1D Cubic NLS (CNLS) and Cubic-Quintic NLS (CQNLS)

Transcription

1 1D Cubic NLS (CNLS) and Cubic-Quintic NLS (CQNLS) Marcello Sani December 5, 2006

2 Contents Introduction Physical applications of the CNLS From the Sine-Gordon to the 1D CNLS Single soliton solution for 1D CNLS Analytical solutions of the CQNLS An example of analytical solution of the CQNLS Numerical computation of a solution of the CQNLS Moving waves

3 Introduction The nonlinear cubic-quintic Schrödinger equation (CQNLS) is the following differential equation: where a 1 and a 2 are real constants. ıu t + u xx + a 1 u u 2 a 2 u u 4 = 0, It takes its name from the fact that the small amplitude approximation is the equation that Schrödinger proposed in the year 1926 for the propagation of a quantum wave packet in free space.

4 Physical applications of the CNLS The CNLS is a generic equation, arising whenever one studies unidirectional propagation of wave packets in a dispersive energy conserving medium at the lowest order of nonlinearity. Applications of the CNLS are Description of non-linear pulses on an optical fiber Two-dimensional self-focusing of a plane wave One-dimensional self-modulation of a monochromatic wave Propagation of a heat pulse in a solid Langmuir waves in plasmas.

5 From the Sine-Gordon to the 1D CNLS Start from the Sine-Gordon equation u tt c 2 0u xx + ω 2 0 sin u = 0. We consider only the first two terms of the Taylor-development of the sinus function sin u = u u A first idea could be to look for a solution of the form of a plane wave with a small perturbating term u(x, t) = εae ı(qx ωt) + ε 2 B(x, t), but in this case we would obtain B(x, t) ε 2 t, i.e. B(x, t) diverges.

6 Introduce multiple scales expansion so the solution will be of the form u(x, t) = ε We use the notation T i = ε i t, X i = ε i x, ε i φ i (X 0, X 1, X 2,..., T 0, T 1, T 2,...). i=0 D i = T i, D Xi = X i.

7 At the order ε we need to solve ( D 2 0 c 2 0DX ω0 2 ) φ0 = 0. The solution is a plane wave: φ 0 = A(X 1, T 1, X 2, T 2,...)e ı(qx0 ωt0) + c.c., where c.c. is the complex conjugate. It holds the dispersion relation ω 2 = ω c2 0 q2.

8 At the order ε 2 we find D 2 0φ 1 + 2D 0 D 1 φ 0 c 2 0D 2 X 0 φ 1 2c 2 0D X0 D X1 φ 0 + ω 2 0φ 1 = 0. We need the following condition to eliminate the secular terms A + qc2 0 A = 0, T 1 ω X 1 where v g := qc2 0 ω is the group velocity. The solution is therefore φ 1 = 0.

9 Order ε 3 : D 2 1φ 0 2D 0 D 2 φ 0 + c 2 0D 2 X 1 φ 0 + 2c 2 0D X0 D X2 φ 0 + ω2 0 6 φ3 0 As above we need the condition 2 A T 2 1 2D 0 D 1 φ 1 + 2c 2 0D X0 D X1 φ 1 = ıω A + c 2 2 A 0 + 2ıqc 2 A T 2 X 2 6 ω2 0 A 2 A = 0. Introducing the new variables X 2 1 ξ i = X i v g T i, τ i = T i and using the condition of the order ε 2, we have ( c 2 0 vg) 2 2 ( ) A A A ξ ıω v g + 2ıqc 2 A τ 2 ξ 2 ξ 2 2 ω2 0 A 2 A = 0.

10 Using the expression for the group velocity we obtain ı A τ 2 + which is just the CNLS with ( c 2 0 v 2 g) 2ω v g = qc 2 0/ω, 2 A ξ ω2 0 4ω A 2 A = 0, ω 2 0 a 1 = ω2 0 2ω ( = 4ω c 2 0 vg) 2 2 ( c 2 0 ). v2 g

11 Single soliton solution for 1D CNLS This is found by looking for solutions of the CNLS, with a 1 = ν, depending on a moving coordinate X = x Ut: u = e ırx ıst v(x), X = x Ut, where r and s are constants. On substitution, the ordinary differential equation for v is We now choose v + ı(2r U)v + (s r 2 )v + ν v 2 v = 0. r = U 2 and s = U 2 4 α, the first being the important one to eliminate the term in v.

12 Then v may be taken to be real and v αv + νv 3 = 0. This gives rise to a cnoidal wave equation for v. It may be integrated once to v 2 = A + αv 2 ν 2 v4, which can be solved in elliptic functions. The limiting case of the solitary wave is possible when ν > 0; we take A = 0, α > 0, and the solution is v = ( ) 1/2 2α ( ) sech α 1/2 (x Ut). ν

13 Analytical solutions of the CQNLS As seen it is possible to find analytic solutions of the CQNLS. One way of doing this is the following: we consider the differential equation in the form ıu t + u xx = a 1 u u 2 + a 2 u u 4 and we use the Ansatz u(x, t) = f(x)e ıat, where a is a real constant and f a complex function. The CQNLS reduces to f xx + af = a 1 f f 2 + a 2 f f 4.

14 We now set f(x) = M(x)e ın(x), where M and N are real functions. Separating the real and the imaginary part of f(x), we get M xx M (N x ) 2 + am = a 1 M 3 + a 2 M 5 and 2M x N x + MN xx = 0.

