Vector Solitons of a Coupled Schrödinger System with Variable Coefficients

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.


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][2][3][4][5][6][7][8] and nonlinear optics [9][10][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 twocomponent condensates with spatially inhomogeneous interactions has been a field of intense research activity in Physics in the last few years [16][17][18][19][20][21][22][23][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-divisionmultiplexing 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 CNLStype system where  ∈ R and  > 0.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 2 Advances in Mathematical Physics 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 (2 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  = 0 and  ≡ 0 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  = (,), V = V(, ) denote the complex envelopes of the propagating beam of the two modes,  1 and  2 are the external potentials,  > 0 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 (, V) of system (1) with  = 0 in the form  (, ) =   ũ () , and in the form for the case () ≡ 0, where ũ and Ṽ are positive real functions and ,  1 , and  2 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  ≡ 0 and  = 0 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  ≡ 0 and  = 0. 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  ≡ 0 and  = 0.
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 fourwave mixing nonlinear terms in the forms V 2  * and  2 V * 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.

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, locallyconvex, linear topological space (over the real numbers).On  a sequence (  )  of seminorms can be defined in such a way that  +1 () ≥   () for every  ∈  and every  = 1, 2, 3, . . .and that the formula Advances in Mathematical Physics 3 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  = B 1 .In general, a set  in a topological linear space  is said to be bounded if, for any neighborhood  of 0 in , there is  > 0 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  > 0 such that  ⊂ B   .If  > 0, then B  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 0 <  <  < ∞, let us denote An operator A defined on  is said to be positive, if A() ⊂ .On the other hand, we say that a positive operator A on  is -compact, if the set A(  ) has a compact closure, for each 0 ≤  < 1.A triplet (, A, ) is said to be admissible, if From Granas' work [47], there is an integer-valued function (, A, ) that satisfies the basic axioms of a fixed point index.Among them, we consider the following ones: (i) Homotopy Invariant.If (, A, ) and (, B, ) are two admissible triplets and the operator A is homotopic to the operator B on , then (, A, ) = (, B, ).
(ii) The Fixed Point Property.If (, A, ) is admissible and (, A, ) ̸ = 0, then A has at least one fixed point in .
(iii) Index of Constant Maps.If (, A, ) is admissible and A is constant (i.e., there is a point  ∈  such that A =  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 A:  →  is continuous, positive, and -compact.
The following theorem is a consequence of the first two lemmas.
An interesting property of system (1), allowing finding new solutions, is described in the following result.
Theorem 4 (Galilean invariance).Let  = 0 and let , , , , ,  1 ,  2 , and  be constants.If (, ) and V(, ) 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  Φ , for any constant Φ.
Proof.It follows directly by substitution into system (1).
Advances in Mathematical Physics

Problem Setting
In the present paper, we are interested in establishing the existence of vector solitons (, V) of system (1) for  = 0 in the form and in the form for the case () ≡ 0.Here ũ and Ṽ are positive real functions, and ,  1 , and  2 are real constants.In each case, we see that the functions ũ and Ṽ must satisfy a system in the form (soliton equations) where the coefficients (), (), (), (),  1 , and  2 are defined in the case that  = 0 by On the other hand, when  ≡ 0, We assume that the coefficients (), (), (), (), and () in system (1) are bounded, continuous, positive, even, and nonincreasing for  ≥ 0,  > 0, the parameters ,  1 ,  2 , ,  1 , and  2 are such that the coefficients (), (), (), and () are positive and bounded, and  1 and  2 are positive.
Abandoning the tildes, we see that to show the existence of a solution  = (, V)  of ( 18) is equivalent to establish the existence of a solution of the fixed point equation: where the operator A is defined by Let us denote by f the Fourier transform of the function .
We define the functions  1 and  2 as Thus K1 and K2 are positive, even, and monotone decreasing on (0, ∞) and belong to  1 (R).Furthermore for  = 1, 2, Therefore Then we can rewrite the operator A as where  *  denotes the convolution between the functions  and .
Hereafter, we consider the space of real valued continuous functions defined on R (denoted by (R)), with the topology of uniform convergence on bounded intervals under the seminorms In this case, the distance is given by Advances The open ball of radius  < 1 centered at zero and its boundary are given, respectively, by Let K ⊂ (R) be the cone defined as Note the that, for  = 1, 2, 3 . .., we have for all  ∈ K that and so we have for 0 <  < 1 that We note that the product space (R) × (R) is also a Fréchet space with generating family of seminorms We also define the cone The metric in the product space (R) × (R) is defined by In particular, for (, ) ∈ K, We also set and its boundary We observe that if (, ) ∈ K ∩ B  (0), that is, d((, ), 0) = , then we have that max { (0) ,  (0 We denote by K  the annular section of the cone K: Hereafter, according to the notation introduced in the previous section, we set, for  > 0, the convex set

