The one-dimensional, cubic, nonlinear Schrödinger Equation [1] intervenes in different physical settings to describe wave propagation in fluids, plasmas… nonlinear optics [2-6] in one of the three forms (c is the light velocity, k the wave number of propagating waves, c 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]

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 [6] easy to check

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 [7], , ,.

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, C₁, C₂ 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, 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 r₀ 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 [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 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.