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Vector solitons of a coupled Schrodinger system with variable coefficients.

1. Introduction

Several physical processes related to wave motion can be described using systems of coupled nonlinear Schrodinger (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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where x [member of] R and [xi] > 0. Here u* denotes the complex conjugate of u. We suppose that the coefficients a, b, c, d, e are positive and depend only on the variable x in order to focus on the influence of x-nonlinear modulations on the multicomponent solitons of the system. Furthermore K, [delta] 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 x, [xi] in system (1) denote time and space, respectively, [delta] means the normalized strength of the linear birefringence (2[delta] is the inverse group velocity difference), and the model's parameters are usually constant or [xi]-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 [delta] = 0 and e [equivalent to] 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 x, [xi] denote space and retarded time, respectively, the functions u = m([xi], x), v = v([xi], x) denote the complex envelopes of the propagating beam of the two modes, [[sigma].sub.1] and [[sigma].sub.2] are the external potentials, K > 0 means the group velocity dispersion coefficient, and the physical parameters a, b, c, and d (depending in this case on both [xi] and x) 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 (u, v) of system (1) with 8 = 0 in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

and in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

for the case e(x) = 0, where w and U are positive real functions and [alpha], [[alpha].sub.1], and [[alpha].sub.2] are real constants.

It is import ant to point out that exact solutions of syst em (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 e [equivalent to] 0 and [delta] [equivalent to] 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 a, b, c, and d have compact support and e [equivalent to] 0 and 8 = 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 e = 0 and [delta] = 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 x-dependent coefficient e(x), and a nonzero value of the parameter [delta]. 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, Munoz Grajales and Quiceno [37] also illustrated the effect of these extra nonlinear terms and the parameter [delta] 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 [v.sup.2]u* and [u.sup.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 Frechet 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 Freechet 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 Frechet space, following the papers by Benjamin et al. [43] and Chen et al. [44, 45]. We must recall that a Frechet space X is a metrizable and complete, locally-convex, linear topological space (over the real numbers). On X a sequence [([p.sub.n]).sub.n] of seminorms can be defined in such a way that [p.sub.n+1] (x) [greater than or equal to] [p.sub.n](x) for every x [member of] X and everyn= 1, 2, 3, ... and that the formula

d{x, y) = [[infinity].summation over (j=1)][1/[2.sup.j]] ([[[p.sub.j](x-y)]/[1 + [p.sub.j](x - y)]]), x, y [member of] X, (4)

provides a metric that generates a topology that coincides with the original topology on X. In this case, we say that X is a Frechet space with generating family of seminorms [([p.sub.n]).sub.n]. Hereafter, we use the notation

[B.sub.r] = {x [member of] X: d (x, 0) < r}, [B.sup.j.sub.r] = {x [member of] X: [p.sub.j] (x) < r}, j [member of] N. (5)

It is clear from (4) that we have that X = [B.sub.1]. In general, a set B in a topological linear space X is said to be bounded if, for any neighborhood U of 0 in X, there is [lambda] > 0 such that [lambda]B [subset] U. In the case of a Frechet space with metric d given by (4), a set B in X is bounded if and only if corresponding to each positive integer j there is R > 0 such that B [subset] [B.sup.j.sub.R]. If r > 0, then [B.sub.r] is usually not bounded (for details see [46]). A closed subset K of a Frechet space X is a cone if the following conditions hold true:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

From (6), K must be convex. On the other hand, we also have a partial ordering on K given by

x [??] y [left right arrow] y - x [member of] K. (7)

For any 0 < r < R < [infinity], let us denote

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

An operator A defined on K is said to be positive, if A(K) [subset] K. On the other hand, we say that a positive operator A on K is K-compact, if the set A([K.sub.r]) has a compact closure, for each 0 [less than or equal to] r < 1. A triplet (K, A, U) is said to be admissible, if

(1) K is a convex subset of X,

(2) U [subset] K is open in the relative topology on K,

(3) A is continuous and K-compact,

(4) there are no fixed points of A on [partial derivative]U, the boundary of the open set U in the relative topology on K.

From Granas' work [47], there is an integer-valued function i(K, A, U) that satisfies the basic axioms of a fixed point index. Among them, we consider the following ones:

(i) Homotopy Invariant. If (K, A, U) and (K, B, U) are two admissible triplets and the operator A is homotopic to the operator B on U, then i(K, A, U) = i(K, B, U).

(ii) The Fixed Point Property. If (K, A, U) is admissible and i(K, A, U) [not equal to] 0, then A has at least one fixed point in U.