Main Results
Before we go further, we establish a general result.
Lemma 5. Let  ∈ (R) ∩  1 (R) be an even bounded positive function on R, which is monotone decreasing on (0, ∞).Then the operator B defined by Proof.We first note that B()() is bounded due to the fact that  ∈  1 (R) and also from the Young inequality, since max Now, we also have that B() ≥ 0 for  ∈ K.In fact, since  ≥ 0 and  ≥ 0. On the other hand, B() is also an even function for  ∈ K.In fact, We claim now that B() for  ∈ K is a continuous function on R. In fact, first note that 0 ≤ () ≤ (0) for any  ∈ R: Using  ∈  1 (R) and the dominated convergence theorem, we conclude that meaning that B() is a continuous function on R, as long as  ∈ K. Finally, we need to establish that B() is a nonincreasing function for  ≥ 0, for  ∈ K.So let  ∈ K be fixed and take  ≥ 0 and ℎ > 0.Then, we have for any  ∈ R that and so we have that Using  = −ℎ/2 in the first formula and  = ℎ/2 in the second one, we get that Now, we note that since  ≥  + (1/2)ℎ ≥ (1/2)ℎ, and the fact that  and  are nonincreasing for  ≥ 0. Now, for the rest of the integral, we use a similar argument, after noting that  and  are even functions.In fact, first note for  ≥ 0 that since  is an even nonincreasing function for  ≥ 0. So, from this fact, we have that since 0 ≤  ≤  + (1/2)ℎ, and the fact that  is an even nonincreasing function for  ≥ 0. In other words, we have shown that for any  ≥ 0 and ℎ > 0, which means that B()() is a nonincreasing function for  ≥ 0.
Lemma 6. Suppose that the coefficients (), (), (), and () in system (18) are continuous bounded positive nonzero functions in R and  1 > 0,  2 > 0. Furthermore, assume that these coefficients are even and nonincreasing for  ≥ 0. Then the operator A defined by ( 25) maps continuously K into K.
For each 0 <  < 1, the set A( K ) is a relative compact subset of K.
Proof.As we mention above,  1 ,  is a nonincreasing function on (0, ∞).Now, let (, V) ∈ K, then we have that  3 + V 2  and V 3 +  2 V belong to the cone K, which means from Lemma 5 that and also that A(, V) for (, V) ∈ K is continuous on R.Moreover, we also have that A 1 (, V) and A 2 (, V) are nonincreasing for  ≥ 0, for all (, V) ∈ K.In other words, we have established that A( K) ⊂ K. Finally, we want to prove that the operator A = (A 1 , A 2 ) maps K continuously to K. To see this, we must recall that convergence in Using this fact, we see easily for  = 1, 2, Advances in Mathematical Physics To analyze the continuity of the first component of the operator A, we decompose the integral on R as Observe that {(, V), Therefore, for  ∈ [, ] and Ω large enough, On the other hand, using the factorizations we have that for  ≥  0 .Analogously, max for  ≥  0 .We conclude that the operator A maps K continuously to K. It remains proving that A( K ) (0 <  < 1) is relative compact subset of K, which is equivalent that A  ( K ) is relative compact subset of K, for  = 1, 2. To see this, we use the Arzela-Ascoli Theorem to establish the compactness of the families M  in (R), Let (, V) ∈ K be such that  = B  (  ) with  1 =  3 +V 2  and  2 = V 3 +  2 V.We first note that, for   ∈ (R), we have that B  (  ) : R → R is a continuous function such that (see Lemma [48]).In other words, we have shown that the set A( K ) is a relative compact subset of K. (71) Clearly, if  = 0, we have that  = 0 ∉ K ∩ B  (0) which is a contradiction.Suppose that  ∈ (0, 1].Then we have that  3 + V 2  and V 3 +  2 V belong to the cone K and therefore Thus Since (, V)  ∈ K ∩ B  (0), Due to max{(0), V(0 Since 0 <  ≤ 1, thus we arrive at which contradicts the selection of . (b) Suppose that there are  = (, V)  ∈ K ∩ B  (0) and  ≥ 0 such that where  = (1, 1)  .Therefore As a consequence, Let us define, for  = 1, 2, Therefore Analogously, we obtain that Substituting these the inequalities above into (79), it follows that We conclude that ∫ 1 0 () 3  and ∫ 1 0 V() 3  are bounded by a constant for all (, V) ∈ K ∩ B  (0), satisfying (77).In consequence, ∫ 1 0 V() 2 () and ∫ 1 0 () 2 V() are also bounded by a constant for all (, V) ∈ K ∩ B  (0), satisfying (77).
On the other hand, from (78) In order to bound the right-hand side of the inequalities above, observe that is periodic and monotone decreasing on [0, 1] and belongs to  1 (0, 1).Therefore, and thus where K2 is the periodic function defined by with Introducing these results into (84), we arrive at From ( 83), we get that  ≤ (∫ Thus if we select  sufficiently close to 1, such that we get that max { (0) , We conclude that which is a contradiction.
Theorem 8. Suppose the same hypothesis on the coefficients , , , ,  1 ,  2 , and  and  as in Lemma 7. Furthermore, assume that system (18) does not have constant solutions (ũ, Ṽ) ̸ = (0, 0).Then the operator A has a nontrivial (i.e., ũ or Ṽ is not constant) fixed point in the cone K. Equivalently, there exists a nontrivial solution of system (18).Moreover, the fixed point index of the operator A on Proof.Observe that (0, 0) is a trivial solution of system (18).
However (0, 0) does not belong to the annular region K  .Thus, Lemmas 6 and 7 and Theorem 3 guarantee the existence of a nontrivial fixed point (i.e., ũ or Ṽ is not constant) of the operator A in the annular region

