Two Dimensional Equations

2.6.1. Two Dimensional Equations

∂w ∂ 2 w

1. i

+ A|w| 2 w = 0.

∂t

∂x 2 ∂y 2

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

2.6.1.3 with

f (u) = Au 2 .

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

w 2 1 = ✿ C 1 w( ✿ C 1 x+C 2 , ✿ C 1 y+C 3 , C

1 t+C 4 ),

− i[λ 1 x+λ 2 y+(λ 1 w 2 2 = + e λ 2 2 ) t+C 5 ] w(x + 2λ 1 t, y + 2λ 2 t, t),

w 3 = w(x cos β − y sin β, x sin β + y cos β, t), where C 1 , ...,C 5 , λ 1 , λ 2 , and β 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 ◦ . Solutions:

1 2 3 2 2 w(x, y, t) = C 2 exp x+C y + (AC 1 − C 2 − C 3 ) t+C 4 ] ,

1 2 2 3 2 C 2 ( x+C ) +( y+C ) −4 AC 1

w(x, y, t) =

exp i

+ iC 4 ,

where C 1 , ...,C 4 are arbitrary real constants.

3 ◦ . “Two-dimensional” solution:

w(x, y, t) = e i(C 1 t+C 2 ) u(x, y),

where C 1 and C 2 are arbitrary real constants, and the function u = u(x, y) is determined by the stationary equation

2 ∂ 2 u ∂ u + 3 + Au − C 1 u = 0. ∂x 2 ∂y 2

4 ◦ . Solution: w(x, y, t) = (f 2 1 x+f 2 y+f 3 ) exp 1 x + g 2 xy + g 3 y 2 + h 1 x+h 2 y+h 3 )

where the functions f k = f k ( t), g k = g k ( t), and h k = h k ( t) are determined by the autonomous system of ordinary differential equations

The prime denotes a derivative with respect to t.

w(x, y, t) = U (ξ 1 , ξ 2 ) e i(k 1 x+k 2 y+at+b) , ξ 1 = x − 2k 1 t, ξ 2 = y − 2k 2 t, where k 1 , k 2 ,

a, and b are arbitrary constants, and the function U = U (ξ 1 , ξ 2 ) is determined by a differential equation of the form 5.4.1.1:

2 + A|U | U − (k 1 + k 2 + a)U = 0.

6 ◦ . “Two-dimensional” solution:

2 2 3 2 2 3 2 w(x, y, t) = Φ(z 2

z 1 = x−k 1 t , z 2 = y−k 2 t , where k 1 , k 2 ,

1 , z 2 ) exp

1 xt + k 2 yt − 3 k 1 t − 3 k 2 t + at + b)

a, and b are arbitrary constants, and the function Φ = Φ(z 1 , z 2 ) is determined by a differential equation of the form 5.4.1.1:

2 + A|Φ| Φ−( k 1 z 1 + k 2 z 2 + a)Φ = 0.

∂z 1 ∂z 2

7 ◦ . “Two-dimensional” solution:

1 x+C 3 y+C 4 w(x, y, t) = √

where C 1 , ...,C 4 are arbitrary constants, and the function u = u(ξ, η) is determined by the differential equation

+ u + A|u| u = 0.

2. 2 i + + A|w| n w = 0. ∂t

∂x 2

∂y 2

Two-dimensional Schr ¨odinger equation with a power-law nonlinearity ;

A and n are real numbers. This is a special case of equation 2.6.1.3 with 2 f (u) = Au n .

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 n y+C 3 , C 1 n t+C 4 ),

2 = i[λ 1 x+λ 2 y+(λ 1 λ 2 w ) e t+C 5 ] w(x + 2λ 1 t, y + 2λ 2 t, t),

w 3 = w(x cos β − y sin β, x sin β + y cos β, t),

where C 1 , ...,C 5 , β, λ 1 , and λ 2 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 ◦ . Solutions:

2 w(x, y, t) = C 2 1 exp 2 x+C 3 y + (A|C 2 1 | n − C

w(x, y, t) =

where C 1 , ...,C 4 are arbitrary real constants.

. For other exact solutions, see equation 2.6.1.3 with 2 f (w) = Aw n .

3. i

2 ∂t + ∂x ∂y 2 + f (|w|)w = 0.

Two-dimensional nonlinear Schr ¨odinger equation of general form .

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

1 = e i[λ 1 x+λ 2 y+(λ 1 λ 2 ) t+A] w(x + 2λ 1 t+C 1 , y + 2λ 2 t+C 2 , t+C 3 ),

w 2 = w(x cos β − y sin β, x sin β + y cos β, t),

where

1 , A, C C 2 , C 3 , λ 1 , λ 2 , and β are arbitrary real constants, are also solutions of the equation.

2 ◦ . Traveling-wave solution:

1 2 3 4 1 2 w(x, y, t) = C 2 exp x+C y + λt + C ) λ = f (|C |) − C 2 − C 3 ,

where C 1 , ...,C 4 are arbitrary real constants.

2 r= 2 px + y and time t are determined by the equation

3 ◦ . Exact solutions depending only on the radial variable

∂w

∂w

+ f (|w|)w = 0,

which is a special case of equation 1.7.5.2 with n = 1.

4 ◦ . “Two-dimensional” solution:

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

where

A and B are arbitrary real constants, and the function u = u(x, y) is determined by a stationary equation of the form 5.4.1.1:

2 + f (|u|)u − Au = 0.

∂x

∂y

5 ◦ . “Two-dimensional” solution: w(x, y, t) = U (ξ, η)e i(A 1 x+A 2 y+Bt+C) , ξ = x − 2A 1 t, η = y − 2A 2 t, where A 1 , A 2 ,

B, and C are arbitrary constants, and the function U = U (ξ, η) is determined by a differential equation of the form 5.4.1.1:

2 + f (|U |)U − (A 1 + A 2 + ∂ξ B)U = 0. ∂η

6 ◦ . “Two-dimensional” solution:

2 2 3 2 2 3 2 w(x, y, t) = Φ(z 2

z 1 = x−k 1 t , z 2 = y−k 2 t , where k 1 , k 2 ,