Schrödinger Equation with a Cubic Nonlinearity Sech-Shaped Soliton Solutions

We first analyze the sech-shaped soliton solutions, either spatial or temporal of the 1D-Schrödinger equation with a cubic nonlinearity. Afterwards, these solutions are generalized to the 2D-Schrödinger equation in the same configuration and new soliton solutions are obtained. It is shown that working with dimensionless equations makes easy this generalization. The impact of solitons on modern technology is then stressed.


Introduction
The one dimensional Schrödinger equation with a cubic nonlinearity has been known for a long time as well as its analytical solutions in terms of sech-shaped functions.Till recently, the situation was different for the two dimensional Schrödinger equation that we shall discuss here.
Using general equations, we start with the spatial and temporal sech-shaped soliton solutions of the 1D-Schrödinger equation with a cubic nonlinearity and it is shown that working with dimensionless equations leads to further types of solitons.Then, the same process with gene-ral and dimensionless equations is applied to the 2D-nonlinear Schrödinger equation which has sech-shaped soliton solutions generalizing 1D-solitons.Finally, because the nonlinear Schrödinger equation is a universal model that describes many physical non linear systems, the importance of solitons in modern technology is stressed.Nonlinear Schrödinger equations in (3D) and in cylindrical coordinates are succinctly discussed in Section 4.

General Equations
The one-dimensional, cubic, nonlinear Schrödinger Equation [1] intervenes in different physical settings to describe wave propagation in fluids, plasmas… nonlinear optics [2][3][4][5][6] in one of the three forms (c is the light velocity, k the wave number of propagating waves,  is a positive dimensionless parameter characterizing the medium in which this propagation takes place).
, , It is known to be one of the simplest partial differential equations with complete integrability, admetting in particular Nth order solitons as solutions and called spatial and temporal when they are solutions of (1a) or (1b).Changing the sign of the last term on the left hand side of Equations (1a)-(1c) gives a second set of cubic nonlinear Schrödinger equations with quasi periodic but no soliton sech-shaped solutions.
It is easy to prove that the first order soliton solution of Equation (1a) with amplitude A is [6]  Indeed: Substituting (3a) and (3c) into (1a) proves the result and, changing z, k into ct, k in (2) gives the first order 0  (1a) soliton solution of Equation (1c) while the solution of (1b) is [6] These solutions have the remarkable feature that their profile does not evolve during propagation.

Dimensionless Equations
Using the dimensionless coordinates  = kz, 2kx   ,  = kct the Equations (1a) and (1c) take the simple form (5a) and (5c) while the Equation (5b) is obtained with [7] 0 But, there exist more general expressions of the first order solitons for instance, for the Equation (5c) rewritten with the coordinates x, z, t, we have in which A, B, C 1 , C 2 are arbitrary real constant with in particular [7]  Similarly, with Equation (5a) also rewritten with x, z, we get as solution in which β is a dimension-less parameter The higher order soliton solutions have more intricate expressions [8] and their profile is no more constant, the solutions being rather periodic than stationary.The profile of a N = 2 soliton is pictured in [3].
The Equation (5b) has the simple solution [6]  but, the comparison of (5b) and (5c) shows that changing x, t,  into , ,  in (6a) gives another solution of (5b) where to avoid confusion  has ben used instead of v.

General Equations
The situation is somewhat different for the two dimensional cubic nonlinear Schrödinger equations (cylindrical coordinates are used in (9b)) , ; 0 , ; , ; 0 , ; 0 They where devoted to some domains, mainly hydrodynamics and mechanics [9][10][11] till that recently nonlinearities became an important topic, specially in optics and photonics, with as consequence to boost works on the analysis of Equations (9).
We prove here that Equation (9a) have soliton-shaped solutions generalizing (2) We first have and according to (3b) together with the second relation (10a) , ; Substituting (11a) and (11c) into (9a) achieves the proof.Changing z, k into ct, −k in (10) gives the soliton-shaped solution   , ; x y t

Dimensionless Equations
The two dimensional generalization of Equation (5c), that is (9c) with dimensionless coordinates, is , ; , ; , ; 0 We look for the solutions of this equation in the form x y x y x y t iv x iv y i v t while , r, s are real parameters and, to symplify we write exp(.) the exponential factor.Then, a simple calculation gives Substituting ( 13) into (12) gives the equation satisfied by  with   0 and we look for the solutions of ( 15) in the form in which , r, s are real parameters to be determined.
Similarly the two dimensional generalization of (5a), that is (9a) with dimensionless coordinates, is

2
, ; , ; , ; , ; 0 with the solutions in which We are left with Equation (9b).Then, using the dimensionless coordinates in which  and r 0 positive.we get We look for the solution of this equation in the form Substituting ( 24) into ( 23) and taking into account (24a) give with the solution [7]      while the solution of (24a) is substituting (25a) and ( 26) into (24) we get finally in which v is an arbitrary real parameter.It does not seem that the sech-shaped soliton (27) is known.But, substituting the dimensionless coordinate gives the sech-shaped pulse , ; 2 exp exp sech

3D-Schrödinger Equation
Using the index 1, 2, 3 j  for the dimensionless coordinates x,y,z together with the sum-mation convention on the repeated indices and , the tridimensional cubic nonlinear Schrödinger equation is Copyright © 2012 SciRes.

OPJ
We look for the solution of this equation in the form the exponential term is written exp(.) to simplify and a simple calculation gives Since , substituting (31) into (30) gives the equation satisfied by We look for its solutions in the form with the real parameters , j  to be determined and writing 1/cosh(.) for

Schrödinger Equation in Cylindrical Coordinates
Using the dimensionless coordinates r, θ, , the Schrödinger equation with a cubic non linelarity is For fields that do not depend on , , this equation reduces to and Equation (38) becomes We look for the solutions of this equation in the form and a simple calculation gives, exp(.)representin exponential term.
g the Substituting ( 42) into (41) and taking into account (43), we get We look for the solutions of this equa with  tion in the form the real parameters β,  to be de-termined

Conclusions termine
The nonlinear Schr processes in which nonlinearity and dispersion cancel giving birth to solitons.This equation [9][10][11] can be applied to hydrodynamics (rogue waves), nonlinear optics (optical solitons in Kerr media), nonlinear aoustics (blood circu-