒ and (at different points). These two points have been indicated by a grey square in the figure. Accordingly, we always find at least an internal homoclinic solution and an internal heteroclinic one.
Figure 6. In the present figure we show the transformation ofunder the Poincaré mapΦfor, the same time considered in Figure5. The parameters used are , , , as before. The point indicated by a cross is precisely the same point of Figure 5. Notice that for this time interval the point makes a little more than a complete turn around the origin. For graphical reasons, a slightly different x and yscaling has been used.
Summarizing the above information, we conclude that, for every time interval , we have at least two solutions which are homoclinic to (one external and one internal) and one internal heteroclinic solution from to .
For the external connections, if the time interval length grows, the situation can be summarized as follows. The number of external homoclinic/heteroclinic solutions increases, depending on the number of winds around the origin of the curve . Indeed, the end point of such a curve on cannot go beyond , while the other end point is free to move on the periodic orbit . If we denote by and the number of external homoclinic and heteroclinic solutions respectively, it holds that . From the above discussion, we can conclude that if, for some nonnegative integer n, we have (where is the period of ), then . An analogous lower bound can be provided for . For a formal proof, see Theorem 2.1.
When grows, the situation for the internal homoclinic/heteroclinic solutions is more intriguing, as illustrated in Figure 7 and Figure 8. A twist effect depending on the different periods of and produces a double spirallike curve when is sufficiently large.
Figure 7. In the present figure we show the transformation ofunder the Poincaré mapΦfor. The parameters used are , , , as before. For graphical reasons, a slightly different x and yscaling has been used.
Figure 8. The present figure shows the evolution ofunder the Poincaré mapΦfor time intervals of different lengths. The parameters used are , , , as before, while for , we have considered the cases 35, 60, 100, 150 (from the left to the right). For graphical reasons, a slightly different x and yscaling has been used.
The fact that looks like a double spiral depends on the different velocities of the points and (which is the intersection of ℒ with the negative yaxis). In fact, since , the points on move faster than those of . By construction, , while . Therefore, when the time gap is sufficiently large, the number of turns of around the origin will exceed the number of turns of . Accordingly, the image through Φ of the right part of ℒ (the subarc of ℒ connecting to R) is a spiral that winds a certain number of times in the clockwise sense around the origin. Indeed, it connects the slower point to the faster one inside the region ℰ, as illustrated in Figure 9. The number of halfturns of this spiral depends on a ‘rotational gap’ that, for our purposes, it will be convenient to define as
Similarly, the image through Φ of the left part of ℒ (the subarc of ℒ connecting R to ) is a spirallike curve winding a certain number of times in the counterclockwise sense around the origin. Indeed, it connects the points and inside the region ℰ (the first point moves at a faster speed than the second one). Again, the number of halfturns of this second curve will depend on . In the end, gluing together the two spirallike curves, we conclude that, if is sufficiently large, the arc will appear like a double spiral with a central ‘hook’. To better describe the structure of , we should observe that the ‘tip of the hook’ is not , but it is the image through Φ of a point (close to R) on the left part of (we owe this remark to the referee). For a formal proof based on the argument outlined above, see Theorem 2.2.
Figure 9. The present figure explains the twist effect due to the different periods ofand. We have considered the evolution of the part of ℒ with connecting to the point R for two different time gaps: and . The corresponding two images are indicated by a cross, while the images are marked by a black dot. The parameters used are , , , as before. Notice that the two images , although close to each other in the figure, have made a different number of turns around the origin. For graphical reasons, a slightly different x and yscaling has been used.
In preparation for this theorem, we introduce the following notation. Given a positive real number S and an energy level , we denote by the lower integer part of , and by the upper integer part of . By definition, during a time interval of length S, a point on the orbit makes more that halfturns around the origin, but less that halfturns. Thus the rotational gap defined above can be written as for . By definition, in the open interval , the smallest integer is , while the largest integer is . Hence, the open interval contains positive integers.
2.4 Conclusion
After the preliminary study of the previous subsections, we are now in a position to express our results for the equation
as statements with a formal proof. Recall that the weight function is defined as follows:
where
We also set and consider the Poincaré map associated to the system
In order to make the exposition more clear, we have decided to consider separately the cases of external and internal connections.
Theorem 2.1 (External connections)
Under the above assumptions, the following results hold for solutions which satisfy the energy condition
• there always exist a (positive) solution homoclinic toand a (negative) solution homoclinic to;
• if, there always exist a heteroclinic solution fromtoand a heteroclinic solution fromto, both with exactly one zero;
• if (for some integer) then, for each integerjwith, there exist at least one solution homoclinic toand one solution homoclinic to, both with exactly 2jzeros;
• if (for some integer) then, for each integerjwith, there exist at least one heteroclinic solution fromtoand one fromto, both with exactlyzeros.
