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A variational description of non-linear Schrodinger equation.

Introduction

The non-linear Schrodinger (NLS) equation arises as the envelope of a dispersive wave system which is almost monochromic and weakly nonlinear. The NLS equation has found numerous applications in physics, in the theory of deep-water waves [1], as well as a model for the non-linear pulse propagation in fibers [2] in the Heisenberg model of magnetics etc. This paper is concerned with the variational method to solve the NLS equation.

The inverse scattering technique discovered by Zabusky and Kruskal [3] is a powerful tool for exact solution of integrable equations, like the KdV equation [4] and the linear Schrodinger equation. The non-linear Schrodinger (NLS) equation can also be solved by this technique for the amplitudes vanishing at plus and minus infinity. Tracy et al. [5] have constructed different quasi-periodic solutions but the corresponding solutions are not explicit except for some special case. Therefore in order to understand the properties of the NLS equation, many numerical studies have been carried out [6,7,8,9]' In view of this it would also be very desirable to obtain approximate analytical results, which could reproduce the important features of the solutions. In this paper we will employ a variational approach, first given by Anderson [2], to solve the NLS equation.

Variational Description of NLS equation

The NLS equation

i[E.sub.t] + [E.sub.xx] + [absolute value of [E.sup.2]] E = 0 (1)

Here E (x, t) is the slowly varying envelope of high-frequency field.

The equation (1) can be formulated as a variational problem corresponding to the Lagrangian

L(t) = [[integral].sup.[lambda]/2.sub.-[lambda]/2] [??](x, t)dx (2)

The Lagrangian density [??](x, t) is given by

[[??].sub.NLS] = 1/2[[E.sup.*][E.sub.t] - c.c] - [[absolute value of [E.sub.x]].sup.2] + 1/2[(E.[E.sup.*]).sup.2] (3)

The asterisk denotes the complex conjugate and the limits of the integration is the periodicity length [lambda] of the solutions, which will later be assumed constant.

We employ the Ritz optimization procedure to the action integral S(t) = [integral]L(t)dt with respect to time dependent parameters of a trial function which admits

(1) the shape of an unmodulated wave with a small sinusoidal disturbance.

(2) provide spatial periodicity of the Lagrangian with period X .

So these features are provided by

E(x, t) = A(t)[dn(z; [beta]) + [beta]cn(z; [beta])] x exp[i{[kz/[alpha]] + c cos(2[pi]z/[alpha][lambda]) + [phi]}] (4)

The time-dependent functions, independent from one another, are A, [beta], c, [phi], [alpha], [x.sub.0] and k.

Here, dn(z; [beta]) and cn(z; [beta]) are the Jacobian elliptic functions for z = [alpha](x - [x.sub.0]) and [alpha] = 4K([beta])/[lambda], and K([beta]) is the complete elliptic integral of first kind. For small parameters [beta] and c, one obtain from equation (4) the following expressions:

E(x, t) = A(t)[1 + ([beta] + ic)cos z] x exp i[kz/[alpha] + [phi]] (5)

Equation (5) describes envelopes of a finite amplitude wave (with wave number k), slightly modulated by a plane wave with wave number [alpha].

Now we proceed to study the variation equations for the parameters A, (, c and 0 with the help of Euler-Lagrange equations. Upon substituting the trial function (4) in (3) with (2), the action integral S becomes:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

Here K'([beta]) = K[[(1 - [[beta].sup.2]).sup.-1/2]] and [C.sub.1], [C.sub.2] and [C.sub.3] are the following combinations of elliptic integrals K, E and their argument [beta]:

[C.sub.1] = 2E -(1 - [[beta].sup.2]) (7a)

[C.sub.2] = (1 + [[beta.sup.]2]) E - (1 - [[beta].sup.2]) k (7b)

[C.sub.3] = 8(1 + [[beta].sup.2]) E - (5 + 3[[beta].sup.2])(1 - [[beta].sup.2]) k (7c)

The details of calculations leading to (6) are in the next section.

Our system of equations, which follows from action integral (6), consists of two conservation laws and two Euler-Langrangian equations:

(1) Integrated variation 0 equation:

[A.sup.2][[2E/K] - (1 - [[beta].sup.2])] = N/[lambda] (8)

(2) The Hamiltonian H:

H = [N[V.sup.2]/4] + [[c.sup.2]/2M] + U (9)

with potential U = [16K[A.sup.2][C.sub.2]/3[lambda]] - [lambda][A.sup.4][C.sub.3]/6K (10)

and the time-dependent mass M = ([lambda]/4[[pi].sup.2])[[[N/[lambda]] - [[[pi].sup.2][A.sup.2]/{[K.sup.2] sinh([pi]K'/K)}]-.sup.1] (11)

(3) Variation c equation:

[[partial derivative]Q/[partial derivative]t] = c/M (12)

where Q is given by

Q = [pi][lambda][A.sup.2] sgn [beta]/[2[K.sup.2] sinh([pi]K'/2K)] (13)

(4) Variation A equation:

N[[phi].sub.t] + [Qc.sub.t] = [N[V.sup.2]/4] - [[c.sup.2]/2M] - [16K[A.sup.2][C.sub.2]/3[lambda]] + [lambda][C.sub.3][A.sup.4]/3K (14)

Equations (8)-(13) forms a closed system which can be solved by a single though non-elementary quadrature. The equation (14) gives the information of phase [phi] only, which the variational criterion leaves undetermined.

