Cylindrical, Spherical, and Modified Korteweg–de Vries Equations

9.1.2. Cylindrical, Spherical, and Modified Korteweg–de Vries Equations

Cylindrical Korteweg–de Vries equation . This is a special case of equation 9.1.2.3 for a = 1, b = −6, and k= 1 2 . The transformation

w(x, t) = −

u(z, τ ), x= , t=−

2 τ 2 leads to the Korteweg–de Vries equation in canonical form 9.1.1.1:

References : R. S. Johnson (1979), F. Calogero and A. Degasperis (1982), G. W. Bluman and S. Kumei (1989).

Spherical Korteweg–de Vries equation . This is a special case of equation 9.1.2.3 for a = 1, b = −6, and k = 1.

1 ◦ . Suppose w(x, t) is a solution of the equation in question. Then the function = 2 3 w −1 1 C 1 wC 1 x + 6C 1 C 2 ln | t| + C 3 , C 1 t + C 2 t , where C 1 , C 2 , and C 3 are arbitrary constants, is also a solution of the equation.

2 ◦ . Degenerate solution linear in x:

w(x, t) =

t(C 2 + 6 ln | t|)

3 ◦ . Self-similar solution:

w(x, t) = t −2 /3 u(z),

z = xt −1 /3 ,

where the function u = u(z) is determined by the ordinary differential equation

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

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

2 ◦ . Degenerate solution linear in x:

(1 − k)x + C 1

w(x, t) =

C 2 t k + bt

3 ◦ . Self-similar solution:

w(x, t) = t −2 /3

u(z), −1 z = xt /3 ,

where the function u = u(z) is determined by the ordinary differential equation

3 –6 w 2 ∂t = 0. ∂x ∂x Modified Korteweg–de Vries equation .

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

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

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

2 ◦ . Self-similar solution ( x 0 and t 0 are arbitrary constants):

( x−x 0 ), where the function f (y) is determined by the third-order ordinary differential equation

−1 w(x, t) = /3 3( t−t

f (y), y= 3( t−t 0 )

f yyy ′′′ − yf y ′ − f − 6f 2 f y ′ = 0.

f yy ′′ −2 f 3 − yf = a.

3 ◦ . Suppose w(x, t) is a solution of the equation in question. Then the function u(x, t) obtained with the Miura transformation

∂w

u(x, t) =

∂x

satisfies the Korteweg–de Vries equation 9.1.1.1:

In general, the converse is not true: if u(x, t) is a solution of the Korteweg–de Vries equation (2), the function w(x, t) linked to it with the Miura transformation (1) satisfies the nonlinear integro-differential equation

= c(t) exp −2

w(x, t) dx .

4 ◦ . Solutions of the modified Korteweg–de Vries equation

may be obtained from solutions of the linear Gel’fand–Levitan–Marchenko integral equation. Any function

F = F (x, y; t) rapidly decaying as x → +∞ and satisfying simultaneously the two linear equations

generates a solution of equation (3) in the form

w = K(x, x; t), where K(x, y; t) is a solution of the linear Gel’fand–Levitan–Marchenko integral equation,

K(x, y; t) = F (x, y; t) + K(x, z; t)F (z, u; t)F (u, y; t) dz du. (5)

✖✂✗ Time t appears in (5) as a parameter. It follows from the first equation in (4) that F (x, y; t) = F (x+y; t).

References : M. J. Ablowitz and H. Segur (1981), F. Calogero and A. Degasperis (1982).

5 ◦ . Conservation laws:

where D t = ∂ and D x = ✖✂✗ ∂ ∂t ∂x .

References : G. B. Whitham (1965), R. M. Miura, C. S. Gardner, and M. D. Kruskal (1968).

Modified Korteweg–de Vries equation .

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

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

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

w(x, t) = a +

z = kx − (6a 2 k+k √ 3

) t + b,

cosh z + 2a

where

a, b, and k are arbitrary constants.

Reference : M. J. Ablowitz and H. Segur (1981).

3 ◦ . Two-soliton solution:

a 1 e θ 1 + a 2 e θ 2 + Aa 2 e 2 θ 1 + θ 2 + Aa 1 e θ 1 +2 θ 2

w(x, t) = 2

1+ e 2 θ 1 + e 2 θ 2 + 2(1 − A)e θ 1 + θ 2 + Ae 2( θ 1 + θ 2 ) ,

1 = a 1 − a 1 t+b 1 , θ 2 = a 2 − a 2 t+b 2 , A=

where a 1 , a 2 , b 1 , and b 2 are arbitrary constants.

Reference : R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. C. Morris (1982).

4 ◦ . Rational solutions (algebraic solitons):

w(x, t) = a −

w(x, t) = a −

a is an arbitrary constant.

References : H. Ono (1976), M. J. Ablowitz and H. Segur (1981).