Equations of General Form Involving Arbitrary Functions of Two Arguments

1.7.6. Equations of General Form Involving Arbitrary Functions of Two Arguments

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

1. i

+ f (x, |w|)w = 0.

∂t

∂x 2

Schr¨odinger equation of general form ; f (x, u) is a real function of two real variables.

1 ◦ . Multiplicative separable solution:

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

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

u ′′ xx − C 1 u + f (x, |u|)u = 0.

2 ◦ . Solution: Z dx w(x, t) = u(x) exp iϕ(x, t) ,

ϕ(x, t) = C 1 t+C 2 + C 3 , u 2 ( x)

where C 1 , C 2 , and C 3 are arbitrary real constants, and the function u = u(x) is determined by the ordinary differential equation

2 u −3 ′′

xx − C 1 u−C 2 u + f (x, |u|)u = 0.

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

2 + ∂t f (t, |w|)w = 0. ∂x Schr¨odinger equation of general form ; f (t, u) is a real function of two real variables.

2. i

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

1 = e i(λx+λ t+C w(x + 2λt + C 2 , t),

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

2 ◦ . Solutions:

w(x, t) = C 1 exp iϕ(x, t) ,

ϕ(x, t) = C 2 x−C 2 2 t+

f (t, |C 1 |) dt + C 3 ;

( x+C 2 ) 2 Z

w(x, t) = C 1 t

exp iψ(x, t) ,

ψ(x, t) =

f t, |C 1 | t −1 /2 dt + C 3 ,

4 t ☞✁✌ where C 1 , C 2 , and C 3 are arbitrary real constants.

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

+ f (x, |w|)w = 0.

∂t x n ∂x

∂x

Schr¨odinger equation of general form ; f (x, u) is a real function of two real variables. To n = 1 there corresponds a two-dimensional Schr ¨odinger equation with axial symmetry and to n = 2,

a three-dimensional Schr ¨odinger equation with central symmetry.

1 ◦ . Multiplicative separable solution:

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

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

x − n ( x n u ′ x ) ′ x − C 1 u + f (x, |u|)u = 0.

2 ◦ . Solution:

dx w(x, t) = u(x) exp iϕ(x, t) ,

ϕ(x, t) = C 1 t+C 2 n 2 + C 3 , x u ( x)

where C 1 , C 2 , and C 3 are arbitrary real constants, and the function u = u(x) is determined by the ordinary differential equation

x − n ( x n ☞✁✌ ′ u x ) ′ x − C 1 u−C 2 x −2 n u 2 −3 + f (x, |u|)u = 0.

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

+ f (t, |w|)w = 0.

∂t ∂x

∂x

Schr¨odinger equation of general form ; f (t, u) is a real function of two real variables. Solution:

− n+1

x 2 − n+1

w(x, t) = C 1 t 2 exp iϕ(x, t) ,

ϕ(x, t) =

f t, |C 1 | t 2 dt + C 2 ,

where ☞✁✌ C 1 and C 2 are arbitrary real constants.

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

+ g(x)

Φ(x, |w|)w = 0.

Schr¨odinger equation of general form ; Φ( x, u) is a real function of two real variables. The case g(x) = f x ′ ( x) corresponds to an anisotropic medium.

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

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

f (x)u ′′ xx + g(x)u ′ x − C 1 u + Φ(x, |u|)u = 0.

2 ◦ . Solution:

w(x, t) = U (x) exp iϕ(x, t) , Z R(x)

Z g(x)

ϕ(x, t) = C 1 t+C 2 2 dx + C 3 , R(x) = exp −

dx ,

f (x) where C 1 , C 2 , and C 3 are arbitrary real constants, and the function U = U (x) is determined by the

U ( x)

ordinary differential equation

x)U −3 + Φ( x, |U |)U = 0. ∂w

f (x)U xx ′′ + g(x)U x ′ −

1 U−C 2 f (x)R (

2 + g 1 ( t, |w|) + ig 2 ( t, |w|) ∂t w. ∂x Solution:

6. = f 1 ( t, |w|) + if 2 ( t, |w|)

w(x, t) = u(t) exp iϕ(x, t) ,

g 2 ( t, |u|) − C 2 1 f 2 ( t, |u|) dt + C 2 , where C 1 and C 2 are arbitrary real constants, and the function u = u(t) is determined by the ordinary

ϕ(x, t) = C 1 x+

differential equation

ug 2

1 ( t, |u|) − C 1 uf 1 ( t, |u|).