Numerical Results
As mentioned above, explicit solutions of full system (1) are not known for general coefficients (), (), (), (), and ().In this section we will describe a numerical strategy to compute approximations of vector solitons (, V) of this system in the form or  (, ) =   1 −(/2) ũ () , V (, ) =   2 +(/2) Ṽ () , where ũ and Ṽ are positive, even, and real solutions of the system The existence of such solutions was already established in the previous section.
To approximate these solutions, following the same collocation-spectral strategy by Pipicano and Muñoz Grajales in [49], let us introduce truncated 2-periodic cosine expansions for ũ and Ṽ: Real and imaginary parts of the vector soliton (, V) of system (1) at  = 20.Solid line: numerical solution (, V) of system (1) using the numerical scheme in [37].Pointed line: expected soliton computed using (108).
and let us introduce analogous expressions for the coefficients Ṽ .This strategy can be used for approximating solutions decaying to zero at infinity, provided that the period 2 is taken large enough.By substituting expressions (100) into (99), evaluating them at the /2 + 1 collocation points we obtain a system of  + 2 nonlinear equations in the form  (ũ 0 , ũ1 , . . ., ũ/2 , Ṽ0 , Ṽ1 , . . ., Ṽ/2 ) = 0, where  + 2 coefficients ũ and Ṽ are the unknowns.Nonlinear system (103) can be solved by Newton's iteration.Computation of the cosine series in (100) and the integrals in (101) is performed using the FFT (Fast Fourier Transform) algorithm.The Jacobian of the vector field  : R +2 → R +2 is approximated by the second-order accurate formula where   = (0, . . ., 1, . . ., 0) and ℎ = 0.01.We stop the iteration procedure when the relative error between two successive approximations and the value of the vector field  are smaller than 10 −12 .Figure 5: Real and imaginary parts of the vector soliton (, V) of system (1) at  = 20.Solid line: numerical solution (, V) of system (1) using the numerical scheme in [37].Pointed line: expected soliton computed using (111).

