Equations with Cubic Nonlinearities Involving Arbitrary Functions

1.7.4. Equations with Cubic Nonlinearities Involving Arbitrary Functions

◮ Throughout this subsection, w is a complex function of real variables x and t; i 2 = −1. ∂w

1. i

f (t)|w| 2 + g(t) w = 0.

∂t ∂x Schr¨odinger equation with a cubic nonlinearity . Here, f (t) and g(t) are real functions of a real variable.

1 ◦ . Solution:

w(x, t) = C 1 exp iϕ(x, t) ,

ϕ(x, t) = C 2 x−C 2 2 t+

C 2 1 f (t) + g(t) dt + C 3 ,

where C 1 , C 2 , and C 3 are arbitrary real constants.

2 ◦ . Solution:

C 1 ( x+C 2 ) 2

w(x, t) = √ exp iϕ(x, t) ,

ϕ(x, t) =

C 2 1 f (t) + tg(t) + C 3 ,

where C 1 , C 2 , and C 3 are arbitrary real constants.

3 ◦ . Solution:

w(x, t) = (ax + b) exp i(αx 2 + βx + γ) , w(x, t) = (ax + b) exp i(αx 2 + βx + γ) ,

a ′ t = −6 aα,

b ′ t = −2 aβ − 2bα, α ′ t = f (t)a 2 −4 α 2 , β ′ t =2 f (t)ab − 4αβ, γ t ′ = f (t)b 2 − β 2 + g(t).

∂w ∂ 2 w

2. i

t)]w|w| 2 2 +[ f 1 ( t) + if 2 ( +[ g 1 ( t) + ig 2 ( t)]w = 0.

∂t ∂x Equations of this form occur in nonlinear optics.

1 ◦ . Solutions:

w(x, t) = ü u(t) exp[iϕ(x, t)], ϕ(x, t) = C 1 x−C 2 1 t+ [ f 1 ( t)u 2 ( t) + g 1 ( t)] dt + C 2 . Here, the function u = u(t) is determined by the Bernoulli equation u

t ′ + f 2 ( t)u + g 2 ( t)u = 0, whose general solution is given by

u(t) = C 3 e G(t) +2 e G(t)

e − G(t) f 2 ( t) dt

, G(t) = 2 g 2 ( t) dt.

2 ◦ . Solutions:

w(x, t) = u(t) exp[iϕ(x, t)], ϕ(x, t) = + [ f 1 ( t)u 2 ( t) + g 1 ( t)] dt + C 2 ,

x+C 1 )

where the function u = u(t) is determined by the Bernoulli equation

t + f 2 ( 3 t)u + g 2 ( t) +

u = 0.

Integrating yields

u(t) = C 3 e G(t) +2 e G(t) e − G(t) f 2 ( t) dt

, G(t) = ln t + 2

g 2 ( t) dt.

Reference : A. D. Polyanin and V. F. Zaitsev (2002).

∂w ∂ 2 w

3. i

+[ f 1 ( x) + if 2 ( x)]w|w| 2 +[ g 1 ( x) + ig 2 ( x)]w = 0.

w(x, t) = ü u(x) exp[iC 1 t + iθ(x)],

where the functions u = u(x) and θ = θ(x) are determined by the system of ordinary differential equations

2 +[ g 1 ( t) + ig 2 ( t)]w|w| 2 +[ h 1 ( t) + ih 2 ( ∂t t)]w = 0. ∂x Solutions:

w(x, t) = ü

2 u(t) exp[iϕ(x, t)], 2 ϕ(x, t) = C

1 x+ [− C 1 f 1 ( t) + g 1 ( t)u ( t) + h 1 ( t)] dt + C 2 .

u ′ t + g 2 ( t)u 3 +[ h 2 ( t) − C 2 1 f 2 ( t)]u = 0,

whose general solution is given by

u(t) = C 3 e +2 F (t) e F (t) e −

Reference : A. D. Polyanin and V. F. Zaitsev (2002).

2 +[ g 1 ( x) + ig 2 ( x)]w|w| 2 +[ h 1 ( x) + ih 2 ( ∂t x)]w = 0. ∂x Solutions:

w(x, t) = ✂ u(x) exp[iC 1 t + iθ(x)],

where the functions u = u(x) and θ = θ(x) are determined by the system of ordinary differential equations

Reference : A. D. Polyanin and V. F. Zaitsev (2002).

∂w

2 +[ g 1 ( t) + ig 2 ( t)]w|w| 2 +[ h 1 ( t) + ih 2 ( ∂t t)]w = 0. ∂x With f n , g n , h n = const, this equation is used for describing two-component reaction-diffusion

6. +[ f 1 ( t) + if 2 ( t)]

systems near a bifurcation point; see Kuramoto and Tsuzuki (1975). Solutions:

w(x, t) = ✂ u(t) exp[iϕ(x, t)],

1 x+ [ C 1 f 2 ( t) − g 2 ( t)u ( t) − h 2 ( t)] dt + C 2 . Here, the function u = u(t) is determined by the Bernoulli equation

2 ϕ(x, t) = C 2

u ′ t + g 1 ( t)u 3 +[ h 1 ( t) − C 2 1 f 1 ( t)]u = 0,

whose general solution is given by

u(t) = C 3 e +2 F (t) e F (t) e −

+[ g ( x)]w|w| 1 2 ( x) + ig 2 +[ h 1 ( x) + ih 2 ( x)]w = 0. ∂t

w(x, t) = ✂ u(x) exp[iC 1 t + iθ(x)],

where the functions u = u(x) and θ = θ(x) are determined by the system of ordinary differential equations

f 1 u ′′ xx − f 1 u(θ x ′ ) 2 − f 2 uθ

xx ′′ −2 f 2 u ′ x θ x ′ + g 1 u + h 1 u = 0,

f 2 u ′′ xx + C 1 u−f 2 u(θ x ′ ) 2 +

1 uθ ′′ xx +2 f 1 u x ′ θ x ′ + g 2 u + h 2 u = 0.