∂w

2 + g 1 ( x, |w|) + ig 2 ( x, |w|) ∂t w. ∂x Solution:

7. = f 1 ( x, |w|) + if 2 ( x, |w|)

ϕ(x, t) = C 1 t + θ(x), where the functions u = u(x) and θ = θ(x) are determined by the system of ordinary differential

w(x, t) = u(x) exp iϕ(x, t) ,

equations

1 u ′′ xx − f 1 u(θ ′ x ) − f 2 uθ ′′ xx −2 f 2 u ′ x θ ′ x + g 1 (| u|)u = 0,

f 1 uθ ′′ xx − f 2 u(θ ′ x ) 2 + f 2 u ′′ xx +2 f 1 u ′ x θ ′ x − C 1 u+g 2 (| u|)u = 0.

Here, f n = f n ( x, |u|), g n = g n ( x, |u|), n = 1, 2.

∂w ∂ 2 w

8. i

f (t, |w|)w = 0.

Solution: w(x, t) = u(t) exp[iv(x, t)], v(x, t) = ϕ(t)x 2 + ψ(t)x + χ(t),

where the functions u = u(t), ϕ = ϕ(t), ψ = ψ(t), and χ = χ(t) are determined by the system of ordinary differential equations

u ′ t +2 ϕu = 0, ϕ ′ t +4 ϕ 2 = 0, ψ ′ t +4 ϕψ + 2ϕf (t, u) = 0,

t + ψ + ψf (t, u) = 0.

Integrating yields

t+C 1 4( t+C 1 ) where C 1 , ...,C 4 are arbitrary real constants.

9. i

+ ∂x 2

f (x, |w|)w = 0.

w(x, t) = U (x) exp[iβt + iV (x)],

where β is an arbitrary real constant, and the real functions U = U (x) and V = V (x) are determined by the system of ordinary differential equations

f (t, |w|)w = 0.

∂t

∂x 2

Solution: w(x, t) = u(t) exp[iv(x, t)], v(x, t) = ϕ(t)x 2 + ψ(t)x + χ(t),

where the functions u = u(t), ϕ = ϕ(t), ψ = ψ(t), and χ = χ(t) are determined by the system of ordinary differential equations

u t ′ +2 uϕf (t, u) = 0, ϕ ′ t +4 2 ϕ f (t, u) = 0, ψ ′ t +4 ϕψf (t, u) = 0, χ ′ t + ψ 2 f (t, u) = 0.

Integrating yields

2 ϕ=C 4

χ = −C 2 u f (t, u) dt + C 3 , where C 1 , C 2 , and C 3 are arbitrary real constants, and u = u(t) is determined by the ordinary

ψ=C 2 u

differential equation u ′ t +2 C 1 u 3 f (t, u) = 0.

f (t, |w|)

Solution: w(x, t) = u(t) exp[iv(x, t)], v(x, t) = ϕ(t)x 2 + ψ(t)x + χ(t),

where the functions u = u(t), ϕ = ϕ(t), ψ = ψ(t), and χ = χ(t) are determined by the system of ordinary differential equations

u ′ t +2 uϕf (t, u) = 0, ϕ ′ t +4 ϕ 2 f (t, u) = 0, ψ t ′ +4 ϕψf (t, u) = 0,

f (t, u) = 0.

Integrating yields

ϕ=C 1 u 2 , ψ=C 2 u 2 , χ = −C 2 2 u 4 f (t, u) dt + C 3 , where C 1 , C 2 , and C 3 are arbitrary real constants, and u = u(t) is determined by the ordinary

differential equation u

f (x, |w|)

There is a solution of the form

w(x, t) = U (x) exp[iβt + iV (x)],

where β is an arbitrary real constant, and the functions U = U (z) and V = V (z) are determined by an appropriate system of ordinary differential equations (which is not written out here).

Chapter 2

Parabolic Equations with Two or More Space Variables