Description of the
The resulting approximations of (ũ, Ṽ) after 10 iterations are presented in Figure 1.To verify that we have computed really a vector soliton of system (1), we run the numerical solver to approximate the solutions to system (1), introduced by Muñoz Grajales and Quiceno in [37], with stepsize Δ = 10 −3 ,  = 2 9 FFT points, using as initial data  (0, ) = ũ () , The result of this computer simulation at  = 20 is presented in Figure 2, superimposed with the expected position of the vector soliton given by  (, ) =   ũ () , We observe a good accordance between the profiles of the components  and V, corroborating that, in fact, we have an approximation of a vector soliton to system (1).
We repeat the numerical experiments above with the same model and numerical parameters, but now for Gaussian-type coefficients, The results are presented in Figure 3.
Experiment Set 2 (nonzero inverse group velocity).We now set  = 1,  1 =  2 = 0, and  1 =  2 = 1.5.Thus  1 =  2 = 0.5.We point out that, in this case,  ̸ = 0, in contrast to the simulations above.The model's coefficients in system (18)  The resulting approximation to the pair (ũ, Ṽ) obtained from Newton's procedure is presented in Figure 4 and the verification of the characteristics of a soliton solution is The results are presented in Figure 6.In all cases, we performed a verification of the approximate soliton solution computed using the numerical scheme developed in [37] for solving coupled system (1).The error observed at  = 20 was around 10 −3 .
Experiment Set 3 (effect of inhomogeneous coefficients).The next step is to analyze the effect of the model's coefficients on the geometry of the computed vector solitons of (18).In the simulations presented in Figure 7 parameters are left unchanged.Observe that the amplitude of the approximate solution decreases when  → 0.
On the other hand, in Figure 8, we let () = () = () = () =  sech( −  0 ) and compute the solution of system (18) for different values of the amplitude .Other parameters are the same as in the previous experiment.
However, for  = 2, 4, we have to start Newton's iteration with the following data:  in order to achieve convergence.From these results, we have numerical evidence on the fact that the amplitude of the solutions of system (18) decreases as the amplitude of the coefficients (), (), (), and () increases.
Experiment Set 4 (single-mode solitons).When V ≡ 0 or  ≡ 0, the cross-modulation terms in system (1) are zero, obtaining that  or V, respectively, satisfy a scalar Schrödinger equation.The single-mode solitons computed are displayed in Figure 9 for the same model and numerical parameters as in the Experiment Set 1.
Experiment Set 5 (influence of the inhomogeneous coefficient ()).In Figure 10, we illustrate the effect of the variable coefficient () on the shape of the vector solitons of system In solid line, we display the vector soliton computed using  = 0, that is,  ≡ 0, superimposed with the profiles obtained for nonzero values of this parameter with  = 3 and  = 10.Observe that the intensity of the extrainhomogeneous nonlinearities in system (1), produced by the coefficient (), affect significantly the amplitude of the vector soliton obtained.

Conclusions
In this paper, using the theory of positive operators in a cone in a Fréchet space [41,42], we established the existence of vector solitons of the system of coupled Schrödinger equations (1).To achieve this, we extended the techniques in [43] for the case of a family of scalar dispersive equations.We further illustrated the geometry of these solutions by approximating Advances in Mathematical Physics  them through a numerical solver, which involves a Newtontype iteration together with a spectral discretization for the spatial variable.Our numerical simulations give evidence on the existence of vector solitons for the two parameter regimes analyzed, in accordance with the theory presented.Further research is needed to study other issues of great interest, such as orbital stability under small initial disturbances of vector solitons and relationship between periodic and nonperiodic finite-energy solutions of system (1).

( 3 )
⊂  is open in the relative topology on , A is continuous and -compact, (4) there are no fixed points of A on , the boundary of the open set  in the relative topology on .

Figure 13 :Figure 14 :
Figure 13: Contour plot of the collision of solitons in Figure 11.

Figure 17 :
Figure 17: Contour plot of the collision of solitons in Figure 15.

Figure 18 :
Figure 18: Contour plot of the collision of solitons in Figure 16.