Dynamical system of optical soliton parameters by variational principle (super-Gaussian and super-sech Pulses)

. The parameter dynamics of super-sech and super-Gaussian pulses for the perturbed nonlinear Schrödinger ’ s equation with power-law nonlinearity is obtained in this article. The variational principle successfully recovers this dynamical system. The details of the variational principle with the implementation of the Euler – Lagrange ’ s equation to the nonlinear Schrödinger ’ s equation with power-law of nonlinearity described in this paper have not been previously reported.


Introduction
The dynamics of optical solitons is a long standing study that has now extended over half-a-century. Various aspects of soliton science have been reported. Notably, most of the papers are from the integrability aspects of a variety of models that arose from wide range of self-phase modulation (SPM) structures. A few papers are from additional, sparingly visible, topics such as conservation laws, quasimonochromatic solitons with the usage of perturbation theory, stochastic perturbation and the corresponding mean free velocity of the soliton and others.
One of the most viable topics that serves as an important foundation stone in optical soliton dynamics is the recovery of the soliton parameter dynamics such as the amplitude, width, center position, phase constant and similar such parameters. This can be achieved in several different ways. A few such mathematical approaches are the soliton perturbation theory, collective variables approach and the moment method. However, for example, soliton perturbation theory has its shortcomings. It fails to recover the variation of the phase constant as well as the variation of the center position of the soliton. The variational principle (VP) overcomes this hurdle. This has been succesfully and widely applied to various areas of Physics and Engineering such as Condensed Matter Physics, Fluid Dynamics and Fiber Optics including dispersion-managed solitons [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20].
The advantages and necessity of obtaining the dynamical system of soliton parameters are multifold. The study of soliton features in optics can be further enhanced through the utilization of these parameter dynamics. Four wave mixing effects, collision-induced timing jitter, and various other phenomena are among those that are included. Therefore, the parameter dynamics with the existence of perturbation terms is being studied by applying the VP to the nonlinear Schrödinger's equation (NLSE). Super-sech and super-Gaussian pulses are the two types of pulses being examined in this context. This would give a generalized flavor to the study of soliton parameters. The details of the VP with the implementation of the Euler-Lagrange's equation to NLSE with power-law of nonlinearity described in this paper have not been previously reported. A quick and succinct introduction is followed by the presentation of results.

Unperturbed NLSE with power-law nonlinearity
The governing model of such equation is written as: where the coefficients b and a are utilized to denote SPM and chromatic dispersion in sequence. The function q = q(x, t) represents the wave profile in a complex-valued form, where i ¼ ffiffiffiffiffiffi ffi À1 p . Equation (1) contains the linear temporal evolution, represented by the first term.

Variational principle
The Lagrangian (L) is associated with equation (1) is written as: One obtains q* by complex-conjugating q. In equation (1), the assumed pulse q = q(x, t) is presented as: We use the symbols h 0 (t), j(t), xðtÞ, B(t), and A(t) to denote the soliton phase, soliton frequency, center position of the soliton, pulse width, and soliton amplitude, respectively. Setting and non-negative integers are the only values that a, b, and c can assume. The integrals of motion can be obtained from the pulse form (5), which can be derived, as presented below The mathematical representation of the Hamiltonian takes the form of

Parameter dynamics of the NLSE
Introducing the following Euler-Lagrange (EL) equation [4,8] in this subsection leads to the derivation of the dynamical system: where the soliton parameters A(t), B(t), xðtÞ, jðtÞ and h 0 ðtÞ are represented by the variable p, where p denotes one of them. The dynamic system below is derived by substituting (9) into (14): where the square roots of the energy are proportional to the constant K in (24). From (21) and (24), we have:

Super-Gaussian pulses
Assuming m > 0, the super Gaussian pulse function can be written as f ðsÞ ¼ e Às 2m . Then, one can obtain the integrals of motion as: and the Hamiltonian is given by: For u > 0, the gamma function is defined as C(u). This compels the parameter m to be bounded below as given by The pulse parameters can be obtained from the evolution equations (20)- (25), which can be expressed in a reduced form as:

Super-sech pulses
For super-sech pulses, we set f(s) = sech 2m s , m > 0: Then, one can address the equations governing the integrals of motion that are expressed as: and we can express the Hamiltonian as: Here, the generalized form of Gauss' hypergeometric function is expressed as: and the symbol for the Pochhammer is: The pulse parameters are governed by the evolution equations (20)- (25), which can be expressed in a simpler form as: 3 Perturbed NLSE with power-law nonlinearity The equation is described by the following governing model: where R½q; q Ã is given by: and , d, a, b, k, h, r, n, g, f, l, r 1 and r 2 are constants, where is from quasimonochromaticity. From (5) and (47), we

Conclusions
Our study recovers the dynamical system of soliton parameters for super-sech and super-Gaussian pulses, as described in this paper. The details of the VP with the implementation of the Euler-Lagrange's equation to the NLSE with powerlaw of nonlinearity indicated in the current work have not been previously reported. These parameter variations, namely the dynamical system opens up with an avalanche of opportunities to study optical soliton sciences further along. This foundation stone of results pave way to further future investigations in this chapter. Later, the dynamical system would be revealed for additional forms of SPM that have not yet been considered. The studies would later be extended to birefringent fibers and DWDM topology. These would give an increased perspective to carry out the analysis further along. This would also be applicable to various additional devices and other forms of waveguides, including optical metamaterials, magnetooptic waveguides, optical couplers, gap solitons and many others. The results of these studies will be reported soon after we align them with the pre-existing ones [21][22][23][24][25]. All of these activities are currently underway.