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.
Parts
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» 6.15. Equations of the Form ∂w i
» Equations of the Form ∂w 2 = f (x, t, w) ∂ w +g x, t, w, ∂w
» Equations with Cubic Nonlinearities Involving Arbitrary Functions
» Equations of General Form Involving Arbitrary Functions of a Single Argument
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