(iii) Index of Constant Maps. If (K, A, U) is admissible and A is constant (i.e., there is a point a [member of] K such that Au = a for all u [member of] K), then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

We refer the reader to [43] (see also [41, 42, 47]) for details in the following results. It is assumed throughout that K is a cone in a Frechet space with generating family of seminorms [([p.sub.j]).sub.j] and the standard metric d as in (4) and that A: K [right arrow] K is continuous, positive, and K-compact.

Lemma 1. Suppose that 0 < [rho] < 1 and either

Ax - x [??] K, [for all]x [member of] [partial derivative][K.sub.[rho]] or (10)

tAx [not equal to] x [for all]x [member of] [partial derivative][K.sub.[rho]] and all t [member of] [0, 1]. (11)

Then one has that (K, A, [K.sub.[rho]]) is admissible and i(K, A, [K.sub.[rho]]) = 1.

Lemma 2. Suppose that 0 < [rho] < 1 and either

x- Ax [??] K, [for all]x [member of] [partial derivative][K.sub.[rho]] or (12)

There exists [??] [not equal to] 0 in K such that x - Ax [not equal to] [??][??] for x [member of] [partial derivative][K.sub.[rho]], [lambda] [greater than or equal to] 0. (13)

Then (K, A, [K.sub.[rho]]) is admissible and i(K, A, [K.sub.[rho]]) = 0.

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

Theorem 3. If either (10) or (11) holds for r satisfying 0 <r < 1 and either (12) or (13) holds for R satisfying 0 < r < R < 1, then A has at least one fixed point in [K.sup.R.sub.r]. Moreover, i(K, A, [K.sup.R.sub.r]) = -1.

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

Theorem 4 (Galilean invariance). Let [delta] = 0 and let a, b, c, d, e, [[sigma].sub.1], [[sigma].sub.2], and K be constants. If u([xi], x) and v([xi], x) are a solution to system (1), then another solution is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

where the velocity V is given by

V = 2[alpha]K. (15)

Furthermore, the solutions of system (1) are indifferent to multiplication by [e.sup.i[PHI]], for any constant [PHI].

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 (u, v) of system (1) for [delta] = 0 in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

and in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

for the case e(x) [equivalent to] 0. Here [??] and [??] are positive real functions, and [alpha], [[alpha].sub.1], and [[alpha].sub.2] are real constants. In each case, we see that the functions [??] and [??] must satisfy a system in the form (soliton equations)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

where the coefficients A(x), B(x), C(x), D(x), [[beta].sub.1], and [[beta].sub.2] are defined in the case that [delta] = 0 by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

On the other hand, when e [equivalent to] 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

We assume that the coefficients a(x), b(x), c(x), d(x), and e(x) in system (1) are bounded, continuous, positive, even, and nonincreasing for x [greater than or equal to] 0, K > 0, the parameters [delta], [[sigma].sub.1], [[sigma].sub.2], [alpha], [[alpha].sub.1], and [[alpha].sub.2] are such that the coefficients A(x), B(x), C(x), and D(x) are positive and bounded, and [[beta].sub.1] and [[beta].sub.2] are positive.

Abandoning the tildes, we see that to show the existence of a solution U = [(u, v).sup.t] of (18) is equivalent to establish the existence of a solution of the fixed point equation:

U = A (U), (21)

where the operator A is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

Let us denote by [??] the Fourier transform of the function f. We define the functions [K.sub.1] and [K.sub.2] as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII](23)

Thus [[??].sub.1] and [[??].sub.2] are positive, even, and monotone decreasing on (0, [infinity]) and belong to [L.sup.1](R). Furthermore for i = 1, 2,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)

Therefore [K.sub.1], [K.sub.2] [member of] C(R) [intersection] [L.sup.1](R).

Then we can rewrite the operator A as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

where f * g denotes the convolution between the functions f and g.

Hereafter, we consider the space of real valued continuous functions defined on R (denoted by C(R)), with the topology of uniform convergence on bounded intervals under the seminorms

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

In this case, the distance is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)

The open ball of radius r < 1 centered at zero and its boundary are given, respectively, by

[B.sub.r](0) = {u [member of] C(R): d(0, u) < r}, [partial derivative][B.sub.r] (0) = {u [member of] C(R): d (0, u) = r}. (28)

Let K [subset] C(R) be the cone defined as

K = {w [member of] C(R): w(x) = w (-x) [greater than or equal to] 0; w is nonincreasing for x [greater than or equal to] 0}. (29)

Note the that, for j = 1, 2, 3 ..., we have for all w [member of] K that

[p.sub.j](w) = [p.sub.1](w), (30)

and so we have for 0 < r < 1 that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (31)

We note that the product space C(R) x C(R) is also a Frechet space with generating family of seminorms

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (32)

We also define the cone [??] [subset] C(R) x C(R) by [??] = K x K.

The metric in the product space C(R) x C(R) is defined by

[??]((f, g), (u, v)) = [[infinity].summation over (j=1)][2.sup.-j][[[P.sub.j](f - u, g - v)]/[1 + [P.sub.j](f - u, g - c)]]. (33)

In particular, for (f, g) [member of] [??],

[??]((f, g), 0) = [[infinity].summation over (j=1)][2.sup.-j] [[[P.sub.j](f, g)]/[1 + [P.sub.j](f, g)]] = [[max {f(0), g(0)}]/[ 1+ max {f(0), 0(0)K]]. (34)

We also set

[B.sub.r](0) = {(f, g)[member of] C(R) x C(R): [??]((f, g), 0) < r}, (35)

and its boundary

[partial derivative] [B.sub.r](0) = {(f, g)[member of] C(R) x C(R): [??]((f, g), 0) = r}. (36)

We observe that if (f, g) [member of] [??] [intersection] [partial derivative][B.sub.r](0), that is, [??]((f, g), 0) = r, then we have that

max {f(0), g(0)} = [r/[1 - r]]. (37)

We denote by [[??].sup.R.sub.r] the annular section of the cone [??]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (38)

Hereafter, according to the notation introduced in the previous section, we set, for r > 0, the convex set

[[??].sub.r] = [??] [intersection] [B.sub.r](0). (39)

4. Main Results

Before we go further, we establish a general result.

Lemma 5. Let k [member of] C(R) [L.sup.1](R) be an even bounded positive function on R, which is monotone decreasing on (0, [infinity]). Then the operator B defined by

B(f)(x) = (k * f)(x)= [[integral].sub.R] k(x - r) f(r)dr (40)

maps K into K.

Proof. We first note that B(f)(x) is bounded due to the fact that k [member of] [L.sup.1](R) and also from the Young inequality, since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (41)

Now, we also have that B(f) [greater than or equal to] 0 for f [member of] K. In fact,

B(f)(x) = [[integral].sub.R] k(x-y)f(y)dy [greater than or equal to] 0, (42)

since k [greater than or equal to] 0 and f [greater than or equal to] 0. On the other hand, B(f) is also an even function for f [member of] K. In fact,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (43)

We claim now that B(f) for f [member of] K is a continuous function on R. In fact, first note that 0 [less than or equal to] f(y) [less than or equal to] f(0) for any y [member of] R:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (44)

Using k [member of] [L.sup.1](R) and the dominated convergence theorem, we conclude that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (45)

meaning that B(f) is a continuous function on R, as long as f [member of] K. Finally, we need to establish that B(f) is a nonincreasing function for x [greater than or equal to] 0, for f [member of] K. So let f [member of] K be fixed and take x [greater than or equal to] 0 and h > 0. Then, we have for any r [member of] R that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (46)

and so we have that

B(f)(x + h) = [[integral].sup.[infinity].sub.0] k(x + h- z-r)f(z + r)dz + [[integral].sup.[infinity].sub.0] k(x + h + z - r) f (z - r) dz. (47)

Using r = -h/2 in the first formula and r = h/2 in the second one, we get that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (48)

Now, we note that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (49)

since z [greater than or equal to] x + (1/2)h [greater than or equal to] (1/2)h, and the fact that k and f are nonincreasing for w [greater than or equal to] 0. Now, for the rest of the integral, we use a similar argument, after noting that k and f are even functions. In fact, first note for z > 0 that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (50)

since f is an even nonincreasing function for w [greater than or equal to] 0. So, from this fact, we have that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (51)

since 0 [less than or equal to] z [less than or equal to] x + (1/2)h, and the fact that k is an even nonincreasing function for w [greater than or equal to] 0. In other words, we have shown that

B (f)(x) - B (f) (x + h) [greater than or equal to] 0, (52)

for any x [greater than or equal to] 0 and h > 0, which means that B(f)(x) is a nonincreasing function for x [greater than or equal to] 0.

Lemma 6. Suppose that the coefficients A(x), B(x), C(x), and D(x) in system (18) are continuous bounded positive nonzero functions in R and [[beta].sub.1] > 0, [[beta].sub.2] > 0. Furthermore, assume that these coefficients are even and nonincreasing for x [greater than or equal to] 0. Then the operator A defined by (25) maps continuously [??] into [??]. For each 0 < r < 1, the set A([[??].sub.r]) is a relative compact subset of [??].

Proof. As we mention above, [K.sub.1], [K.sub.2] [member of] [L.sup.1](R) are positive functions and also from the Young inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (53)

If we set [B.sub.i](f) = [K.sub.i] * f, then from Lemma 5, we have that [B.sub.i](f) [member of] K for any f [member of] K. In other words, [B.sub.i](f) is an even, continuous, positive, bounded function on R. Further, [B.sub.i](f) is a nonincreasing function on (0, [infinity]).

Now, let (u, v) [member of] [??], then we have that A[u.sup.3] + B[v.sup.2]u and C[v.sup.3] + D[u.sup.2]v belong to the cone K, which means from Lemma 5 that

[A.sub.1] (u, v) [greater than or equal to] 0, [A.sub.2] (U, v) [greater than or equal to] 0 (54)

and also that A(u, v) for (u, v) [member of] [??] is continuous on R. Moreover, we also have that, [A.sub.1](u, v) and [A.sub.2] (u, v) are non-increasing for x [greater than or equal to] 0, for all (u, v) [member of] [??]. In other words, we have established that A([??]) [subset] [??].

Finally, we want to prove that the operator A = ([A.sub.1], [A.sub.2]) maps [??] continuously to [??]. To see this, we must recall that convergence in (C(R) x C(R), [??]) is equivalent to uniform convergence on closed bounded intervals I [subset] R. Assume that ([u.sub.n], [v.sub.n]) [right arrow] (u, v) in (C(R) x C(R), [??]), as n [right arrow] [infinity]. Let I = [a, b] [subset] R bea fixed closed bounded interval. For [member of] > 0 given, we choose [OMEGA] > 0 sufficiently large such that, for i = 1, 2,

[[integral].sup.b-[OMEGA].sub.-[infinity]][K.sub.i](y)dy + [[integral].sup.[infinity].sub.a + [OMEGA]][K.sub.i](y)dr < [member of]. (55)

Using this fact, we see easily for i = 1, 2,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (56)

To analyze the continuity of the first component of the operator A, we decompose the integral on R as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (57)

Observe that {(u, v), ([u.sub.n], [v.sub.n]), n [member of] N} is a bounded set in (C(R) x C(R), d). Thus there exists a constant [bar.C] > 0 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (58)

Let us denote

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (59)

Therefore, for x [member of] [a, b] and [OMEGA] large enough,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (60)

On the other hand, using the factorizations

[u.sup.3.sub.n] - [u.sup.3] = ([u.sub.n] - u) ([u.sup.2.sub.n] + [u.sup.2.sub.n] + [u.sub.n]u + [u.sup.3]), [v.sup.2.sub.n][u.sub.n] - [v.sup.2]u = ([v.sub.n] - v) ([v.sub.n] + v)[u.sub.n] + [v.sup.2]([u.sub.n] - u), (61)

we have that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (62)

Due to ([u.sub.n], [v.sub.n]) [right arrow] (u, v) in (C(R) x C(R), [??]) as n [right arrow] [infinity], given [member of] > 0, there exists [n.sub.0] [member of] N such that if n [greater than or equal to] [n.sub.0],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (63)

Therefore, it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (64)

Substituting the results above into inequality (57), it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (65)

for n [greater than or equal to] [n.sub.0]. Analogously,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (66)

for n [greater than or equal to] [n.sub.0]. We conclude that the operator A maps [??] continuously to K.

It remains proving that A([[??].sub.r]) (0 < r < 1) is relative compact subset of [??], which is equivalent that [A.sub.i]([[??].sub.r]) is relative compact subset of [??], for i = 1, 2. To see this, we use the Arzela-Ascoli Theorem to establish the compactness of the families [M.sub.i] in C(R),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (67)

Let (u, v) [member of] [[??].sub.r] be such that w = [B.sub.i]([f.sub.i]) with [f.sub.1] = A[u.sup.3] + B[v.sup.2]u and [f.sub.2] = C[v.sup.3] + D[u.sup.2]v. We first note that, for [f.sup.i] [member of] C(R), we have that [B.sub.i]([f.sup.i]) : R [right arrow] R is a continuous function such that (see Lemma 5)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (68)

as h [right arrow] 0 (uniformly in h), meaning that [B.sub.i](f) is equicontinuous in R due to uniformity of the last estimates in h. Moreover, the families [M.sub.i] are equicontinuous in R, since [B.sub.i]([f.sub.i]) are equicontinuous in R and the uniform estimate for f(0) in (31). On the other hand, for each x [member of] R, the set

[M.sub.i](x) = {w(x): w [member of] [M.sub.i]} (69)

has a compact closure in R since [M.sub.i](x) [subset] [0, [C.sub.i](r)] for any x [member of] R. Here [C.sub.i](r), i = 1, 2, are constants which depend only on r. From the Arzela-Ascoli Theorem, the families [M.sub.i] are normal (see Theorem 1.23, chapter VII in [48]). In other words, we have shown that the set A([[??].sub.r]) is a relative compact subset of [??].

