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Yamgoué Serge Bruno, Deffo Guy Roger, Tala-Tebue Eric, Pelap François Beceau. Exact solitary wave solutions of a nonlinear Schrödinger equation model with saturable-like nonlinearities governing modulated waves in a discrete electrical lattice. Chinese Physics B, 2018, 27(12): 126303  
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Exact solitary wave solutions of a nonlinear Schrödinger equation model with saturable-like nonlinearities governing modulated waves in a discrete electrical lattice
Yamgoué Serge Bruno1, †, Deffo Guy Roger2, Tala-Tebue Eric3, Pelap François Beceau2
Department of Physics, Higher Teacher Training College Bambili, The University of Bamenda, P. O. Box 39, Bamenda Cameroon
Unité de Recherche de Mécanique et de Modélisation des Systèmes Physiques (UR-2MSP), Faculté des Sciences, Université de Dschang, BP 69 Dschang, Cameroun
Department of Telecommunication and Network Engineering, Fotso Victor University Institute of Technology, The University of Dschang, P. O. Box 134, Bandjoun, Cameroon

 

† Corresponding author. E-mail: sergebruno@yahoo.fr

Abstract

In this paper, we introduce and propose exact and explicit analytical solutions to a novel model of the nonlinear Schrödinger (NLS) equation. This model is derived as the equation governing the dynamics of modulated cutoff waves in a discrete nonlinear electrical lattice. It is characterized by the addition of two terms that involve time derivatives to the classical equation. Through those terms, our model is also tantamount to a generalized NLS equation with saturable; which suggests that the discrete electrical transmission lines can potentially be used to experimentally investigate wave propagation in media that are modeled by such type of nonlinearity. We demonstrate that the new terms can enlarge considerably the forms of the solutions as compared to similar NLS-type equations. Sine–Gordon expansion-method is used to derive numerous kink, antikink, dark, and bright soliton solutions.

PACS: 63.20.Pw
Keyword:nonlinear Schrödinger equation;nonlinear time derivative terms;saturable nonlinearity;exact solitary solutions
1. Introduction

Nonlinear equations are widely used as models to describe many important dynamical phenomena in various fields of sciences, particularly in nonlinear optics,[1–4] in plasma physics,[5,6] in biophysics,[7] in Bose–Einstein condensates,[8] in atomic chain,[9,10] in Fermi–Pasta–Ulam lattice,[11,12] in crystals,[13–16] and in discrete electrical transmission lines.[17–24] Among these equations, the famous nonlinear Schrödinger (NLS) equation is usually derived from fundamental principles as a primary approximation. As such, its suitability for the accurate description of the phenomena of interest in these fields can be limited. This has aroused the interest in investigation approaches that relax some of the assumptions that are usually considered in a first approximation. Such approaches often lead to generalized forms of the nonlinear Schrödinger equation rather than the standard one. Consider for instance a discrete nonlinear electrical transmission line (DNLTL) that consists of a number of identical LC blocks connected in sequence one to the other. Each block, whose model is presented in Fig. 1, contains two linear inductors Ls and Lp of which one is in the series branch and the other in shunt branch, and a bias-dependent capacitor Cp responsible for the nonlinearity.

Fig. 1. Schematic representation of a unit block of the nonlinear transmission line.

We assume that the capacitance–voltage relationships are approximated by , and respectively for the nonlinear capacitor Cp(Vn) in the parallel branch and for the one in the series branch, Cs(V). Thus, C0p and C0s are the respective limiting values of these capacitances when the voltages across them are infinitesimally small. α and β, and similarly η and λ, represent respectively the quadratic and cubic coefficients of nonlinearity.

Applying Kirchhoff laws to this circuit leads to the following set of differential equations governing the dynamics of signals in the network[15]

with n being the total number of blocks. Here, C0r = C0s/C0p is a dimensionless constat while and are characteristic frequencies of the system. Equation (1) has been used to investigate analytically and numerically the peak solitary wave and the modulational instability phenomenon in network.[26] Similarly, it has been used also to describe the dynamics of modulated waves in the specific case of the linear capacitor Cs in the series branch, that is, for η = λ = 0.[18,27,28] Other special cases of this equation abound in the study of nonlinear discrete electrical transmission lines.[21,29]

A basic approximation that is commonly made in investigating equations of this type consists in taking their solutions in the form A0exp(i(kn–ωt)) while assuming A0 to be infinitesimally small. Such solutions are characterized the so-called linear dispersion relation, given in our case by

where ω (rad/s) and k (rad/cell) are, respectively, the angular frequency and wave number. The graph of the above relation, shown in Fig. 2, presents a lower cutoff mode frequency when k = 0 and an upper cutoff frequency when k = π; which indicates that the network is a bandpass waveguide.

