Kadomtsev–Petviashvili Equation

10.3.1. Kadomtsev–Petviashvili Equation

Kadomtsev–Petviashvili equation in canonical form (Kadomtsev and Petviashvili, 1970). It arises in the theory of long, weakly nonlinear surface waves propagating in the x-direction, with the variation in y being sufficiently slow.

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

1 = C 1 wC 1 x + 6C 1 ϕ(t), ✛ C 1 y+C 2 , C 1 t+C 3 + ϕ ′ t ( t), where C 1 , C 2 , and C 3 are arbitrary constants and ϕ(t) is an arbitrary function, are also solutions of

the equation.

2 ◦ . The time-invariant solutions satisfy the Boussinesq equation 10.2.1.2 (see also 10.2.1.1). The y-independent solutions satisfy the Korteweg–de Vries equation 9.1.1.1.

3 ◦ . One-soliton solution:

2 + ap ln 2 1+ kx+kpy−k(k w(x, y, t) = −2 )

A, k, and p are arbitrary constants.

4 ◦ . Two-soliton solution:

w(x, y, t) = −2

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

5 ◦ . N -soliton solution:

w(x, y, t) = −2

ln det A,

∂x 2

where A is an N × N matrix with entries

exp[( p n + q m ) x]

and the p n , q m , and C n are arbitrary constants ( C n > 0).

6 ◦ . Rational solutions:

w(x, y, t) = −2 2

2 ln( x + py − ap ∂x t),

2 2 12 w(x, y, t) = −2

2 ln ( x+p 1 y − ap 1 t)(x + p 2 y − ap 2 t) +

∂x

a(p 1 − 2 ) p 2

where p, p 1 , and p 2 are arbitrary constants.

7 ◦ . Two-dimensional power-law decaying solution ( a = −1):

2 w(x, y, t) = 4 2 e x + βe e

y) 2 − 2 γ 2 ( 2 y) −3 /γ

x = x − (β e + γ ) t, y = y + 2βt, e

( e x + βe y) + γ ( e y) +3 /γ

where ✜✂✢ β and γ are arbitrary constants.

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

where λ is an arbitrary constant and the function U = U (z, t) is determined by a third-order differential equation of the form 9.1.4.1:

with ϕ(t) being an arbitrary function. For ϕ = 0 we have the Korteweg–de Vries equation 9.1.1.1.

9 ◦ . “Two-dimensional” solution:

w = V (ξ, t), 2 ξ=x+C 1 y − aC

1 t,

V = V (ξ, t) is determined by a third-order differential equation of the form 9.1.4.1:

where C 1 and C 2 are arbitrary constants and the function

∂V 3 ∂ V ∂V

−6 V = ϕ(t),

with ϕ(t) being an arbitrary function. For ϕ = 0 we have the Korteweg–de Vries equation 9.1.1.1.

10 ◦ +. “Two-dimensional” solution:

w(x, y, t) = u(η, t),

η=x+

4 at

where the function u(η, t) is determined by the third-order differential equation

u = ψ(t),

with ψ(t) being an arbitrary function. For ψ = 0 we have the cylindrical Korteweg–de Vries equation

References : R. S. Johnson (1979), F. Calogero and A. Degasperis (1982).

11 ◦ +. There is a degenerate solution quadratic in x:

w=x 2 ϕ(y, t) + xψ(y, t) + χ(y, t).

12 ◦ . The Kadomtsev–Petviashvili equation is solved by the inverse scattering method. Any rapidly decaying function

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

generates a solution of the Kadomtsev–Petviashvili equation in the form

w = −2

K(x, x; y, t),

dx

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

K(x, z; y, t) + F (x, z; y, t) + K(x, s; y, t)F (s, z; y, t) ds = 0.

The quantities ✣✂✤ y and t appear here as parameters.

References : V. S. Dryuma (1974), V. E. Zakharov and A. B. Shabat (1974), I. M. Krichever and S. P. Novikov (1978), M. J. Ablowitz and H. Segur (1981), S. P. Novikov, S. V. Manakov, L. B. Pitaevskii, and V. E. Zakharov (1984), V. E. Adler, A. B. Shabat, and R. I. Yamilov (2000).

6 a Unnormalized Kadomtsev–Petviashvili equation . The transformation w=− U (x, y, τ ), τ = at

b leads to an equation of the form 10.3.1.1: