Anti Einstein Technique : Analytic Solution of Nonlinear Schrödinger Equation by Means of A New Approach

The nonlinear Schrödinger equation (NSE) has served as the governing equation of optical soliton in the study of its applications to optical communication and optical switching. Various schemes have been employed for the solution of this nonlinear equation as well as its variants. We report in this paper a relatively simpler new approach for the analytic solution of NSE. In this scheme the equation was first transformed into an arctangent differential equation, which was then separated into the linear and nonlinear parts, with the linear part solved in a straight forward manner. The solution of the nonlinear equation was written in the form of modulation function characterized by its amplitude function A and phase function F(A). Substituting the linear solution for A, the arctangent differential equation was solved for a certain initial value of A. It is shown that this method is applicable to other first-order nonlinear differential equation such as the Korteweg de Vries equation (KdV), which can be transformed into an arctangent differential equation.

I. Introduction

The phenomenon of the solitary wave propagation was observed for the first time by the Scottish scientist John Scott Russell in 1844, when one day he was watching water waves of a certain shape kept on traveling without changing their shape for a distance as far as his eye could see. To explain the behavior of such unusual wave, Korteweg and de Vries governed a model for the wave propagation in shallow water in form a partial differential equation called as KdV differential equation, which its solution appropriates to the features of the solitary wave called as soliton[1]. The existence of solitons in optical fiber was predicted by Zakarov and Zabat (1972) after they derived a differential equation for the light propagating in an optical fiber, that demonstrated later by Hazagawa in 1973 at Bell Laboratory. Next, Mollenauer and Stolen employed the solitons in optical fiber for generating subpicosecond pulses.

Keywords : nonlinear Schrödinger equation, KdV, arctangent, Anti Einstein Technique

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Anti Einstein Technique : Analytic Solution of Nonlinear Schrödinger Equation by Means of A New Approach (Indonesia)

Persamaan Schrödinger Nonlinear (NSE) bertindak sebagai solusi optik dalam studi aplikasi komunikasi optik dan switch optik. Berbagai usaha telah dilakukan untuk mencari penyelesaian dari persamaan non linier ini seperti halnya terhadap variannya. Pada tulisan ini kita berikan suatu pendekatan yang relatif lebih sederhana mudah dan baru mengenai penyelesaian persamaananalitik dari NSE. Di rencana ini persamaan yang pertama diubah ke dalam suatu persamaan arctangent yang diferensial, yang kemudian dipisahkan ke dalam linier dan nonlinear , dengan bagian yang linier memecahkan suatu cara lurus kedepan. Solusi dari persamaan nonlinear ditulis dalam format dari fungsi modulasi ditandai oleh F(A fungsi fase dan A fungsi amplitudo nya). menggantikan solusi yang linier untuk A, persamaan arctangent yang diferensial dipecahkan untuk suatu nilai awal yang tertentu dari A. Ditunjukkan bahwa metoda ini sesuai persamaan nonlinear orde 1 seperti persamaan Korteweg de Vries equation ( Kdv), yang dapat diubah ke dalam suatu penyamaan arctangent yang diferensial.

I. Introduction (pengenalan)

Peristiwa dari perambatan gelombang yang tunggal diamati untuk pertama kali oleh Scott Russell Yohanes ilmuwan Scottish di 1844, ketika suatu hari ia sedang menyaksikan ombak air dari suatu bentuk yang tertentu yang disimpan ketika bepergian tanpa mengubah bentuknya untuk suatu jarak sejauh mata melihat. Untuk menjelaskan perilaku dari gelombang yang tidak biasa seperti itu, Korteweg de Vries membuat suatu model untuk perambatan gelombang di air yang dangkal seperti persamaan diferensial KdV, solusi nya seperti pada persamaan [1].

Keberadaan dari solitons di serabut yang berhubung dengan mata diramalkan oleh Zabat dan Zakarov ( 1972) setelah mereka memperoleh suatu persamaan diferensial untuk cahaya menyebarkan di suatu serabut optik, yang ditunjukkan kemudian oleh Hazagawa di 1973 pada Bell Laboratory. yang berikutnya, Stolen dan Mollenauer yang dipekerjakan solitons di serabut optik untuk membangkitkan denyut nadi subpicosecond.

Kata Kunci: nonlinear Schrödinger equation, KdV, arctangent, Anti Einstein Technique

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Introducing Stable Modulation Technique for Solving an Inhomogeneous Bernoulli Differential Equation

The Bernoulli differential equation that has been used as primary modeling in many application branches commonly contains harmonic force function in the inhomogeneous term. The nhomogeneous Bernoulli differential equation (IBDE) is specified by nonlinearity of nth order. For instance, 3rd order IBDE is called a stochastic differential equation that has been commonly applied for describing the corrosion mechanism, the transport of fluxon, the generation of squeezed laser, etc. Due to the difficulty of applying a linearization procedure for IBDE, it has commonly been solved numerically. In this paper we introduced a so-called Stable Modulation Technique (SMT) which is able to solve a first order nonlinear differential equation that transformable into the homogeneous Bernoulli differential equation (BDE). SMT is employed by splitting BDE into linear and nonlinear parts. The solution of its nonlinear part has been found to be AF(A) where A is the initial value of nonlinear solution part and F is the modulation function whose phase is a function of A. The general solution of BDE obtained by substituting the linear solution part into initial value of the nonlinear solution part. Although IBDE can not be transformed into BDE completely, SMT gives nevertheless an approximation solution in AF(A) form, where the homogeneous solution part becomes its amplitude term. The AF(A) formula for stochastic differential equation with cosine function as inhomogeneous term can be decomposed as well into transient and steady state solutions. In addition, a special example of solving 2 ogeneous Bernoulli differential equation (BDE) has en used as a primary modeling scheme in many ation branches. A nonlinear BDE is specified by a nonlinearity of n. Although the stochastic differential equation is the first order differential equation, nevertheless the procedure of obtaining its analytical solution is very complicated as shown in utilizing of BWK (Brillouin, Wenzel, Kramer) and Reversion methods nd order BDE in creating a new Planck’s formula of black-body radiation is also presented.

Keywords–Stable modulation technique, modulation function, stochastic differential equation.

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Analytical Solution Of The Ricatti Differential Equation For High Frequency Derived By Using The Stable Modulation Technique

The Ricatti differential equation dy/dt = P(t)y2+Q(t)y+R(t) is a nonlinear differential equation which is of contemporary interest in various fields, including particle dynamics, optics, and petroleum exploration. The Ricatti differential equation is easily solved by numerical methods. But in order to obtain the exact solution in analytical form, the first order of nonlinear inhomogeneous differential equation is commonly convert into a second order linear differential equation by use of a change of the dependent variable. In this paper we introduce the stable modulation technique (SMT) to solve the Ricatti differential equation without of the use linearization procedure. The main principle of the SMT in solving a first order nonlinear differential equation is modulate the solution of the linear part into the initial value of the nonlinear part solution. Important to be stressed here that the solution of nonlinear part must be written in the modulation function, where the initial value acts as amplitude and also including in the total phase shift. For a special case, dy/dt = -by2+ay+Acos(2πft) where a and b of both are constants, while t is variable of the time (in s), frequency f in Herzt (Hz) we find that the analytical solution of the SMT is appropriate with numerical solution espescially for high frequency f≥10 Hz, amplitude value A≤1, and initial value of the y in the range 0.1≤y0≤1. The analytical solution above can be used as trial function when the Ricatti differential equation will be solved by using combination of the modulational instability technique and variational approximation.

Keywords : Nonlinear inhomogeneous differential equation, linearization procedure, stable modulation technique, modulational instability

I. Introduction

The general form of the Ricatti differential equation (DE) is of the form :
dy/dt = P(t)y2+Q(t)y+R(t), .......................................(1)
where y and t are respectively dependent and independent variables, both of P(t) and Q(t) are homogeneous coefficients, where R(t) is the inhomogeneous term[1]. The Ricatti DE is mother of all ordinary differential equations (ODE’s) second order generating special functions like Airy, Bessel and etc,[2] even the Helmholtz equation[3] which has broad applications including optics and geophysics. Common procedure in solving the Ricatti DE is by transforming into a linear ODE second order[1],[2],[4]. For special case, the analytical exact solution of the Ricatti DE can not be obtained, because the analytical solution of the corresponding linear ODE second order is not available.

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