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This paper shows that the centered fractional derivatives introduced by Manuel Duarte Ortigueira in 2006 are useful in the description of optical solitons. It is shown that we can construct a fractional extension of the nonlinear Schr ödinger (NLS) equation which incorporates Ortigueira’s derivatives and has soliton solutions. It is also shown that this fractional NLS equation has a Lagrangian density and can be derived from a variational principle. Finally, a fractional extension of Noether’s theorem is formulated to determine the conserved quantities associated to the invariances of the action integral under infinitesimal transformations.

In 2010, it was found that the famous nonlinear Schrödinger (NLS) equation:

which occupies a central role in the study of light pulses propagating in optical fibers, has a fractional extension which has soliton-like solutions [

Equation (1) is adequate to describe optical pulses when the power transmitted along the fiber is low (a few milliwatts), and the width of the pulses is in the range of a few picoseconds. However, when the pulses are shorter and the power is higher, the NLS equation has to be modified by adding higher-order dispersive and nonlinear terms, as in the equation:

where

The existence of exact solitons in these two equations suggests that perhaps soliton solutions may also exist in a fractional equation of the form:

where α is a real number in the interval

where

and the coefficient

Unexpectedly, the results found in [

does indeed have stable soliton solutions.

We can see that the derivative defined in (6) can be considered as an alternative to define a centered fractional derivative, as it combines left- and right-sided Grünwald-Letnikov derivatives. However, other possibilities exist. One of them is the Riesz derivative [

where

The first of these derivatives is adequate when α is close to an even integer, and it is called “type 1” fractional centered derivative in Ortigueira’s papers. The second one is the “type 2” centered derivative, and it is appropriate when α is closer to an odd integer.

In the present communication we investigate if it is possible to replace the integer-order derivatives and

The paper is structured as follows: in Section 2 we show that it is indeed possible to replace the integer-order derivatives

To begin this section, we should mention that Equations (11) and (12) were obtained by extrapolating the finite difference approximation for the n-th derivative of f(t) with centered differences. However, a careful derivation of these extrapolations shows that in Equation (11) a factor

which shall be called “even” and “odd” centered fractional derivatives from now on.

Now, let us focus our attention on our first goal: to find out if it is possible to generalize Equation (2) by replacing the integer-order derivatives

In

where

In a similar way, we can now compare the solutions of the equations:

with

can see that the pulse slowly disperses, and it moves along the t axis, which is a consequence of being near to 3 (we know that the effect of a third derivative

It is worth observing that the divergence of the solutions of Equations (16) and (19) might be considered as a posteriori proof that Ortigueira’s decision of eliminating the complex factors

There is, however, a small discrepancy that has to be clarified. In

Therefore, if we want to generalize Equation (2) by including centered fractional derivatives, we should replace

and consequently

The above results seem to imply that a reasonable fractional generalization of Equation (2) would be:

But we can improve this equation by introducing weight factors in front of the fractional derivatives. We desire that the influence of the type 1 derivative diminishes as

in front of the type 1 derivative, and a factor:

in front of the type 2 derivative. Therefore, a good candidate to generalize Equation (2) to fractional orders seems to be:

To find out if this equation has soliton-like solutions we will solve it numerically with an initial condition that has a chance to be near to a soliton. And a promising initial condition could the exact soliton solution of Equation (2), which has the form [

where:

Therefore, in

and this is the initial condition that we will use to solve Equation (25).

In

We should observe, however, that the evolution of the initial condition (32) is different if

It is worth remembering that in the case of Equation (9) it was necessary to include the nonlinear term

shows that the initial condition (32) is dispersed away quite rapidly if

can see the evolution of the pulse when

The fact that Equation (25) does not require the nonlinear term

To close this section we would like to observe that in Equations (9) and (25) we put the coefficients of

Now, let us investigate if it is possible to formulate a generalized least action principle which applies to Lagrangian densities which involve Ortigueira’s centered fractional derivatives. Therefore, let us begin by supposing that we have a functional (that we shall call “action”, as usual) defined as follows:

where

Once with our action integral, we would like to obtain the conditions that the function

where the variation of the Lagrangian is given by:

Now, in order to obtain a fractional differential equation from the condition (35), it is necessary to rearrange the integrand in (35) in such a way that each of its terms contains a factor

dard calculation, but the integration of the terms containing

which can be obtained directly from the definitions of the derivatives

where

Using the Parseval relations (37) and (38) we can integrate by parts all the terms in the integrand of Equation (35) and then, collecting the terms which contain

and a similar equation holds with

If we now consider the Lagrangian density:

and we substitute it into Equation (40), we obtain Equation (25). Therefore, the fractional equation (25), in addition of having soliton-like solutions, can be obtained from the least action principle using the Lagrangian density (41).

