_{1}

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We first analyze the sech-shaped soliton solutions, either spatial or temporal of the 1D-Schr?dinger equation with a cubic nonlinearity. Afterwards, these solutions are generalized to the 2D-Schr?dinger 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.

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 1DSchrödinger 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.

The one-dimensional, cubic, nonlinear Schrödinger Equation [

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 [

with

Indeed:

while

Substituting (3a) and (3c) into (1a) proves the result and, changing z, k into ct, -k in (2) gives the first order soliton solution of Equation (1c) while the solution of (1b) is [

These solutions have the remarkable feature that their profile does not evolve during propagation.

Using the dimensionless coordinates z = kz, , t = kct the Equations (1a) and (1c) take the simple form (5a) and (5c)

while the Equation (5b) is obtained with [

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 [

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 [

The Equation (5b) has the simple solution [

but, the comparison of (5b) and (5c) shows that changing x, t, y into t, z, f in (6a) gives another solution of (5b)

where to avoid confusion h has ben used instead of v.

The situation is somewhat different for the two dimensional cubic nonlinear Schrödinger equations (cylindrical coordinates are used in (9b))

They where devoted to some domains, mainly hydrodynamics and mechanics [9-11] 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)

with

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 of Equation (9c).

The two dimensional generalization of Equation (5c), that is (9c) with dimensionless coordinates, is

We look for the solutions of this equation in the form

in which while a, 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 f with

and we look for the solutions of (15) in the form

in which l, r, s are real parameters to be determined. Writing to simplify, we get

and

substituting (17) and (18a,b) into (15) gives

implying

so that the solution (16) becomes with

to be compared with (6a).

Similarly the two dimensional generalization of (5a), that is (9a) with dimensionless coordinates, is

with the solutions in which and

We are left with Equation (9b). Then, using the dimensionless coordinates, , , in which r and r_{0} positive. we get

We look for the solution of this equation in the form

with satisfying the equation

Substituting (24) into (23) and taking into account (24a) give

with the solution [

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 to into (27) gives the sech-shaped pulse

Using the index 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

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 f

We look for its solutions in the form with the real parameters l, to be determined

and writing 1/cosh(.) for, we get

and

Substituting (34) into (32) gives

implying

which achieves to determine (33) and consequently the solution (30) of the three dimensional cubic nonlinear Schrödinger equation

Using the dimensionless coordinates r, θ, f, the Schrödinger equation with a cubic non linelarity is

For fields that do not depend on q, f, this equation reduces to

and assuming, we get

so that

and Equation (38) becomes

We look for the solutions of this equation in the form

and a simple calculation gives, exp(.) representing the exponential term.

Substituting (42) into (41) and taking into account (43), we get since

We look for the solutions of this equation in the form with the real parameters β, l to be de-termined

Writing, for hyperbolic functions, a simple calculation gives

Substituting (45) and (46) into (44) gives

that is

We consider an asymptotic approximation of this equation for _{ } with so that to the order Equation (48) becomes

with the solution, which achieves to determine the spherical solution of the cubic nonlinear Schrödinger equation.

The nonlinear Schrödinger equation describes physical processes in which nonlinearity and dispersion cancel giving birth to solitons. This equation [9-11] can be applied to hydrodynamics (rogue waves), nonlinear optics (optical solitons in Kerr media), nonlinear aoustics (blood circulation), quantum condensates (Bose-Einstein), heath waves… All these processes lead to the generation of solitons along pulse propagation: An example is supplied by the optical solitons that travel without distortion justifying their importance [12-15] for laser pulse propagation in optical fibers.

Two dimensional solitons present a great interest since they propagate in lattices [16,17] as well as surface waves [18-21]. Some works were recently devoted to the 2Doptical solitons [22,23] and the sech-shaped solutions (20), (21) of Equations (9a) and (9c) are a particular case of the spatial temporal solutions discussed in [

No doubt that some of the 2D sech-shaped solitons discussed here will find practical applications in a near future.