Lemma 7. Suppose the same hypothesis in Lemma 6. Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (70)

and let R, r be constants such that 0 < r < 1/(1+ [square root of (2M)]) < R < 1, and let R be sufficiently close to 1. Then

(a) U [not equal to] tA(U) for each U [member of] [??] [intersection] [partial derivative][B.sub.r](0), and t [member of] [0, 1],

(b) U - A(U) [not equal to] aV for each U [member of] [??] [intersection] [partial derivative][B.sub.R](0), and a [greater than or equal to] 0, where V is the constant function given by V = [(1, 1).sup.t].

Proof. (a) Suppose that there exist U = [(u, v).sup.t] [member of] [??] [intersection] [partial derivative][B.sub.r](0) and t [member of] [0, 1] such that [??](U, 0) = r and

U = tA (U). (71)

Clearly, if t = 0, we have that U = 0 [??] [??] [intersection] [partial derivative][B.sub.r](0) which is a contradiction. Suppose that t [member of] (0, 1]. Then we have that A[u.sup.3] + B[v.sup.2]u and C[v.sup.3] + D[u.sup.2]v belong to the cone K and therefore

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (72)

Thus

[1/t]u(0) [less than or equal to] A(0)u[(0).sup.3] + B (0) V [(0).sup.2] u(0), 1 (73)

[1/t]v (0) [less than or equal to] C(0) v[(0).sup.3] + D(0)u[(0).sup.2]v(0).

Since [(u, v).sup.t] [member of] [??] [intersection] [partial derivative][B.sub.r](0),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (74)

Due to max{u(0), v(0)} = r/(1 - r),

[1/t]([r/[1 - r]]) [less than or equal to] 2M[([r/[1 - r]]).sup.3], [([[1 - r]/r]).sup.2] [less than or equal to] 2Mt. (75)

Since 0 < t [less than or equal to] 1, thus we arrive at

r [greater than or equal to] [1/[1 + [square root of (2Mt)]]] [greater than or equal to] [1/[1 + [square root of (2M)]]], (76)

which contradicts the selection of r.

(b) Suppose that there are U = [(u, v).sup.t] [member of] [??] [intersection] [partial derivative][B.sub.R](0) and a [greater than or equal to] 0 such that

U - A (U) = aV, (77)

where V = [(1, 1).sup.t]. Therefore

u(x) = a + [K.sub.1] * (A[u.sup.3] + B[v.sup.2]u), v(x) = a + [K.sub.2] * (C[v.sup.3] + D[u.sup.2]v). (78)

As a consequence,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (79)

Let us define, for i = 1, 2,

[k.sub.i] = [[integral].sup.2.sub.0][K.sup.i](z)dz. (80)

Therefore

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (81)

Analogously, we obtain that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (82)

Substituting these the inequalities above into (79), it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (83)

We conclude that [[integral].sup.1.sub.0] u[(y).sup.3]dy and [[integral].sup.1.sub.0] v[(y).sup.3]dy are bounded by a constant for all (u, v) [member of] [??] [intersection] [partial derivative][B.sub.R](0), satisfying (77). In consequence, [[integral].sup.1.sub.0] v[(y).sup.2]u(y)dy and [[integral].sup.1.sub.0] u[(y).sup.2]v(y)dy are also bounded by a constant for all (u, v) [member of] [??] [intersection] [partial derivative][B.sub.R](0), satisfying (77).

On the other hand, from (78)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (84)

In order to bound the right-hand side of the inequalities

above, observe that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (85)

Now, since [K.sub.1] [member of] [L.sup.1](R) [intersection] [L.sup.[infinity]](R) is monotone decreasing on [0, [infinity]), the function defined by

[[??].sub.1] (x) := [[infinity].summation over (j=-[infinity])] [K.sub.1] (x + 2j) (86)

is periodic and monotone decreasing on [0, 1] and belongs to [L.sup.1](0, 1). Therefore,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (87)

and thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (88)

Analogously,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (90)

where [[??].sub.2] is the periodic function defined by

[[??].sub.2](x) := [[infinity].summation over (j=-[infinity])][K.sub.2] (x + 2j), (90)

with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (91)

Introducing these results into (84), we arrive at

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (92)

From (83), we get that a [less than or equal to] [([[integral].sup.1.sub.0] u[(y).sup.3]dy).sup.1/3] and a [less than or equal to] [([[integral].sup.1.sub.0] v[(y).sup.3]dy).sup.1/3]. Therefore

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (93)

Thus if we select R sufficiently close to 1, such that

[[max {k, [mu]}]/[1 + max{k, [mu]}]] < R < 1, (94)

we get that

max {u(0), v(0)} < max {K, [mu]} < [R/[1 - R]]. (95)

We conclude that

[??]((u, v), 0) = [[max {u(0), v (0)}]/[1 + max{u(0), v(0)}]] < R, (96)

which is a contradiction.

Theorem 8. Suppose the same hypothesis on the coefficients A, B, C, D, [[beta].sub.1], [[beta].sub.2], and r and R as in Lemma 7. Furthermore, assume that system (18) does not have constant solutions ([??], [??]) [not equal to] (0, 0). Then the operator A has a nontrivial (i.e., u or [??] is not constant) fixed point in the cone [??]. Equivalently, there exists a nontrivial solution of system (18). Moreover, the fixed point index of the operator A on [[??].sup.R.sub.r] is i([??], A, [[??].sup.R.sub.r]) = -1.

Proof. Observe that (0, 0) is a trivial solution of system (18).

However (0, 0) does not belong to the annular region [[??].sup.R.sub.r]. Thus, Lemmas 6 and 7 and Theorem 3 guarantee the existence of a nontrivial fixed point (i.e., [??] or [??] is not constant) of the operator A in the annular region [[??].sup.R.sub.r] = {U [member of] K: r < [??](U, 0) < R}[subset] K.

5. Numerical Results

As mentioned above, explicit solutions of full system (1) are not known for general coefficients a(x), b(x), c(x), d(x), and e(x). In this section we will describe a numerical strategy to compute approximations of vector solitons (u, v) of this system in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (97)

or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (98)

where [??] and [??] are positive, even, and real solutions of the system

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (99)

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 Munoz Grajales in [49], let us introduce truncated 2/-periodic cosine expansions for [??] and [??]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (100)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (101)

and let us introduce analogous expressions for the coefficients [[??].sub.n]. This strategy can be used for approximating solutions decaying to zero at infinity, provided that the period 21 is taken large enough.

By substituting expressions (100) into (99), evaluating them at the N/2 + 1 collocation points

[x.sub.j] = [[2l(j - 1)]/N], j=1, ..., [N/2] + 1, (102)

we obtain a system of N + 2 nonlinear equations in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (103)

where N + 2 coefficients [[??].sub.n] and [[??].sub.n] 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 F : [R.sup.N+2] [right arrow] [R.sup.N+2] is approximated by the second-order accurate formula

[J.sub.i, j]F(x) [approximately equal to] [[F(x + h[e.sub.j]) - [F.sub.i](x - h[e.sub.j])]/[2h]] j=1, ..., N + 2, (104)

where [e.sub.j] = (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 F are smaller than [10.sup.-12].

5.1. Description of the Numerical Experiments. In all numerical experiments the number of FFT points is N = [2.sup.9] and l = 35. Thus the computational domain of the variable x is [0, 70].

Experiment Set 1 (vanishing inverse group velocity). We will first consider the regime where [delta] = 0, [[sigma].sub.1] = 0.5, [[sigma].sub.2] = -0.5, [alpha] = 1.5, and K = 0.5. Therefore [[beta].sub.1] = 0.5 and [[beta].sub.2] = 0.25. The coefficients in system (1) are taken as

a(x) = b (x) = c(x) = d(x) = e (x) = [sech.sup.p] (x - [x.sub.0]), (105)

with p = 3 and [x.sub.0] = 35. Note that the parameter [x.sub.0] regulates the position of the irregular section of the coefficients a(x), b(x), c(x), d(x), and e(x). The initial values for the Newton's iteration are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (106)

The resulting approximations of ([??], [??]) 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 Munoz Grajales and Quiceno in [37], with stepsize [DELTA][xi] = [10.sup.-3], N = [2.sup.9] FFT points, using as initial data

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (107)

The result of this computer simulation at [xi] = 20 is presented in Figure 2, superimposed with the expected position of the vector soliton given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (108)

We observe a good accordance between the profiles of the components u 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,

a(x) = b (x) = c(x) = d(x) = e (x) = exp (-0.5 [(x - [x.sub.0]).sup.2]). (109)

The results are presented in Figure 3.

Experiment Set 2 (nonzero inverse group velocity). We now set [delta] = 1, [[sigma].sub.1] = [[sigma].sub.2] = 0, and [[alpha].sub.1] = [[alpha].sub.2] = 1.5. Thus [[beta].sub.1] = [[beta].sub.2] = 0.5. We point out that, in this case, [delta] [not equal to] 0, in contrast to the simulations above. The model's coefficients in system (18) are taken as

a(x) = b (x) = c(x) = d (x) = [sech.sup.p] (x - [x.sub.0]), e(x) = 0. (110)

The resulting approximation to the pair ([??], [??]) obtained from Newton's procedure is presented in Figure 4 and the verification of the characteristics of a soliton solution is displayed in Figure 5. Note that, within this parameter regime, the expected soliton solution is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (111)

We repeat the previous numerical experiment with the same model and numerical parameters but instead use the coefficients

a(x) = b (x) = c(x) = d(x) = e (x) = exp (-0.5 [(x - [x.sub.0]).sup.2]). (112)

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 [xi] = 20 was around [10.sup.-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, we set e [equivalent to] 0, [delta] = 1, [[alpha].sub.1] = [[alpha].sub.2] = 1.5, [[sigma].sub.1] = [[sigma].sub.2] = 0, K = 0.5, [x.sub.0] = 35, and a(x) = b(x) = c(x) = d(x) = [sech.sup.p](x - [x.sub.0]). The other numerical parameters are left unchanged. Observe that the amplitude of the approximate solution decreases when p [right arrow] 0.

On the other hand, in Figure 8, we let a(x) = b(x) = c(x) = d(x) = Asech(x - [x.sub.0]) and compute the solution of system (18) for different values of the amplitude A. Other parameters are the same as in the previous experiment. However, for A = 2, 4, we have to start Newton's iteration with the following data:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (113)

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 a(x), b(x), c(x), and d(x) increases.

Experiment Set 4 (single-mode solitons). When v [equivalent to] 0 or u [equivalent to] 0, the cross-modulation terms in system (1) are zero, obtaining that u or v, respectively, satisfy a scalar Schrodinger 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 e(x)). In Figure 10, we illustrate the effect of the variable coefficient e(x) on the shape of the vector solitons of system (1). The model's parameters are K = 0.5, [alpha] = 1.5, [[sigma].sub.1] = 0.5, [[sigma].sub.2] = -0.5, [delta] = 0, and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (114)

with p = 3 and [x.sub.0] = 35. The initial values for the Newton iteration in this case are taken as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (115)

In solid line, we display the vector soliton computed using A = 0, that is, e [equivalent to] 0, superimposed with the profiles obtained for nonzero values of this parameter with A = 3 and A = 10. Observe that the intensity of the extrainhomogeneous nonlinearities in system (1), produced by the coefficient e(x), affect significantly the amplitude of the vector soliton obtained.

Experiment Set 6 (interaction of vector solitons). We are interested in illustrating the collision of coupled-mode waves in the regime where [delta] = 0 and all model's coefficients are constants. By using the Galilean invariance of system (1) (see Theorem 4), we can write the vector solitons in a more general form, by considering a nonzero translational velocity. Thus, we run again the numerical solver in [37] with an initial profile consisting of two well-separated solitons:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (116)

where the initial separation between the two coupled-mode solitons with frequencies [V.sub.i]/2K, i = 1, 2, is [x.sub.0] = 15, and [V.sub.1] = 1 and [V.sub.2] = -1 are the differing velocities of the two solitary waves produced by this initial condition. Furthermore, [[phi].sub.1] and [[phi].sub.2] are the initial phases of the left-going pulses. Here [??] and [??] are a coupled-mode solution of (18) obtained by using Newton's iteration with K = 0.5, [[sigma].sup.1] = [[sigma].sup.2] = 0, [alpha] = 1.5, a = c = 1, b = d = 2/3, e = 1/3, andinitialvalues [u.sub.0](x) = sech([xi]-35) and [v.sub.0](x) = sech([xi] - 35). The numerical parameters for computing the evolution of these pulses with respect to the variable [xi] are L = 70 (length of the spatial computational domain), [DELTA]t = 1e - 3 (time step size), and N = [2.sup.10] (number of FFT points). In Figures 11 and 12 we see the collision of the coupled-mode solitons given in (116) with [[phi].sub.1] = [[phi].sub.2] = 0, and in Figures 13 and 14 are the corresponding contour plots. We observe that the two solitons propagate with the expected velocities [V.sub.1] = 1 and [V.sub.2] = -1, and they elastically (lose little energy) pass through one another; the emergent solitons appear to be unaltered (with approximately the same amplitude and velocity) after the collision. In Figures 15, 16, 17, and 18, we repeat the previous numerical simulation to illustrate the collision of the coupled-mode solitons given in (116) when the initial phases are changed to [[phi].sub.1] = [pi] and [[phi].sub.2] = 0. As in the previous computer simulation, the solitons cross each other, but now some exchange of energy is present between the two coupled components [absolute value of (u([xi], x))], [absolute value of (v([xi], x))]. As a consequence, we find that they does not maintain their amplitudes after the collision.