Noether’s theorem states that if the action integral is invariant under an infinitesimal transformation, then a conservation law exists. In the following we will investigate if this theorem also holds when we have action integrals which involve Lagrangian densities which depend on integer-order derivatives and also on centered fractional ones.

In this communication, we will only consider infinitesimal transformations of the form:

where

and from these equations it follows that:

Now, in order to arrive at Noether’s theorem, it is necessary to substitute

It should be noticed that Equation (50) differs from (36) because the first two terms on the r.h.s of (50) did not appear in Equation (36). In the derivation of the Euler-Lagrange equations from the least action principle, only the functions

Once we have substituted

where we have defined:

The form of Equation (51) is interesting because it does not have the form of a conservation law due to the presence of the last term (the term P). This term disappears when the Lagrangian density does not contain the fractional derivatives

We should now observe that even when the term P is present in Equation (51), this equation may imply the existence of a conserved quantity, because when we integrate this equation over t (from

and consequently for any solution which satisfies the boundary condition:

there is a conserved quantity since Equation (55)reduces to:

Therefore we have the following fractional extension of Noether’s theorem:

If we have a fractional partial differential equation which can be obtained from a Lagrangian density which depends on two functions

1)

2) The condition (56) is satisfied [where

It should be observed that other fractional generalizations of Noether’s theorem have been formulated in the past [

Now, we will apply the theorem presented above to determine the conserved quantities associated with three infinitesimal transformations. The first one is a infinitesimal gauge transformation:

It can be verified that the action integral associated to the Lagrangian density shown in Equation (41) is invariant under this transformation (i.e.

In other words: the invariance of the action under a gauge transformation implies that the energy of the pulse is conserved.

As a second example we can consider the following infinitesimal transformation:

A straightforward calculation shows that also in this case we have

If we now substitute the Lagrangian (41) in this equation, it reduces to:

where:

is the Hamiltonian density corresponding to the Lagrangian given in (41). It is worth mentioning that this Hamiltonian does not contain the term

It may be a surprise that the conservation of the Hamiltonian is a consequence of the invariance of the action integral under translations in z. We are used to think that the Hamiltonian conservation is associated to invariances under time translations. However, we must remember that in the context of soliton propagation in optical fibers, the evolution variable is the spatial coordinate z, and therefore, in this context,z plays the same role that is usually played by the time in mechanical problems. This is the reason for the Hamiltonian conservation to be associated to translations in z.

As a third example we can consider a time translation:

A direct calculation shows that also in this case the variation of the Lagrangian (41) associated to this transformation vanishes (i.e.

It is worth observing that this conservation law also holds in the case of the standard NLS equation [

In this communication, we show that there exists a fractional generalization of the NLS equation [Equation (25)] which admits soliton-like solutions, and employs Ortigueira’s centered fractional derivatives [Equations (11)- (12)] to describe the dispersion of light pulses travelling along an optical fiber. It is found that Ortigueira’s centered derivatives are more adequate to describe the dispersion of optical pulses than the Grünwald-Letnikov derivatives used in [

Therefore, we have seen that Ortigueira’s centered fractional derivatives can be incorporated in a generalized NLS equation [Equation (25)] which describes the propagation of light pulses in optical fibers, and this new fractional equation has the following five characteristics:

a) It is an interesting physical model.

b) It has soliton solutions (fractional optical solitons).

c) It is superior to other models which accept fractional optical solitons because Equation (25) does not require additional nonlinear terms to describe the propagation of solitons.

d) It can be obtained from a Lagrangian density, via the least action principle.

e) Some of its conserved quantities can be obtained by means of a generalized fractional Noether’s theorem.

It is worth mentioning that Ortigueira has recently proposed a new unified centered fractional derivative [

with this unified centered derivative, in order to find out if the resulting equation admits soliton solutions and can be derived from a variational principle.

As a final remark we would like to add that the theory of optical solitons is not only related to the fractional derivatives and the fractional calculus (as we have seen in this paper), but also to the concept of fractional dimensions [

We thank DGTIC-UNAM (Dirección General de Cómputo y de Tecnologías de Información y Comunicación de la Universidad Nacional Autónoma de México) for granting us access to the computer Miztli through the Project SC16-1-S-6, in order to carry out this work.

Jorge Fujioka,Manuel Velasco,Argel Ramírez, (2016) Fractional Optical Solitons and Fractional Noether’s Theorem with Ortigueira’s Centered Derivatives. Applied Mathematics,07,1340-1352. doi: 10.4236/am.2016.712118