Abstract

We show the existence of waveforms of finite-energy (vector solitons) for a coupled nonlinear Schrödinger system with inhomogeneous coefficients. Furthermore, some of these solutions are approximated using a Newton-type iteration, combined with a collocation-spectral strategy to discretize the corresponding soliton equations. Some numerical simulations concerned with analysis of a collision of two oncoming vector solitons of the system are also performed.

1. Introduction

Several physical processes related to wave motion can be described using systems of coupled nonlinear Schrödinger (CNLS) equations. Recently there has been a great interest on the study of CNLS systems with nonlinear terms modulated by coefficients which depend either on space, time, or both. This research is motivated by the potential applications of these models to the fields of Bose-Einstein condensates (BECS) [1–8] and nonlinear optics [9–11]. For instance, in the case of BECS [12, 13], a description of the effect of the Feshbach resonances in the mean field limit can be developed using CNLS systems of Gross-Pitaevskii equations [14, 15]. The study of wave propagation in Bose-Einstein two-component condensates with spatially inhomogeneous interactions has been a field of intense research activity in Physics in the last few years [16–24]. In particular, the investigation of multicomponent solitons (also known as vector solitons) has attracted a great deal of attention, starting with the classical work by Manakov [25]. This type of permanent finite-energy waveform arises in CNLS systems due to the interplay between the second-order dispersion and cubic or high-order nonlinearity.

The CNLS systems may also model beam propagation inside crystals, water wave interactions, biophysics [26], finance [27], and oceanography [28], and in the field of communications such equations have been employed to describe pulse propagation along orthogonal polarization axes in nonlinear optical fibers and in wavelength-division-multiplexing systems [29]. Further physical phenomena in nonlinear optics can be described by this family of equations (see [10] and references therein).

In this paper, we will consider theoretically the CNLS-type system where and . Here denotes the complex conjugate of . We suppose that the coefficients , , , , are positive and depend only on the variable in order to focus on the influence of -nonlinear modulations on the multicomponent solitons of the system. Furthermore , are positive real constants. As mentioned above, such type of CNLS system can be used to model a variety of physical phenomena. In the case that all coefficients are constants, system (1) is a model to describe one-dimensional light propagation through a linearly birefringent lossless optical fiber, taking into account the Kerr effect (see Menyuk [30, 31], Agrawal [32], and Evangelides Jr. [33]). In applications to optical fibers, the variables , in system (1) denote time and space, respectively, means the normalized strength of the linear birefringence ( is the inverse group velocity difference), and the model’s parameters are usually constant or -varying (i.e., they vary along the fiber axis). However, space-time-dependent parameters can be encountered in CNLS systems applied to BECS. Recently, Cardoso et al. [34] and Han et al. [35] used systems in the form of (1) with and to describe the interaction among the modes in one-dimensional Bose-Einstein condensates modulated in space and time. In this type of application, the variables , denote space and retarded time, respectively, the functions , denote the complex envelopes of the propagating beam of the two modes, and are the external potentials, means the group velocity dispersion coefficient, and the physical parameters , , , and (depending in this case on both and ) describe the strength of the cubic nonlinearities. The motivation of the study of the interaction of propagating waves in Bose-Einstein condensates with spatial inhomogeneities comes from the discovery in the last years of novel experimental ways to control experimentally the interactions through optical manipulation of the Fechbach resonances [36].

The first aim of the present paper is to address the physically relevant question of the existence of vector solitons of system (1) with in the form and in the form for the case , where and are positive real functions and , , and are real constants.

It is important to point out that exact solutions of system (1) have been obtained only in particular cases. We mention the work by Belmonte-Beitia et al. [38], where explicit solutions of system (1) for and and some examples of variable coefficients were computed using Lie group theory. However, the integrations involved are very long and can not be evaluated as a closed form expression for general model’s coefficients. In [39], Belmonte-Beitia et al. also proved the existence of vector solitons of the system above, in the case that the inhomogeneous coefficients , , , and have compact support and and . Furthermore, Kartashov et al. [11] computed numerically and analyzed the stability of two-component solitons in a medium with a periodic modulation of the nonlinear coefficients for system (1) with and .

In the present paper, we wish to generalize the previous results by establishing analytically existence of vector solitons in forms (2) and (3), of full system (1), considering the extra cross-mode nonlinear terms preceded by the -dependent coefficient , and a nonzero value of the parameter . In the case of CNLS models for pulse propagation in optical fibers, this class of cross-mode terms account for the coherent nonlinear interaction between two linear polarizations of the electromagnetic waves. As pointed out by Menyuk [30], these modulation terms may play an important role in a fiber with very low birefringence. Recently, Muñoz Grajales and Quiceno [37] also illustrated the effect of these extra nonlinear terms and the parameter on modulation instability of a pulse along an optical fiber modelled by full system (1) but with constant coefficients. On the other hand, in [40], some analytical vector bright solitons were calculated for a generalized CNLS system to model BECS including self-phase modulation, cross-phase modulation coefficients, a time-dependent anti-trapping parabolic potential, and four-wave mixing nonlinear terms in the forms and with a time-dependent coefficient.

To study existence of vector solitons of system (1), we apply the positive operator theory introduced originally by Krasnosel’skii [41, 42], following the ideas by Benjamin et al. [43] in the framework of solitary wave solutions of a family of scalar dispersive models for water wave propagation.

In second place, we compute numerically some vector solitons of system (1) in forms (2) and (3) using a Newton iteration, combined with a collocation-spectral strategy to discretize the corresponding soliton equations. This strategy allows us to compute approximations to new vector solitons of the system for a variety of inhomogeneous model’s coefficients. Some numerical simulations concerned with the collision of two oncoming vector solitons of the CNLS system are also performed.