6. Conclusions

In this paper, using the theory of positive operators in a cone in a Frechet space [41, 42], we established the existence of vector solitons of the system of coupled Schrodinger 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 them through a numerical solver, which involves a Newton-type 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).

http://dx.doi.org/10.1155/2016/5787508

Competing Interests

The author declares that there are no competing interests regarding the publication of this paper.

Acknowledgments

This research was supported by Universidad del Valle, Calle 13, No. 100-00, Cali, Colombia, under the research project C.I. 71020.

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Juan Carlos Munoz Grajales

Departamento de Matematicas, Universidad del Valle, Calle 13, No. 100-00, Cali, Colombia

Correspondence should be addressed to Juan Carlos Munoz Grajales; jcarlmz@yahoo.com

Received 26 April 2016; Accepted 26 June 2016

Academic Editor: Antonio Scarfone

Caption: Figure 1: Vector soliton of system (1) obtained after 10 Newton's iterations and [delta] = 0, [[sigma].sub.1] = 0.5, [[sigma].sub.2] = -0.5, [alpha] = 1.5, and K = 0.5.

Caption: Figure 2: Real and imaginary parts of the vector soliton (u, v) of system (1) at [xi] = 20. Solid line: numerical solution (u, v) of system (1) using the numerical scheme in [37]. Pointed line: expected soliton computed using (108).

Caption: Figure 3: Vector soliton of system (1) obtained after 10 Newton's iterations and [delta] = 0, [[sigma].sub.1] = 0.5, [[sigma].sub.2] = -0.5, [alpha] = 1.5, and K = 0.5.

Caption: Figure 4: Vector soliton of system (1) obtained after 10 Newton's iterations and [delta] = 1, [[sigma].sub.1] = [[sigma].sub.2] = 0, and [[alpha].sub.1] = [[alpha].sub.2] = 1.5.

Caption: Figure 5: Real and imaginary parts of the vector soliton (u, v) of system (1) at [xi], = 20. Solid line: numerical solution (u, v) of system (1) using the (111)

Caption: Figure 6: Vector soliton of system (1) obtained after 10 Newton's iterations and [delta] = 1, [[sigma].sub.1] = [[sigma].sub.2] = 0, and [[alpha].sub.1] = [[alpha].sub.2] = 1.5.

Caption: Figure 7: Effect of the model's coefficients a, b, c, d on the shape of the soliton solutions. Here e [equivalent to] 0, [delta] = 1, [[alpha].sub.l] = [[alpha].sub.2] = 1.5, [[alpha].sub.l] = [[alpha].sub.2] = 0, K = 0.5, [x.sub.0] = 35, and a(x) = b(x) = c(x) = d(x) = [sech.sup.p](x - [x.sub.0]), with p = 1, 4, 1/16.

Caption: Figure 8: Effect of the model's coefficients on the shape of the soliton solutions. Here e [equivalent to] 0, [delta] = 1, [[alpha].sub.1] = [[alpha].sub.2] = 1.5, [[sigma].sub.1] = [[sigma].sub.2] = 0, K = 0.5, [x.sub.0] = 35, and a(x) = b(x) = c(x) = d(x) = A sech(x - [x.sub.0]), with A = 1 (solid line), A = 2 (pointed line), and A = 4 (dashed line).

Caption: Figure 9: Single-mode solitons for system (18).

Caption: Figure 10: Effect of the inhomogeneous coefficient e(x) on the shape of the vector soliton. Here [delta] = 0, [alpha] = 1.5, [[sigma].sub.1] = 0.5, [[sigma].sub.2] = -0.5, K = 0.5, [x.sub.0] = 35, a(x) = c(x) = [sech.sup.p](x - [x.sub.0]), b(x) = d(x) = 2 [sech.sup.p](x - [x.sub.0]), e(x) = A [sech.sup.p](x - [x.sub.0]), and p = 3. Solid line: A = 0; that is, e [not equal to] 0. Pointed line: A = 3. Dashed line: A = 10.

Caption: Figure 11: Collision of coupled-mode solitons with [[phi].sup.1] = [[phi].sup.2] = 0.

Caption: Figure 12: Collision of coupled-mode solitons with [[phi].sup.1] = [[phi].sup.2] = 0.

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

Caption: Figure 14: Contour plot of the collision of solitons in Figure 12.

Caption: Figure 15: Collision of coupled-mode solitons with [[phi].sub.1] = [pi] and [[phi].sub.2] = 0.

Caption: Figure 16: Collision of coupled-mode solitons with [[phi].sub.1] = [pi] and [[phi].sub.2] = 0.

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

Caption: Figure 18: Contour plot of the collision of solitons in Figure 16.
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Title Annotation:Research Article
Author:Grajales, Juan Carlos Munoz
Publication:Advances in Mathematical Physics
Article Type:Report
Date:Jan 1, 2016
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