Proof As a first step, we focus our attention on the search of solutions homoclinic to the point . The case of homoclinics to is analogous, thus it will be omitted.
Recalling that , our goal is to find points on which are moved by the Poincaré map on .
We introduce in the phaseplane for (2.7), a system of polar coordinates , with center in the origin. The initial points in the arc (which is the closure of the intersection of with the external region ℱ) are parameterized as , for , with , and strictly increasing with s. The target set is the symmetric of with respect to the xaxis. It will be denoted by and parameterized by reversing the angle .
Using the same polar coordinates to represent the solutions of (2.7), we can express the final points as
with
We notice that
is the solution of (2.7) (at the time t), which departed from the point at the time . Observe that, for every , the map
is strictly decreasing on (this is an equivalent way to express the fact that the solutions turn around the origin in the clockwise sense).
With these positions, we obtain an external solution homoclinic to if and only if there exists such that . This happens if and only if
for some nonnegative integer j. In this case, if we denote by the corresponding homoclinic solution of (2.1) such that , we have that for all . Moreover, if , for all , while has precisely 2j zeros in if .
Using the fact that the energy level lines of (2.7) in the region are strictly starshaped with respect to the origin and symmetric with respect to the xaxis, we find that (2.12) holds (for some nonnegative integer j) if and only if
is satisfied (for the same j).
By virtue of (2.13), we can refer only to the angular coordinates, hence we will obtain solutions as follows. Since the point lies on (which is the trajectory of a heteroclinic solution of (2.7)), it can never reach the point . Accordingly, the angle satisfies
On the other hand, as , we have that , therefore
As a consequence, and the intermediate value theorem ensures the existence of an such that . In this way we have found a positive homoclinic solution to , independently of the length T of the time interval .
Suppose now that for some positive integer n. In this case,
As a consequence, for every integer j with , we have , and again the intermediate value theorem ensures the existence of an such that . This ends the proof for homoclinic solutions.
For the search of heteroclinic solutions from to , we follow a similar procedure, choosing as a target the set , which is the symmetric of with respect to the yaxis (here we exploit the oddness of the nonlinear term ). In terms of polar coordinates, the desired solutions will be obtained if and only if there exists such that
for some integer . Then the proof can be concluded as above, via the intermediate value theorem. We observe that the case of heteroclinic solutions from to is analogous, thus it will be omitted. □
The next theorem deals with the internal connections. For this result, it is useful to recall the rotational gap defined in (2.11).
Theorem 2.2 (Internal connections)
Under the above assumptions, the following results hold for solutions which satisfy the energy condition
• there always exist a solution homoclinic toand a solution homoclinic to;
• there always exist a heteroclinic solution fromtoand a heteroclinic solution fromto;
• if, there exist at leastsolutions homoclinic toandsolutions homoclinic to;
• if, there exist at leastheteroclinic solutions fromtoandfromto.
Proof As the first step, we focus our attention on the search of solutions homoclinic to the point . The case of homoclinics to is analogous, thus it will be omitted.
Our goal is to find points on which are moved by the Poincaré map on .
As a preliminary remark, we note that is a simple arc contained in ℰ, connecting its end points and on through the ‘intermediate’ point on . Observe also that and are antipodal. In fact, and are antipodal and is an odd map (indeed, is a solution of (2.7) if and only if is a solution of the same equation). Now, if the trivial situation occurs, we have . Otherwise, we find that and belong to the two different components of . Since and are the end points of the arc , by an elementary connectivity argument, we conclude that . This proves the first assertion of the theorem.
For the search of heteroclinic solutions from to , we adopt a similar procedure, choosing as a target the set ℒ itself. The fact that, for every , there is always at least one intersection point between and ℒ follows by the same argument developed for homoclinic solutions. Namely, if the trivial situation occurs, we have . Otherwise, we find that and belong to the two different components of . Therefore, it follows that .
We study now the problem of multiplicity of solutions. As before, we consider at first the case of solutions homoclinic to .
We introduce in the inner region ℰ of a phaseplane for (2.7) a system of modified polar coordinates with center in the origin. In this system, every point is determined by its angular coordinate θ and its energy E defined in (2.8). Since the energy level lines in ℰ are strictly starshaped with respect to the origin, we obtain a coordinate system equivalent to the polar one.