15 Multiplying the second equation by M 2MM x N x + M 2 N xx = 0, i.e. ( M 2 ) x N x + M 2 N xx = 0, and integrating twice we obtain N in terms of M N = S M 2 dx + N 0, where S and N 0 are real integration constants. N 0 represents a constant change of phase. Substituting this last equation into the first, leads to M xx S 2 M 3 + am = a 1 M 3 + a 2 M 5.

16 We have now to multiply by M x and integrate to get (M x ) 2 + S 2 M 2 + am 2 = a 1 M a 2 M K 0, where K 0 is an integration constant. This equation leads to a standard elliptic integral by the substitution M(x) = [pw (y)] 1/2, pw > 0, where p is a nonvanishing constant and y = y = ( pa1 ) 1/2 x, for a2 = 0, 2 ( ) 1/2 4a2 px, for a

17 Then we have for the cubic case and for the quintic case (W y ) 2 = 4W 3 8a a 1 p W 2 + 4KW 8S2 a 1 p 3 4(W W 1 )(W W 2 )(W W 3 ) (W y ) 2 = W 4 + 3a 1 2a 2 p W 3 3a a 2 p 2 W 2 + 4KW 3S2 a 2 p 4 (W W 1 )(W W 2 )(W W 3 )(W W 4 ), where K := pk 0 y 2 /x 2 is a constant and the W i s are the roots of the right-hand sides.

18 An example of analytical solution of the CQNLS The equation has the solution for ıu t + u xx + u u 2 δu u 4 = 0 u(x, t) = sech x 2[ C + 1 C 2 sech2 x ] 1/2 eıt, C = 4δW 3 3 = 4δ 3 = 1 + 2δ 3 ( ) 3 4δ δ 2 (3 16δ) 3 4δ 2 (3 16δ).

19 Where W 3 is one of the roots of the polynomial W δ W δ W 2, i.e. W 1 = 0, W 2 = 0 W 3 = 3 4δ D, W4 = 3 4δ 1 2 D, with D = δ 4 δ 2.

20 Note that the constant C is complex for δ > 3/16. In fact D < 0 for δ > 3 16 = , because D = δ 4 δ 2 < δ δ 2 < δ < 0 δ > 3 16.

21 Numerical computation of a solution of the CQNLS We look for a solution of the CQNLS for δ = 1, using a known solution, ıu t + u xx + u u 2 δu u 4 = 0, z(x, t) = 2 sech(x)e ıt, of the CNLS (δ = 0). We perform numerical homotopy continuation of z(x, t) by slowly increasing the value of δ up to 1. Suppose at first the wanted solution has the form u(x, t) = v(x)e ıt, for a real function v(x) that we need to determine numerically.

22 This leads to the following differential equation for v v + v + v 3 δv 5 = 0, that we discretize with the finite difference method and solve, at each value of δ, using the Newton iteration. We are starting from the function v 0 (x) := z(x, 0) = 2 sech(x) that sastisfies the differential equation v 0 + v 0 + v 3 0 δv 5 0 = 0. As yet seen, analitically the obtained solution would be u(x, t) = sech(x) 2[ C + 1 C/2 sech 2 (x) ] 1/2 eıt, for a constant C depeding on δ.

23 Moving waves The above numerically calculated wave is a stationary one. We are now going to see how it is possible to derive a moving one. We assume solves the CQNLS It follows that v satisfies u(x, t) = Ae ıωt v(x) ıu t + u xx + u u 2 δu u 4 = 0. Aωv + Av + A 3 v 3 δa 5 v 5 = 0. Use Galilean boost to obtain a moving wave, with speed w, starting from a stationary one. Let x := x wt t := t.

24 We would like to know how frequency and amplitude get transformed ω =? We define à =? ũ(x, t) := Ã(w)e ı( ω(w) t+µx) v( x) = Ã(w)e ı( ω(w)t+µx) v(x wt), for a constant µ. Inserting, we get a new differential equation for v ( ı ı ω(w)ã(w)e ı( ω(w)t+µx) v wã(w)e ı( ω(w)t+µx) v ) +Ã(w)e ı( ω(w)t+µx) v +2ıµÃ(w)e ı( ω(w)t+µx) v µ 2 Ã(w)e ı( ω(w)t+µx) v + à 3 (w)v 3 e ı( ω(w)t+µx) δã 5 (w)v 5 e ı( ω(w)t+µx) = 0.

25 It follows Ã(w)v Ã(w) ω(w)v ıã(w)wv + 2ıÃ(w)µv Ã(w)µ 2 v Taking now w = 2µ, we get +Ã 3 (w)v 3 δã 5 (w)v 5 = 0. v ω(w)v µ 2 v + Ã 2 (w)v 3 δã 4 (w)v 5 = 0. So ũ is still a solution of the CQNLS if Then also solves the CQNLS. ω = ω µ 2 = ω w2 4 Ã = A. ũ (x, t) = Ae ı (ω w 2 4 )+ w 2 x v(x wt).

26 References Cagnon, L.: Exact travelling-wave solutions for optical models based on the nonlinear cubic-quintic Schrödinger equation. JOSA A. 6, (1989) Peyard, M., Dauxois, T.: Physique des solitons. Les Ullis: EDP Eciences, (2004) Scott, A.: Nonlinear science. Oxford Univ. Press, Oxford, 56 (2003) Whitham G. B.: Linear and Nonlinear waves. Wiley- Interscience, Sec. 17.8, (1974)