Fig. 2. The linear dispersion curve of the network according of the wave vector k (rad/cell) for ω0 = 3.77 × 106 rad/s, u0 = 2.58 × 106 rad/s, and C0r = 0.03.

Focusing our attention on waves with frequencies approaching the lower cutoff frequency of the lattice but with amplitude of arbitrary magnitude, we look for a solution of Eq. (1) in the form

where “c.c.” stands for complex conjugate, θ = −ωt is the phase variable, x = εn and T = ε2t with a dimensionless parameter ε are introduced to account for the slow temporal and spatial variations of the amplitude induced by the mutual coupling between the oscillators. It is obvious from Eq. (3) that the voltage at any site shifted by an integer m from a reference site n is given by
In order to reduce the mathematical intractability of the manipulations involved in the analysis, we assume that 0 ˂ ε ≪ 1 so that Taylor expansions of the following form
are warranted. Here the specific values m = 1 and m = −1 are relevant for Eq. (1). The use of the variables θ, x, and T also implies that the time derivative operators are transformed according to
Considering the above relations, the dynamics of the envelop A(x,T) is governed by the following nonlinear partial differential
with , and s2 = −6β.

This equation differs from the classical NLS equation by the two terms of cubic nonlinearity in its right-hand-side. It is worth to observe that these terms involve derivatives with respect to the variable attached with the first order derivative in the standard NLS equation. Although the s2-term can be considered to be common in NLS type equations with saturable nonlinearity, the s1-term as well as the full right-hand-side of Eq. (3) have, to the best of our knowledge, rarely been considered. Equation (3) has been derived above as governing the transmission of modulated signals with carrier frequency in the lower gap of the spectrum of this system. However, the legendary universality of NLS type equations makes it legitimate to expect that its applications can extend beyond discrete electrical transmission line. For instance, this equation may also be written in the form

which shows that it is a generalized NLS equation with saturable type nonlinearities. Accordingly, waves propagation in media modeled by such nonlinearities,[30–33] including metamaterials in particular, can possibly be investigated experimentally using basic nonlinear discrete electrical transmission lines.[34] This further motivates the interest in the solutions of Eq. (3). In the literature, these solutions exist on several forms[4,8,35–42] and are very significant for the description of the phenomena in diverse fields of Physics and Engineering.

Our investigations in this work then aim to determine exact and explicit analytical solutions to this new model of NLS equation. To this end, our paper is organized as follows. The next section begins with the reduction of the extended NLS equation to a single nonlinear ordinary differential equation (ODE). Sine–Gordon expansion-method, which is among the most powerful and effective method devised by mathematicians and physicists to find exact solutions of ODEs is then briefly outlined. In Section 3, the exact solitary wave solutions of this last equation will be given. The last section will be devoted to the conclusion.

2. The methods
2.1. ODE equation

Over the past several decades, many research works have focused on techniques for solving partial differential equations (PDEs). A first step in most of these techniques consists in reducing the PDEs in consideration into ODEs. In this view, we are interested in this paper by the class of solutions of Eq. (3) which can be represented by

where qe, qe, and Ω are real constants to be determined as part of solutions while ρ and ϕ and real-valued functions of a single variable. Substituting Eq. (4) into Eq. (3) and separating the real and imaginary part leads to the following system of coupled nonlinear ODEs:
It is easy to verify that equation (5b) becomes an exact differential if multiplied by ρ. Doing so leads in effect to
or
It is then clear that
where C0 is an arbitrary constant. This last equation is a first integral of Eq. (5b). It can be easily solved for dϕ/dτ. Considering the particular case where C0 = 0, we obtain
For simplicity, we fix the qe so that the derivative of the phase ϕ(τ) is proportional to ρ2. Thus
By introducing Eq. (6) into Eq. (5a) while accounting for Eq. (7), we obtain the following single nonlinear ODE which governs the real amplitude of the wave:
with c1 = s1 − s2, c2 = s1 + s2, and c3 = 3s2 − 5s1. Equation (8) is the celebrated cubic–quintic Duffing oscillator equation whose periodic solutions have been the focus of several research papers.[43–45] It is worth to notice that the quintic term in this equation is induced by the two nonlinear temporal terms of Eq. (3), though they are only of cubic order nonlinearity.

The next step after transforming the PDEs into ODEs is the effective determination of solutions to the latter. A first approach toward this goal for our problem is a direct integration from the first integral

which underlines the analogy of Eq. (8) with the Newtonian equation of motion of a point particle of unit mass. Here, the wave amplitude ρ(τ) plays the same role as the relative displacement of the particle; which moves freely in the following non-harmonic potential
This approach leads to an improper integral which contains a square root of a six-degree polynomial in the denominator[20] and so is quite difficult to evaluate.

The alternative approach preferred herein rely on the variety of methods available in the literature to deal with this task and which avoid the direct integration. Examples are the (G′/G)-expansion method,[46,47] the exp-expansion method,[48] the extended F-expansion method,[49] the bifurcation method of dynamical system,[50,51] the Jacobi elliptic function rational expansion method,[52] the tanh-function method,[53,54] the tan-expansion method,[55] the transformed rational function method,[56,57] the semi-inverse variational principle,[58] the Hirota bilinear method,[59–62] the variable coefficient Jacobian elliptic function method,[63] the sine-Gordon expansion approach,[64–67] and others. For our problem, the sine-Gordon expansion-method has revealed to be very effective. An important advantage of this techniques is that the computations involved can be systematically performed by a computer algebra system such as Maple or Mathematica. Below, it is briefly reviewed and used later to find various forms of solution to Eq. (8).

2.2. The sine-Gordon expansion-method

In this subsection we present the sine-Gordon expansion method which will later be utilized to investigate the solitary waves that can propagate through our discrete electric transmission line. This method is based on sine-Gordon equation and the traveling wave transformation.[68–70]

In fact, consider a nonlinear partial differential equations (NLPDEs) with two independent variables x and t,

In general, the left-hand side of Eq. (9) is a polynomial in U and its various derivatives. Assume that U(x,t) = U(ξ), ξ = kx − υct, where k and υc are constants to be determined later. Then equation (9) is reduced to an ordinary differential equation
where G is a polynomial of U and its various derivatives. We consider that the sought solution can be expressed as follows:
According to Refs. [65]–[67], equation (11) can be rewritten as
where the function w(ξ) = U(ξ)/2 satisfies the following relation
Equation (13) is variables separable. We obtain the following two significant equations upon solving it
The parameter M is, in most cases, a positive integer that can be determined by considering the homogeneous balance between the highest order derivative and the highest order nonlinear terms appearing in ODE (10). Collecting and setting the coefficients of terms of the form sini(w)cosj(w) to zero yields a system of coupled nonlinear algebraic equations. Solving this system by using Maple gives the values of ai, bi, a0, k1, and υc. Finally, substituting these values of Eq. (11), we obtain the traveling wave solutions to Eq. (10).

3. Exact solutions

We organize our search of solutions according to whether the coefficient of the quintic term is nonzero or not.

3.1. The full cubic–quintic Duffing oscillator equation

We deal first with the most general situation where the coefficient of the quintic term is nonzero, i.e., . This is the case when s1 ≠ s2 and 3s2 ≠ 5s1 and ve ≠ 0. Considering the sine-Gordon expansion-method, the homogenous balance process gives M = 1/2. Then we make the change of dependent variable and equation (8) becomes

Repeating the homogeneous balance for this new equation, we now obtain M = 1. Thus, it will be used in the continuation to determine the solutions of Eq. (3). Equation (12) for the value M = 1 obtained in the balancing process becomes
Substituting Eqs. (17), (18), and (19) into Eq. (16), and equating the coefficients of each power of sini(w)cosj(w) to zero, we obtain a system of algebraic equations for the parameters a0, a1, b1, ve, Ω, and μ, namely:

It appears that there are three more equations than the six variables sought for. Such a large number of coupled nonlinear algebraic equations can be obviously very challenging for human manipulation. On the contrary, computer algebra tools such as Maple, Mathematica or MuPAD are very powerful for such tasks, especially as these equations are all polynomial. For the explicit solutions that are of interest to us in this paper, and disregarding the trivial (constant) solutions, we find with the help of Maple that they depend on the combination of the physical parameters as considered below.

According to Eqs. (4), (17), and Eqs. (21), we obtain the following exact bright solitary wave solutions of Eq. (3)
where Θ = (qex − ΩT + ϕ(τ)) with ϕ and qe are given in Eqs. (6) and (7). The intensity of these solutions is given by
and the graphical representation is given in Fig. 3 for some arbitrary chosen values of parameters.

According to Eqs. (4), (17), and Eqs. (24), we obtain the following exact kink and anti-kink solitary wave solutions of Eq. (3)
where Θ = (qex − ΩT + ϕ(τ)) with ϕ and qe being given in Eqs. (6) and (7). The intensity of these solutions is given by
and the graphical representation is given in Fig. 4 for some arbitrary chosen values of parameters.

According to Eqs. (4), (17), and Eqs. (27a) and (27b), we obtain the following exact bright and dark solitary wave solutions of Eq. (3):
where Θ = (qex − ΩT + ϕ(τ)) with ϕ and qe being given in Eqs. (6) and (7). The intensity of these solutions is given by the following expression
and the graphical representation is given in Fig. 5 for some arbitrary chosen values of parameters.

According to Eqs. (4), (17), and Eqs. (30a) and (30b), we obtain the following exact kink and anti-kink solitary wave solutions of Eq. (3):

where Θ = (qex − ΩT + ϕ(τ)) with ϕ and qe being given in Eqs. (6) and (7). The intensity of these solutions is given by the following expression
and the graphical representation is given in Fig. 6 for some arbitrary chosen values of parameters.

According to Eqs. (4), (17), and Eqs. (33a) and (33b), we obtain the following exact kink and anti-kink solitary wave solutions of Eq. (3):

where Θ = (qex − ΩT + ϕ(τ)) with ϕ and qe being given in Eqs. (6) and (7). The intensity of these solutions is given by the following equation
and the graphical representation is given in Fig. 7 for some arbitrary chosen values of parameters.

Fig. 3. (color online) The three-dimensional (3D) and two-dimensional (2D) intensities of the bright solitary wave solution Eq. (22) with the parameters P = Q = 1, s1 = -1.25, s2 = −2.5, b1 = 1, and −10 ˂ x ˂ 10: (a) 0 ˂ T ˂ 2, (b) T = 0.
Fig. 4. (color online) The three-dimensional and two-dimensional intensities of the kink (a1 = a0) and anti-kink (a1 = −a0) solitary wave solutions of Eq. (25) with the parameters chosen as P = Q = 1, s1 = −1, s2 = −2, a0 = 0.4, and −20 ˂ x ˂ 20: (a) 0 ˂ T ˂ 6, (b) T = 0.
Fig. 5. (color online) The 3D and 2D intensities of the bright (b1 = a0) and dark (b1 = −a0) solitary wave solutions Eq. (28) with the parameters P = −Q = 1, s1 = −1, s2 = −2, a0 = 0.5, and −40 ˂ x ˂ 40: (a) 0 ˂ T ˂ 20, (b) T = 0.
Fig. 6. (color online) The 3D and 2D intensities of the kink (a1 = a0) and anti-kink (a1 = −a0) solitary wave solutions Eq. (31) with the parameters P = −Q = 1, s1 = −0.5, s2 = −0.75, a0 = 0.5, and −40 ˂ x ˂ 40: (a) 0 ˂ T ˂ 6, (b) T = 0.
Fig. 7. (color online) The 3D and 2D intensities of the kink (a1 = a0) and anti-kink (a1 = −a0) solitary wave solutions Eq. (34) with the parameters P = −Q = 1, s1 = −0.5, s2 = −0.75, and −40 ˂ x ˂ 40: (a) 0 ˂ T ˂ 2, (b) T = 0.
3.2. The standard Duffing equation

When either of the conditions ve = 0 or s1 = −s2 or s1 = 36s2/5 is realized, equation (8) degenerates to the standard Duffing equation. This is one of the most studied equations and its solutions (periodical and non-periodic solutions) are well examined and expressed by using the elliptic functions, hyperbolic functions, and trigonometric functions.[52,71,72] In the present article, we will restrict ourselves here to only point out that the new terms can be used to induce solutions of some desired nature in parameter regions where such solutions would not be obtained in the standard NLS equation. For example, it is well known that the solution of the cubic Duffing equation corresponds to bright soliton only if the coefficient of the linear term is negative while that of the cubic term is positive. From Eq. (8), this last condition cannot be met for the classical NLS equation when PQ ˂ 0. However, for either of s1 = −s2 or s1 = 3s2/5 with s1 ≠ 0, it is possible to first fix the value of the free parameter Ω so as to make the cubic coefficient positive and then also fix ve so that the linear coefficient is negative, thereby realizing the conditions of existence of bright solitons.

4. Conclusion

In this paper, we have introduced a novel model of the nonlinear Schrödinger equation containing nonlinear time-derivative terms which make the model to be equivalent to a generalized NLS equation with saturable nonlinearities. This new type of NLS equation has been obtained herein as the one governing the dynamics of modulated gap waves in a basic discrete electrical transmission line.

Seeking analytical solutions of our model, we have shown that the new terms induce an additional quintic term in the Duffing oscillator when the classical travelling wave transformation is used to reduce the NLS equation to an ODE. The sine-Gordon expansion-method has been employed to obtain varied exact analytical solutions of the model. Four types of solutions, namely the kink soliton, the antikink soliton, the dark soliton, and the bright soliton have been obtained. Such solutions give rise to particle-like structures, such as magnetic monopoles, and extended structures, like, domain walls and cosmic strings, that have implications in cosmology of the early universe.[73] As such, the results reported here can be of significant importance for such problems as the transmission of information in nonlinear waveguide, in nonlinear electrical transmission lines and many other domains. Likewise, the sine-Gordon expansion-method used here is powerful and can be also applied to other nonlinear equations in mathematical physics.

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