A new numerical scheme based on Haar wavelets for the numerical solution of the Chen–Lee–Liu equation

Abstract

This work based on a new numerical scheme for the numerical solutions of the Chen–Lee–Liu equation using Haar wavelet collocation method. In this work, we combined the backward Euler difference (BED) formula with the Haar wavelet collocation method (HWCM). The BED formula approximates the time derivative of the Chen–Lee–Liu equation, while the HWCM approximates the space derivatives of the Chen–Lee–Liu equation, consequently converting it into a finite system of algebraic equations. Furthermore, we validate the accuracy of the proposed method numerically and graphically with the help of several examples. Moreover, we have also presented an error analysis to prove the convergence of the proposed method.

Introduction

Nonlinear Schrödinger (NLS) equations play a vital role in applied mathematics and physics, such as plasma physics, nonlinear optics, hydrodynamics, the surface of water waves, quantum mechanics, molecular biology, pulse in biological chains, fluid dynamics and elastic media [1], [2], [3], [4], [5], [6], [7], [8], [9]. But these are low-order approximation models of nonlinear effects and since, in reality, numerous physical phenomena are too complicated, so it is indispensable to introduce higher-order nonlinear terms in the model for obtaining a good approximation of nonlinear effects. Derivative nonlinear Schrödinger (DNLS) equations are one of the useful equations in the nonlinear model by which these type of difficulties can be removed [10]. The popular form of DNLS equations are Kaup–Newell equation [11], the Chen–Lee–Liu equation [12], and the Gerjikov–Ivanov equation [13] which are usually known as DNLS-I, DNLS-II, and DNLS-III equations, respectively.

In 1979, Chen et al. [12] used the inverse scattering method to linearize the nonlinear Hamiltonian (NLH) systems and identified them using the time part of the Lax equation to examine the integrability of NLH systems. Furthermore, numerous integrable and non-integrable equations were introduced in which integrable derivative-type nonlinear equation is given by

$$i{\mathrm{\Psi }}_t+{\mathrm{\Psi }}_{xx}+i|\mathrm{\Psi }{|}^2{\mathrm{\Psi }}_x=0,$$

which is usually called the Chen–Lee–Liu equation. In Optics, the Chen–Lee–Liu equation also provides a real physical model like NLS equation. The Kaup–Newell equation is given by [11]

$$i{\mathrm{\Psi }}_t+{\mathrm{\Psi }}_{xx}+i{\left(\right|\mathrm{\Psi }{|}^2\mathrm{\Psi })}_x=0.$$

There exists a gauge transformation [14], [15] which connects these two equations is $\tilde{\mathrm{\Psi }}=\mathrm{\Psi }exp\left(-\frac{i}{2}{\int }^x|\tilde{\mathrm{\Psi }}{|}^2\mathrm{dx}\right)$. Note that, $|\mathrm{\Psi }{|}^2=|\tilde{\mathrm{\Psi }}{|}^2$ which is a trivial but fundamental observation.

In 1967, DeMartini et al. [16] first introduced the self-steepening of light pulses, which originate from their propagation in an intensity dependent index of refraction medium. This was also observed later in optical pulses with possible shock formation [17]. However, the isolated self-steepening is a rare phenomenon and is often studied along with self-phase-modulation [18], [19]. Consequently, self-steepening is hardly observed alone, although, in theory, a controllable self-steepening was suggested in 2006 and demonstrated experimentally using the wave vector mis-match [20]. In 2007, Moses et al. [21] proved through an experiment that optical pulse propagation involving self-steepening without self-phase-modulation. The Chen–Lee–Liu equation models this experiment, which gives the first experimental evidence of the Chen–Lee–Liu equation. In 2015, Zhang et al. [22] used determinant representation of n-fold Darboux transformation to obtain various explicit solutions of the Chen–Lee–Liu equation. In 2017, Ekici et al. [23] used the extended trial function method to obtain soliton solutions in magneto-optic waveguides. In 2017, Liu et al. [24] investigated periodic solitons interaction with controllable parameters. In 2018, Yu et al. [25] used Hirota method to investigate the soliton transmission of NLS equation. In 2020, Biswas [26] discussed the property of Quasi-monochromatic dynamics of optical solitons for NLS equation with quadratic-cubic nonlinearity. In 2020, Bansal et al. [27] applied lie symmetry analysis to construct optical soliton solutions of Eq. (1).

In the present study, we have made an effort to find the numerical solution of the Chen–Lee–Liu equation. We consider the propagation of an optical pulse within a monomode fiber modelled by a family of the following Chen–Lee–Liu equation:

$$i{\mathrm{\Psi }}_t+\alpha {\mathrm{\Psi }}_{xx}+i\lambda |\mathrm{\Psi }{|}^2{\mathrm{\Psi }}_x=0,x\in [c,d],t\ge 0,$$

with initial condition

$$\mathrm{\Psi }(x,0)=\varphi (x),$$

and Dirichlet boundary conditions

$$\mathrm{\Psi }(c,t)={\gamma }_0(t),\mathrm{\Psi }(d,t)={\gamma }_1(t),$$

where Ψ = Ψ(x, t) is a complex-valued function and $\alpha ,\lambda \in ℝ$ are arbitrary constants. In optical fiber, terms involving the parameter α and λ are usually associated with group velocity dispersion and self-steepening phenomena, respectively [2]. The coordinates x and t are referred to as spatial coordinate traveling with the group velocity and slow time, respectively, in hydrodynamics. But in optical fiber physics, they denote distance and retarded time, respectively [28], [29].

In [30], wavelets are defined as functions that satisfy some mathematical conditions and are used in representing data or other functions. The fundamental idea behind this is to analyze data or signal as per scale. Wavelets were developed independently by mathematicians, electrical engineers, geologists, and quantum physicists, and collaborations among these fields in the last three decades have led to new and varied applications such as image compression, radar, earthquake prediction, turbulence, and human vision. Wavelets have been used to solve partial differential equations since the early 1980s. They are widely used for numerical approximation and have multiple applications in approximation theory. Many variants of wavelets [31], [32], [33], [34], [35] have been used to find the numerical solutions of ordinary differential equations, partial differential equations, numerical integration, fractional partial differential equations, and integral equations.

Haar wavelet is one of the most straightforward wavelet, which is defined by an analytical expression and is a handy tool for finding the numerical solutions of integral and differential equations. However, it has a drawback that it is discontinuous and consequently non-differentiable. Therefore, it is impractical to use the Haar wavelet directly for determining the numerical solutions of differential equations. It is a piecewise constant function, and thus, its derivatives of any order vanish. Therefore, several researchers have used its integrated approach for solving the differential equations. Chen and Hsiao [36] solved lumped and distributed parameter systems using Haar wavelet and also applied them in linear stiff systems [37]. Lepik further developed the Haar wavelet approach to solve partial and higher order differential equations [32], [38].

To the best of our knowledge, no attempt on this equation has been made by the proposed method in literature for finding the numerical solution of Eq. (3). The work is ordered as follows. In Section 2, Haar wavelet, function approximation, and integrals of Haar wavelet are discussed. In Section 3, we introduced a method by combining the Haar wavelet with the BED formula for the numerical solution of Eq. (3). In Section 4, we discussed the convergence of the proposed method and its error analysis. In Section 5, numerical examples are given in support of our approach. In Section 6, the conclusion of our work is presented.