The rest of this paper is organized as follows. In Section 2, we review some known results on fixed points of positive operators in a Fréchet space, necessary in order to develop the existence theory of vector solitons of system (1). In Section 3 we reduce the problem to find a fixed point of a nonlinear positive operator defined on a cone in an appropriate Fréchet space. In Section 4, we use the theory of fixed point index and positive operators to establish the existence of a family of solitons of system (1). In Section 5, we introduce the numerical solver employed to compute solitons of the system and illustrate the theoretical results. Finally, Section 6 contains the conclusions of our work.

2. Preliminary Results

In this section we include a brief review of some results from the functional analysis of positive operators whose domain constitutes a subset of a Fréchet space, following the papers by Benjamin et al. [43] and Chen et al. [44, 45]. We must recall that a Fréchet space is a metrizable and complete, locally-convex, linear topological space (over the real numbers). On a sequence of seminorms can be defined in such a way that for every and every and that the formula provides a metric that generates a topology that coincides with the original topology on . In this case, we say that is a Fréchet space with generating family of seminorms . Hereafter, we use the notation It is clear from (4) that we have that . In general, a set in a topological linear space is said to be bounded if, for any neighborhood of 0 in , there is such that . In the case of a Fréchet space with metric given by (4), a set in is bounded if and only if corresponding to each positive integer there is such that . If , then is usually not bounded (for details see [46]). A closed subset of a Fréchet space is a cone if the following conditions hold true: From (6), must be convex. On the other hand, we also have a partial ordering on given by For any , let us denote An operator defined on is said to be positive, if . On the other hand, we say that a positive operator on is -compact, if the set has a compact closure, for each . A triplet is said to be admissible, if (1)is a convex subset of ,(2) is open in the relative topology on ,(3) is continuous and -compact,(4)there are no fixed points of on , the boundary of the open set in the relative topology on .From Granas’ work [47], there is an integer-valued function that satisfies the basic axioms of a fixed point index. Among them, we consider the following ones: (i)Homotopy Invariant. If and are two admissible triplets and the operator is homotopic to the operator on , then .(ii)The Fixed Point Property. If is admissible and , then has at least one fixed point in .(iii)Index of Constant Maps. If is admissible and is constant (i.e., there is a point such that for all ), then We refer the reader to [43] (see also [41, 42, 47]) for details in the following results. It is assumed throughout that is a cone in a Fréchet space with generating family of seminorms and the standard metric as in (4) and that is continuous, positive, and -compact.

Lemma 1. Suppose that and either Then one has that is admissible and .

Lemma 2. Suppose that and eitherThen is admissible and

The following theorem is a consequence of the first two lemmas.

Theorem 3. If either (10) or (11) holds for satisfying and either (12) or (13) holds for satisfying , then has at least one fixed point in . Moreover, .

An interesting property of system (1), allowing finding new solutions, is described in the following result.

Theorem 4 (Galilean invariance). Let and let , , , , , , , and be constants. If and are a solution to system (1), then another solution is given by where the velocity is given by Furthermore, the solutions of system (1) are indifferent to multiplication by , for any constant .

Proof. It follows directly by substitution into system (1).

3. Problem Setting

In the present paper, we are interested in establishing the existence of vector solitons of system (1) for in the form and in the form for the case . Here and are positive real functions, and , , and are real constants. In each case, we see that the functions and must satisfy a system in the form (soliton equations) where the coefficients , , , , , and are defined in the case that by On the other hand, when , We assume that the coefficients , , , , and in system (1) are bounded, continuous, positive, even, and nonincreasing for , , the parameters , , , , , and are such that the coefficients , , , and are positive and bounded, and and are positive.

Abandoning the tildes, we see that to show the existence of a solution of (18) is equivalent to establish the existence of a solution of the fixed point equation: where the operator is defined by Let us denote by the Fourier transform of the function . We define the functions and as Thus and are positive, even, and monotone decreasing on and belong to . Furthermore for , Therefore .

Then we can rewrite the operator as where denotes the convolution between the functions and .

Hereafter, we consider the space of real valued continuous functions defined on (denoted by ), with the topology of uniform convergence on bounded intervals under the seminorms In this case, the distance is given by The open ball of radius centered at zero and its boundary are given, respectively, by Let be the cone defined as Note the that, for , we have for all that and so we have for that We note that the product space is also a Fréchet space with generating family of seminorms We also define the cone by .

The metric in the product space is defined by In particular, for , We also setand its boundary We observe that if , that is, , then we have that We denote by the annular section of the cone : Hereafter, according to the notation introduced in the previous section, we set, for , the convex set

4. Main Results

Before we go further, we establish a general result.

Lemma 5. Let be an even bounded positive function on , which is monotone decreasing on . Then the operator defined by maps into .

Proof. We first note that is bounded due to the fact that and also from the Young inequality, since Now, we also have that for . In fact, since and . On the other hand, is also an even function for . In fact, We claim now that for is a continuous function on . In fact, first note that for any : Using and the dominated convergence theorem, we conclude that meaning that is a continuous function on , as long as . Finally, we need to establish that is a nonincreasing function for , for . So let be fixed and take and . Then, we have for any that and so we have that Using in the first formula and in the second one, we get that Now, we note that since , and the fact that and are nonincreasing for . Now, for the rest of the integral, we use a similar argument, after noting that and are even functions. In fact, first note for that since is an even nonincreasing function for . So, from this fact, we have that since , and the fact that is an even nonincreasing function for . In other words, we have shown that for any and , which means that is a nonincreasing function for .