Observe that the set intersects every halfline from the origin exactly in one point. Accordingly, we can parameterize those points using the angular coordinate. With this convention, the initial points in the arc ℒ are parameterized as for (s is the angle) with , and . As a consequence, is strictly decreasing for and strictly increasing in . The target set (the symmetric of ℒ with respect to the xaxis) is parameterized with the angle .
For the specific case of system (2.6), an analytic expression for ℒ as a graph in the plane, is given as follows:
with
Hence, the natural parametrization of ℒ in the plane is given by for .
Using the same modified polar coordinates to represent the solutions of (2.7), we can express the final points in the plane by means of their angular coordinate and energy . As in the previous proof, we observe that, for every , the map is strictly decreasing on , while the energy is constant with respect to t. Observe also that in the new coordinates , system (2.7) becomes
with
In the plane, the points of ℒ move with a negative angular speed along the lines of constant energy. Thus, under the action of the flow associated to (2.16), the points of ℒ shift from the right to the left in the strip
As we have explained before, we obtain an internal solution homoclinic to whenever there is a point such that , with . In the plane, this target set is expressed as the union of the graphs
where
with
(see Figures 10, 11 for a graphical description).
Figure 10. The initial setand the target setrepresented in theplane. The present figure is drawn for the parameters , , . The internal region ℰ corresponds to the strip . Since we are interested in the evolution of the set ℒ through the Poincaré map, we consider only the halfstrip . For the flow associated to (2.16), all the points of ℒ move from the right to the left on lines parallel to the θaxis. The point R (as well as the points on the line ) moves faster than the points which are on the line . During all the evolution, the distance between the images of and remains constantly equal to π.
Figure 11. The present figure describes the same situation of Figure7in the angleenergy coordinates (with the same choice of the coefficientsk,,and time gap).
In this setting, we obtain an internal solution homoclinic to if and only if there exists such that , where we have denoted by Φ the Poincaré map for (2.16) in the time interval . Of course Φ is exactly (which was the Poincaré map associated to (2.7)) in the new coordinates. Using the parameterized curve for the initial points, we can express Φ as
The points and are antipodal, lie on the same energy line and move with the same angular speed. In the plane, and are expressed by , . Therefore,
Moreover, for every integer , we have that
(this follows from the fact that the orbit has a period and is symmetric with respect to the xaxis).
In the same plane, the point R is indicated by . Using the fact that the orbit has a period and is symmetric with respect to the yaxis, we find that, for every integer ,
Suppose now that
Then, the open interval contains at least one odd integer.
Let be the largest odd integer contained in . From , we have that
Suppose also that j is a positive integer with . In this case, from , we have that
Let
and
with be the left and the right parts of ℒ in the plane. Similarly, we define
and
By definition, for any nonnegative integer i, the set is a simple arc connecting the points and in the rectangle and, similarly, is a simple arc connecting the points and in the rectangle . On the other hand, the set is a simple arc connecting the points and in and, similarly, is a simple arc connecting the points and in the same strip.
Suppose now that (in this case, j is odd). In such a situation, from (2.19) and (2.20) we have that
for an even integer. Hence the points and are separated by the arc , while the points and are separated by and . As a consequence, contains at least three points, precisely the nonempty intersections of , and .
Suppose that (in this case, j is even). In such a situation, from (2.19) and (2.20) we have that
for an even integer. Hence the points and are separated by the arcs and . The points and are separated by , and . As a consequence, contains at least five points: two points coming from the intersections of with the target set Ξ and three from the intersections of .
Proceeding by induction, with the same argument, the following result is obtained.
Claim 1has at leastsolutions, whereis the number of integers less than or equal towhich are contained in the open interval.
A completely symmetric argument leads to the same multiplicity result for solutions which are homoclinic to .
At last, we look for a multiplicity result for heteroclinic solutions from to . We follow step by step the argument described in the part of the proof devoted to the search of multiple internal homoclinic solutions and, therefore, we transform our equation in the coordinates. As we have explained before, we obtain an internal heteroclinic (from to ) whenever there is a point such that , with . In the plane, this target set is expressed as the union of the graphs
where is the function defined in (2.15).
As before, we assume the validity of condition (2.18). Then the open interval contains at least one even integer.
Let be the largest even integer contained in . From , we have that
For any positive integer j with , we have that (2.20) holds too. At this point, we have simply to repeat (with obvious changes due to the fact that the target set is which is a shift of Ξ by ) the argument presented above for the case of homoclinic connections and obtain the following conclusion.
Claim 2has at leastsolutions, whereis the number of integers less than or equal towhich are contained in the open interval.
A completely symmetric argument leads to a multiplicity result for heteroclinic solutions from to .
