Exact Solutions > Nonlinear Partial Differential Equations > Second-Order
Parabolic ... Equations > Nonlinear Schrodinger Equation (Schrödinger Equation
). 3. i.
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i
∂w
∂2w
+ f (|w|)w = 0. ∂t ∂x2 Nonlinear Schrodinger equation (Schr¨odinger equation) of general form; f (u) is a real function of a real variable. 3.
+
1◦ . Suppose w(x, t) is a solution of the Schrodinger equation in question. Then the function w1 = e−i(λx+λ
2
t+C1 )
w(x + 2λt + C2 , t + C3 ),
where C1 , C2 , C3 , and λ are arbitrary real constants, is also a solution of the equation. 2◦ . Traveling-wave solution: £ ¤ w(x, t) = C1 exp iϕ(x, t) ,
ϕ(x, t) = C2 x − C22 t + f (|C1 |)t + C3 .
3◦ . Multiplicative separable solution: w(x, t) = u(x)ei(C1 t+C2 ) , where the function u = u(x) is defined implicitly by Z du p = C4 ± x, 2 C1 u − 2F (u) + C3
Z F (u) =
uf (|u|) du.
Here, C1 , . . . , C4 are arbitrary real constants. 4◦ . Solution:
w(x, t) = U (ξ)ei(Ax+Bt+C) ,
ξ = x − 2At,
(1)
where the function U = U (ξ) is determined by the autonomous ordinary differential equation 00 Uξξ + f (|U |)U − (A2 + B)U = 0. Integrating yields the general solution in implicit form: Z Z dU p = C2 ± ξ, F (U ) = U f (|U |) dU . (2) (A2 + B)U 2 − 2F (U ) + C1 Relations (1) and (2) involve arbitrary real constants A, B, C, C1 , and C2 . 5◦ . Solution (A, B, and C are arbitrary constants): £ ¤ w(x, t) = ψ(z) exp i(Axt − 32 A2 t3 + Bt + C) ,
z = x − At2 ,
00 where the function ψ = ψ(z) is determined by the ordinary differential equation ψzz + f (|ψ|)ψ − (Az + B)ψ = 0.
6◦ . Solutions: w(x, t) = ± √
£ ¤ 1 exp iϕ(x, t) , C1 t
ϕ(x, t) =
(x + C2 )2 + 4t
Z
¡ ¢ f |C1 t|−1/2 dt + C3 ,
where C1 , C2 , and C3 are arbitrary real constants. 7◦ . Solution: £ ¤ w(x, t) = u(x) exp iϕ(x, t) ,
Z ϕ(x, t) = C1 t + C2
dx + C3 , u2 (x)
where C1 , C2 , and C3 are arbitrary real constants, and the function u = u(x) is determined by the autonomous ordinary differential equation u00xx − C1 u − C22 u−3 + f (|u|)u = 0. 1
2
NONLINEAR SCHRODINGER EQUATION OF GENERAL FORM
8◦ . There is an exact solution of the form £ ¤ w(x, t) = u(z) exp iAt + iϕ(z) ,
z = kx + λt,
where A, k, and λ are arbitrary real constants. See also special cases of the nonlinear Schrodinger equation: • Schrodinger equation with a cubic nonlinearity , • Schrodinger equation with a power-law nonlinearity . References Gagnon, L. and Winternitz P., Lie symmetries of a generalized nonlinear Schrodinger equation. I. The symmetry group and its subgroups, J. Phys. A, Vol. 24, p. 1493, 1988. Gagnon, L. and Winternitz P., Lie symmetries of a generalized nonlinear Schrodinger equation. II. Exact solutions, J. Phys. A, Vol. 22, p. 469, 1989. Sulem, C. and Sulem, P.-L., The Nonlinear Schrodinger Equation. Self-Focusing and Wave Collapse, Springer-Verlag, New York, 1999. Akhmediev, N. N. and Ankiewicz, A., Solitons. Nonlinear Pulses and Beams, Chapman & Hall, London, 1997. Polyanin, A. D. and Zaitsev, V. F., Handbook of Nonlinear Partial Differential Equations , Chapman & Hall/CRC, Boca Raton, 2004.
Nonlinear Schrodinger Equation of General Form c 2004 Andrei D. Polyanin Copyright °
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