Three and n Dimensional Equations

2.6.2. Three and n Dimensional Equations

∂w ∂ 2 w

1. i

+ A|w| w = 0.

Three-dimensional Schr ¨odinger equation with a cubic nonlinearity . This is a special case of equation

2.6.2.2 with

f (u) = Au 2 .

1 ◦ . Suppose w(x, y, t) is a solution of the Schr ¨odinger equation in question. Then the functions

1 = C 1 w( C 1 x+C 2 , C 1 y+C 3 , C 1 z+C 4 , C 1 t+C 5 ), − i[λ 1 x+λ 2 y+λ 3 z+(λ 2 1 + λ 2 2 + λ 3 w 2

e ) t+C 6 ] w(x + 2λ 1 t, y + 2λ 2 t, z + 2λ 3 t, t), where C 1 , ...,C 6 , λ 1 , λ 2 , and λ 3 are arbitrary real constants, are also solutions of the equation. The

plus or minus signs in the expression for w 1 are chosen arbitrarily.

2 ◦ . There is an exact solution of the form

where f k = f k ( t), g k = g k ( t), and h k = h k ( t).

3 ◦ . Solution: w(x, y, z, t) = U (ξ 1 , ξ 2 , ξ 3 ) e i(k 1 x+k 2 y+k 3 z+at+b) , ξ 1 = x − 2k 1 t, ξ 2 = y − 2k 2 t, ξ 3 = z − 2k 3 t, where k 1 , k 2 , k 3 ,

a, and b are arbitrary constants, and the function U = U (ξ 1 , ξ 2 , ξ 3 ) is determined by the differential equation

2 + A|U | U − (k 1 + k 2 + k 3 + a)U = 0.

4 ◦ . “Three-dimensional” solution:

1 x+C 3 y+C 4 z+C 5 w(x, y, z, t) = √

C t+C 2 C 1 t+C 2 C 1 t+C 2 C 1 t+C 2 where C 1 , ...,C 5 are arbitrary constants, and the function u = u(ξ, η, ζ) is determined by the

differential equation

+ 2 + − iC 1 ξ + η + ζ + u + A|u| u = 0.

References : L. Gagnon and P. Winternitz (1988, 1989), N. H. Ibragimov (1995), A. M. Vinogradov and I. S. Krasil’shchik (1997).

∂w ∂ 2 w

2. i

+ f (|w|)w = 0.

∂t

∂x 2 ∂y 2 ∂z 2

Three-dimensional nonlinear Schr ¨odinger equation of general form . It admits translations in any of the independent variables.

1 ◦ . Suppose w(x, y, z, t) is a solution of the Schr ¨odinger equation in question. Then the function − i[λ 1 x+λ 2 y+λ 3 z+(λ 1 2 + 2 + w 2

e ) λ 2 λ 3 t+A] w(x + 2λ 1 t+C 1 , y + 2λ 2 t+C 2 , z + 2λ 3 t+C 3 , t+C 4 ), where

1 , A, C ...,C 4 , λ 1 , λ 2 , and λ 3 are arbitrary real constants, is also a solution of the equation.

2 ◦ . Exact solutions depending only on the radial variable r= 2 px + 2 y 2 + z and time t are determined by the equation

∂w

2 ∂w

+ f (|w|)w = 0,

which is a special case of equation 1.7.5.2 with n = 2.

w(x, y, z, t) = e i(At+B) u(x, y, z),

where

A and B are arbitrary real constants, and the function u = u(x, y, z) is determined by the stationary equation

∆ u + f (|u|)u − Au = 0.

4 ◦ . Axisymmetric solutions in cylindrical and spherical coordinates are determined by equations where the Laplace operator has the form

5 ◦ . “Three-dimensional” solution:

2 2 2 w = U (ξ, η, t), 2 ξ=y+ , η = (C − 1) x −2 Cxy + C z ,

where

C is an arbitrary constant (C ≠ 0), and the function U = U (ξ, η, t) is determined by the differential equation

6 ◦ . “Three-dimensional” solution: w = V (ξ, η, t), p ξ = Ax + By + Cz, η= ( Bx − Ay) 2 +( Cy − Bz) 2 +( Az − Cx) 2 ,

where

A, B, and C are arbitrary constants and the function V = V (ξ, η, t) is determined by the equation

References : L. Gagnon and P. Winternitz (1988, 1989), N. H. Ibragimov (1995).

∂w

∆w + |w| 2 w. ∂t This is an n-dimensional Schr¨odinger equation with a cubic nonlinearity. Conservation laws: | 2 w|

3. i

t + i∇ ⋅ ¯ w∇w − w∇ ¯ w x = 0,

w)∇ ¯ w − (∆ ¯ w + |w| w)∇w ¯ x = 0. The bar over a symbol denotes the complex conjugate. ❄☎❅

2 1 4 2 |∇ 2 w| −

2 | w| t + i∇ ⋅

Reference : A. M. Vinogradov and I. S. Krasil’shchik (1997).

Chapter 3

Hyperbolic Equations with One Space Variable