Equations of General Form Involving Arbitrary Functions of a Single Argument

1.7.5. Equations of General Form Involving Arbitrary Functions of a Single Argument

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

1. i

+ f (|w|)w = 0.

∂t

∂x 2

Schr¨odinger equation of general form ; f (u) is a real function of a real variable.

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+C 3 ),

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

2 ◦ . Traveling-wave solution:

w(x, t) = C 1 exp

iϕ(x, t) 2 , ϕ(x, t) = C

2 x−C 2 t + f (|C 1 |) t+C 3 .

3 ◦ . Multiplicative separable solution:

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

where the function u = u(x) is defined implicitly by

uf (|u|) du.

C 1 u −2

F (u) + C 3

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

4 ◦ . Solution:

(1) where the function U = U (ξ) is determined by the autonomous ordinary differential equation

w(x, t) = U (ξ)e i(Ax+Bt+C) ,

ξ = x − 2At,

U ξξ ′′ + f (|U |)U − (A 2 + B)U = 0. Integrating yields the general solution in implicit form: Z

Relations (1) and (2) involve arbitrary real constants

A, B, C, C 1 , and C 2 .

5 ◦ . Solution (

A, B, and C are arbitrary constants):

z = x − At 2 , where the function ψ = ψ(z) is determined by the ordinary differential equation

w(x, t) = ψ(z) exp i(Axt − 2 3 A 2 t 3 + Bt + C) ,

ψ ′′ zz + f (|ψ|)ψ − (Az + B)ψ = 0.

6 ◦ . Solutions:

1 ( x+C 2 ) 2 Z

w(x, t) = ✄

exp iϕ(x, t) ,

ϕ(x, t) =

f |C −1

1 t| √ /2 dt + C 3 ,

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

7 ◦ . Solution: Z dx

w(x, t) = u(x) exp iϕ(x, t) , ϕ(x, t) = C 1 t+C 2 2 + C 3 , u ( x)

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

u ′′ xx − C 1 u−C 2 2 u −3 + f (|u|)u = 0.

w(x, t) = u(z) exp iAt + iϕ(z) ,

z = kx + λt,

where

A, k, and λ are arbitrary real constants, and the functions u = u(z) and ϕ = ϕ(z) are determined by the system of ordinary differential equations

k 2 uϕ ′′

zz +2 2 k u ′ z ϕ z ′ + λu ′ z = 0, k 2 u ′′ zz − k 2 u(ϕ ′ z ) 2 − λuϕ ′ z − Au + f (|u|)u = 0.

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

+ f (|w|)w = 0.

Schr¨odinger equation of general form ; f (u) is a real function of a real variable. 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:

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

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 (|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 −3 n ( x n u ′

x ) ′ x − C 1 u−C 2 x n u + f (|u|)u = 0.

w(x, t) = C 1 t − 2 exp iϕ(x, t) ,

x 2 n+1

ϕ(x, t) =

f | 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).

∂w

3. =( a + ib)

f (|w|) + ig(|w|) w.

∂t

∂x

Generalized Landau–Ginzburg equation ; f (u) and g(u) are real functions of a real variable, a and b are real numbers. Equations of this form are used for studying second-order phase transitions in superconductivity theory (see Landau and Ginzburg, 1950) and to describe two-component reaction- diffusion systems near a point of bifurcation (Kuramoto and Tsuzuki, 1975).

1 ◦ . Suppose w(x, t) is a solution of the generalized Landau–Ginzburg equation. Then the function

= iC w 1 1 e w(x + C 2 , t+C 3 ),

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

2 ◦ . Traveling-wave solution:

f (|C 1 |)

w(x, t) = C 1 exp iϕ(x, t) ,

ϕ(x, t) = ✝ x

+ t g(|C 1 |) −

f (|C 1 |) + C 2 ,

a a where C 1 and C 2 are arbitrary real constants.

w(x, t) = u(t) exp iϕ(x, t) , ϕ(x, t) = C 1 x−C 2 1 bt + g(|u|) dt + C 2 , where u = u(t) is determined by the ordinary differential equation u ′ t = f (|u|)u − aC 2 1 u, whose

general solution can be represented in implicit form as

Z du

= t+C 3 .

f (|u|)u − aC 2

4 ◦ . Solution:

w(x, t) = U (z) exp iC 1 t + iθ(z) ,

z = x + λt,

where C 1 and λ are arbitrary real constants, and the functions U = U (z) and θ = θ(z) are determined by the system of ordinary differential equations

z θ z − λU z + f (|U |)U = 0, aU θ ′′ zz − bU (θ 2 z ′ ) + bU zz ′′ +2 aU z ′ θ z ′ − λU θ ′ z − C 1 U + g(|U |)U = 0.

References : V. S. Berman and Yu. A. Danilov (1981), A. D. Polyanin and V. F. Zaitsev (2002).

∂w ∂ 2 w

4. i

f (|w|)w = 0.

∂t ∂x

∂x

1 ◦ . 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 2 ϕ = 0, ψ t ′ +4 ϕψ + 2ϕf (u) = 0, χ ′ t + ψ 2 + ψf (u) = 0.

Integrating yields

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

2 ◦ . Solution:

w(x, t) = U (z) exp[iβt + iV (z)], z = kx + λt,

where k, β, and λ are arbitrary real constants, and the functions U = U (z) and V = V (z) are determined by the system of ordinary differential equations

f (|w|)w = 0.

∂t

∂x 2

1 ◦ . Suppose w(x, t) is a solution of this equation. Then the functions

w 1 = ✠ e iC 1 w( ✠

2 x+C 3 , C 2 t+C 4 ),

where C 1 , ...,C 4 are arbitrary real constants, are also solutions of the equation. The plus or minus signs are chosen arbitrarily.

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

Integrating yields

2 u , χ = −C 2 u f (u) dt + C 3 , where C 1 , C 2 , and C 3 are arbitrary real constants, and the function u = u(t) is defined implicitly as

2 ϕ=C 2

2 ψ=C 4

( C 4 is an arbitrary constant)

3 ◦ . There is a solution of the form

w(x, t) = U (z) exp[iβt + iV (z)], z = kx + λt,

where k, β, and λ are arbitrary real constants, 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).

4 ◦ . There is a self-similar solution of the form w(x, t) = V (ξ), where ξ = x 2 /t.