Equations Involving Arbitrary Parameters of the Form

3.5.1. Equations Involving Arbitrary Parameters of the Form

∂ 2 w = f (w) ∂x∂y

1. = aw n .

∂x∂y This is a special case of equation 3.5.3.1 with f (w) = aw n .

1 ◦ . Suppose w(x, y) is a solution of the equation in question. Then the function

w 1 =( C 1 C 2 ) n−1 wC 1 x+C 3 , C 2 y+C 4 , where C 1 , ...,C 4 are arbitrary constants, is also a solution of the equation.

w(x, y) =

C 1 x+

2(1 + n)

w(x, y) = 2 a(1 − n) 1− n xy + C

1 x+C 2 y+C 1 C 2 1− n .

3 ◦ . Traveling-wave solution in implicit form (generalizes the first solution of Item 2 ◦ ):

4 ◦ . Self-similar solution:

w=x n−1 U (ξ), ξ = yx β ,

where β is an arbitrary constant, and the function U (ξ) is determined by the modified Emden–Fowler equation

For exact solutions of this equation, see the book by Polyanin and Zaitsev (2003).

2. = ae λw .

∂x∂y Liouville equation . This is a special case of equation 3.5.3.1 with f (w) = ae λw .

1 ◦ . Suppose w(x, y) is a solution of the equation in question. Then the function

w 1 = wC 1 x+C 2 , C 3 y+C 4 +

ln( C 1 C 3 ),

λ where C 1 , ...,C 4 are arbitrary constants, is also a solution of the equation.

2 ◦ . General solution:

aλ Z

w=

f (x) + g(y) −

ln

exp f (x) dx +

exp g(y) dy

where f = f (x) and g = g(y) are arbitrary functions and k is an arbitrary constant.

3 ◦ . The Liouville equation is related to the linear equation ∂ xy u = 0 by the B¨acklund transformation

4 ◦ . The original equation can also be linearized with either of the differential substitutions

1 2 ∂v ∂v

w= ln

v = v(x, y);

z = z(x, y).

cos 2

z ∂x ∂y

5 ◦ . Solutions (for a = λ = 1):

w = ln

f (x)g(y) cosh

C 1 + C 2 g(y) dy −

f (x) dx ,

w = ln

f (x)g(y) sinh

C 1 + C 2 g(y) dy +

f (x)g(y) cos

C 1 + C 2 g(y) dy +

f (x) dx ,

where f (x) and g(y) are arbitrary functions, and C 1 and C 2 are arbitrary constants.

References : J. Liouville (1853), R. K. Bullough and P. J. Caudrey (1980), S. V. Khabirov (1990), N. H. Ibragimov (1994).

3. –2 = e w – e w .

∂x∂y

This is a special case of equation 3.5.3.1 with f (w) = e w − e −2 w .

1 ◦ . Solutions:

∂ 2 (ln ζ

w = ln 1−2

∂x∂y

1 , A, A A 2 , k, k 1 , and k 2 are arbitrary constants.

2 ◦ . On passing to the new independent variables z = x − y and t = x + y, one obtains an equation of the form 3.2.1.4:

= −2 + e − e w

∂t 2 ∂z 2

3 ◦ . The substitution u=e w leads to the Tzitz´eica equation:

Reference : S. S. Safin and R. A. Sharipov (1993), O. V. Kaptsov and Yu. V. Shan’ko (1999, other exact solutions are also given there).

4. = a sinh w. ∂x∂y

Sinh-Gordon equation . On passing to the new independent variables z = x − y and t = x + y, one obtains an equation of the form 3.3.1.1:

2 ∂t = ∂z 2 + a sinh w.

References : S. P. Novikov, S. V. Manakov, L. B. Pitaevskii, and V. E. Zakharov (1984), A. Grauel (1985).

5. = a sin w. ∂x∂y

Sine-Gordon equation . This is a special case of equation 3.5.3.1 with f (w) = a sin w.

1 ◦ . Traveling-wave solution:

 a  4 arctan exp

if aAB > 0, w(x, y) =

( Ax + By + C)

AB

 4 arctanh exp

A, B, and C are arbitrary constants.

2 ◦ . Solution:

1 a w(x, y) = 4 arctan

C 1 + C 2 sinh( v 1 − v 2 )

C k x−

y , k = 1, 2,

2 C k ➌✝➍ where C 1 and C 2 are arbitrary constants.

C 1 − C 2 cosh( v 1 + v 2 )

Reference : R. K. Bullough and P. J. Caudrey (1980).

w = U (ξ),

ξ = xy,

where the function U = U (ξ) is determined by the second-order ordinary differential equation ξU ξξ ′′ + U ξ ′ = a sin U .

4 ◦ . The B¨acklund transformation

brings the original equation to the identical equation

∂ 2 u = a sin u. ∂x∂y

Given a single exact solution, the formulas of (1) allow us to successively generate other solutions of the sine-Gordon equation.

5 ◦ . The sine-Gordon equation has infinitely many conservation laws. The first three of them read as follows:

D x w 2 y + D y 2 a cos w = 0,

y −4 w yy + D y 4 aw y cos w = 0,

3 6 − 12 2 2 + 16 3 2 4 D 2 x w y w y w yy w y w yyy + 24 w yyy + D y a(2w y − 24 w yy ) cos w = 0, where D x = ∂ ∂x and D y = ∂ ∂y (analogous laws can be obtained by swapping the independent

variables x ⇄ y).

References : A. C. Scott, F. Y. Chu, and D. W. McLaughlin (1973), J. L. Lamb (1974), R. K. Dodd and R. K. Bullough (1977).

6 ◦ . The equation in question is related to the equation

by the transformation

References for equation 3.5.1.5: R. Steuerwald (1936), I. M. Krichever (1980), R. K. Bullough and P. J. Caudrey (1980), S. P. Novikov, S. V. Manakov, L. B. Pitaevskii, and V. E. Zakharov (1984), N. H. Ibragimov (1994).

2 w ∂x∂y . On passing to the new independent variables z = x − y and t = x + y, one obtains an equation of the

Reference : F. Calogero and A. Degasperis (1982)