A simple qualitative analysis is also possible as a result of the Hamiltonian structure of the equations (8)-(14). It can be shown that the "mass" M of equation (11) is positive and finite for all [absolute value of [beta]] < 1, while the generalized coordinate Q is an

increasing function of [beta], continuously differentiable for [beta] [member of] [-1,1], including [beta] = 0. These facts enable us to analyze the evolution of the system on the basis of a phase diagram [partial derivative][beta]/[partial derivative]t vs [beta] rather than c vs Q using the "potential" U([beta]) after substituting A from (8). Thus the qualitative behavior of the system is entirely determined by its parameters in the linear regime: [beta] [congruent to] 0.

Derivation of Action Integral

The following formulas for derivation of complete elliptic integrals will be used:

1. D = [-2E/s([[beta].sup.2])] = (K - E)/[[beta].sup.2] > 0;

2. B = -2dK/d([[beta].sup.2])] = (E - [[beta].sup.2]K)/([[beta].sub.2][[beta].sup.2]) > 0; (15)

3. C = -2dB/s([[beta].sup.2])] = [[(2 - [beta]).sup.2] K - 2E)]/[[beta].sup.4] > 0 where [beta] = 1 - [[beta].sup.2]

We substitute our trial function (4) to the NLS Langragian [(2) with (3)] and we put [alpha](x - [x.sub.0]) = z and k(t) = V/2 (constant);

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

[I.sub.1] = [[integral].sup.2k.sub.-2k] dz[[dn(z; [beta]) + [beta] cn(z; [beta])].sup.2] cos([pi] z/2K) = 2[[pi].sup.2] sgn([beta])/K sinh([pi]K/2K) (17)

[I.sub.1] = [[integral].sup.2k.sub.-2k] dz[[dn(z; [beta]) + [beta] cn(z; [beta]).sup.]2] = 4[2E - [[beta]'.sup.2]K] = 4[C.sub.1] (18)

[I.sub.3] = [[integral].sup.2k.sub.-2k] dz[[dn(z; [beta]) + [beta] cn(z; [beta]).sup.]2] [sin.sup.2] ([pi] z/2K) = 2[2 E - [[beta]'.sup.2] K - [[pi].sup.2]/(K sinh [pi]K'/2K)] (19)

[I.sub.4] = [[integral].sup.2k.sub.-2k] dz[[d/dz]]{dn(z; [beta]) + [beta] cn(z; [beta])}].sup.2] = 4/3[(1 + [[beta].sup.2])E - (1 - [[beta].sup.2])K] = 4[C.sub.2]/3 (20)

[I.sub.5] = [[integral].sup.2k.sub.-2k] dz[[dn(z; [beta]) + [beta] cn(z; [beta])].sup.4] = 4/3 [8(1 + [[beta].sup.2])E - (5 + 3[[beta].sup.2])(1 - [[beta].sup.2])K] = 4[C.sub.3]/3 (21)

In calculating [I.sub.1] and [I.sub.3] we assume [alpha] = 4k/[lambda] and A = [alpha]/[square root of 2], c = 0, [x.sub.0] = 2kt.

Inserting this value of [alpha] and formula (17)-(21) into equation (16) we get the described integral (6).

Numerical Solution

The non-linear Schrodinger Equation i[E.sub.t] + [E.sub.xx] + [[absolute value of E].sup.2] E = 0

can be directly solved by finite difference scheme, where E(x, t) is the slowly varying envelope of the high frequency field given by

E(x, t) = A(t)[1 + ([beta] + ic) cos z] x exp i[[kz/[alpha]] + [phi]]

In order to illustrate the theoretical discussion of the previous sections we will analyze results of the variational methods and compare with the outcome from the numerical integration of the NLS equation.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

Fig. 1 is an example of time evolution of a homogeneous field with the amplitude A = 0.5 and periodicity length [alpha] = 0.6. Here one period of the nonlinear evolution is shown. Fig. 2 is the time evolution of the trial function (4) as predicted by equations (8)-(14) for the same initial conditions with [beta] = c = 0.01 and [phi] = 0. They both agree very well with a slight discrepancy between the periods of nonlinear evolution.

For a different set of parameters A = 1, [alpha] = 1.2, [beta] = c = 0.1 and V = 0, z = [alpha](x - [x.sub.0]), [x.sub.0] = 0.

The solution is given in Figs.3 and 4

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

Conclusion

We applied Ritz variational method based on the NLS Lagrangian to construct a model function between linear and non-linear evolution of modulational instability. Spatial variance of trial function was assumed a priori while time dependence of its parameters was subject to optimization. We choose two solutions in the form of a combination of Jacobian Elliptic functions (with an appropriate phase factor). The periodicity length and group velocities were assumed constant. The results of theoretical model compare well with numerical solutions to NLS equation.

References

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[11] A. V. Porubov and D. F. Parker, Some general periodic solutions to coupled nonlinear Schrodinger equations, Wave Motion 29, no. 2, 97-109 (1999).

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Arun Kumar

Department of Mathematics, Government College, Kota (Rajasthan), India
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Author:Kumar, Arun
Publication:International Journal of Computational and Applied Mathematics
Date:Aug 1, 2009
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