As final step, we have just to show how the condition on the rotational gap allows us to achieve the conclusion from Claim 1 and Claim 2 that we have obtained along the proof. Now, it is sufficient to observe that implies (2.18) and, therefore, the open interval contains at least two integers. More precisely, by the definition of , the interval contains at least integers and, therefore,
This concludes the proof of the theorem. □
3 Remarks and related results
We end the paper with a list of remarks about possible variants and extensions of the main results obtained for equation
1. A problem which naturally arises from the analysis that we have performed concerns what happens if, for a stepwise weight function satisfying (2.5), we suppose that . Repeating the preliminary phaseplane analysis of Section 2.2, one can easily check that for any gap , there always exist a solution homoclinic to and another homoclinic to , as well as a heteroclinic from to and another one from to . However, in general, if , one cannot obtain multiplicity results like those achieved in Section 2 without some extra assumptions on . This is the reason for which, in the study of equation (2.1), we have considered only the case . For different examples on related equations in which a weight coefficient can be above or below its limits at infinity, see, for instance, [46].
On the other hand, if we assume that the weight function changes its sign, with , some interesting multiplicity results could be produced. In fact the phase portrait of (1.1) in the time interval shows a global center, hence if is large, a lot of connections between the unstable and stable manifolds of the points can be obtained. For the sake of conciseness, we omit the study of this latter situation, which is beyond the goal of the present paper.
2. We notice that the same argument of the proofs applies to an equation of the form
for , , and with satisfying condition (∗), if we assume that is a stepwise weight function, playing the same role of in (2.1). In such a case, the saddle points become with , where is the inverse of f restricted to . In this way, we can apply our results to nonlinearities like those considered in [2,27]. More precisely, Theorem 2.1 holds without any further assumption, while for Theorem 2.2 we need to require a gap between the periods of the orbits and . In order to obtain this gap, we can apply (for instance) some results ensuring the monotonicity of the timemap (like [[28], Theorem A]).
3. Due to the special form of the weight coefficient, it is standard to verify (via a simple rescaling procedure) that (1.1) is equivalent to an equation of the form
where and is a stepwise coefficient. In this manner, one can deal with some nonlinear Schrödingertype equations related to the case of potential wells or potential walls [29,30].
4. The approach used in the proofs, based on the properties of the Poincaré map, guarantees that our results are stable with respect to small perturbations. More precisely, fixed a suitable length for the time interval, Theorems 2.1 and 2.2 provide a lower bound for the number of solutions. We can state that the same lower bound persists for a small perturbation of the coefficient in the norm on . Therefore, the assumption that the weight in (1.1) or in (3.1) are stepwise functions can be slightly relaxed, so we can ‘smooth’ them.
5. With reference to equations (1.1) or (3.1) with stepwise coefficients, we observe that our approach can be adapted to boundary value problems on a compact interval like, e.g., the Dirichlet (twopoint) or the Neumann problem. In these cases, we have to find solutions connecting given lines which depart from the origin. For the sake of conciseness, we cannot describe the most general situation, but we just outline a possible application for the Neumann problem
with a stepwise function such that for and for , for . Let us denote by the Poincaré map associated to system (2.6) on the time interval , and by the Poincaré map associated to system (2.7) on the time interval . We observe that there exists a maximal compact interval with such that for all the points with . In this manner . The curve
represents the set of all the points in the region , which are images (by ) of initial points of the xaxis (therefore ). Figure 12 illustrates the curve for a short time interval (). For a larger τ, the line becomes a double spiral with a certain number of turns around the origin, while the part of contained in the region ℱ gets very close to and . Then we can repeat the same argument developed in the previous sections by looking for the intersections of with the xaxis.
Figure 12. In the present figure the lineis the transformation of the segmentunder the Poincaré mapfor. The parameters used are , , . For graphical reasons, a slightly different x and yscaling has been used.
6. As a final remark, we mention the fact that combining our technique with Ważewski’s method [31], following the approach developed by Conley in [18], one can deal with some more general classes of weight functions. For example, one could tackle with these techniques the cases of asymptotically constant or asymptotically periodic coefficients. These extensions, however, need a more delicate analysis and they are beyond the goals of the present paper.
We hope that the abundance of multiplicity results found in the present work (in the special case of stepwise coefficients) may suggest possible directions for extending Theorem 2.1 and Theorem 2.2 to more general weight functions. This will be our goal for a future investigation of the problem.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the manuscript and read and approved the final manuscript.
Acknowledgements
The authors are deeply indebted with the referee for the careful checking of the manuscript and for his/her remarks, including a correction to an erroneous argument in the previous version of the proof. This research was partially supported by the project PRIN2009 Equazioni Differenziali Ordinarie e Applicazioni.
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