Khokhlov–Zabolotskaya Equation

7.1.2. Khokhlov–Zabolotskaya Equation

Two-dimensional Khokhlov–Zabolotskaya equation . It describes the propagation of a sound beam in a nonlinear medium; t and y play the role of the space coordinates and x is a linear combination of time and a coordinate.

The equation of unsteady transonic gas flows (see 7.1.3.1 with

a = b = 1/2)

2 u xτ + u x u xx − u yy =0

can be reduced to the Khokhlov–Zabolotskaya equation; see Lin, Reissner, and Tsien (1948). To this end, one should pass to the new variable τ = 2t, differentiate the equation with respect to x, and then substitute w = −∂u/∂x.

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

w 2 = w(x + λy + ϕ(t), y + 2λt, t) + ϕ ′

2 t ( t) − λ ,

where C 1 , ...,C 5 and λ are arbitrary constants and ϕ = ϕ(t) is an arbitrary function, are also solutions of the equation.

2 ◦ . Solutions:

w(x, y, t) = −

+ ϕy + ψ,

t+C 1

2 w(x, y, t) = 2ϕx + (ϕ 2

t −2 ϕ ) y + ψy + χ,

1 4 1 3 1 2 w(x, y, t) = (ϕy + ψ)x −

w(x, y, t) = C 2

1 x+C 2 y+ϕ+ϕ ′ t − C 2 ,

w(x, y, t) = 2 4 t(x + ϕ) − (y + C 2 ) + ϕ ′

where ϕ = ϕ(t), ψ = ψ(t), χ = χ(t), and θ = θ(t) are arbitrary functions, the prime stands for the

differentiation, and C 1 and C 2 are arbitrary constants.

3 ◦ . Solution in implicit form:

z=w−ϕ ′ t ( t), where ϕ(t) and F (z) are arbitrary functions. With λ = 0, this relation determines the general

tz + x + λy + λ 2 t + ϕ(t) = F (z),

y-independent solution of the original equation.

4 ◦ . “Two-dimensional” generalized separable solution quadratic in x:

w = f (y, t)x 2 + g(y, t)x + h(y, t),

where the functions f = f (y, t), g = g(y, t), and h = h(y, t) are determined by the system of differential equations

f 2 yy = −6 f ,

g yy = −6 f g + 2f t ,

h 2 yy = −2 fh+g t − g .

system is given by

R ( g t − g ) dy, R = y + ϕ(t), R

+ C 4 ( t)R +

( g t − g ) dy −

where ϕ(t), C 1 ( t), . . . , C 4 ( t) are arbitrary functions.

5 ◦ . “Two-dimensional” solution:

w = xu(ξ, t), −1 ξ = yx /2 ,

where the function u = u(ξ, t) is determined by the differential equation

6 ◦ . “Two-dimensional” solution:

α ′ t +4

w = v(ζ, t) + 2 x, ζ=y + αx,

where α = α(t) is an arbitrary function and the function v = v(ζ, t) is determined by the differential equation

α The last equation has a particular solution of the form v = ζϕ(t), where the function ϕ = ϕ(t) is

determined by the Riccati equation

′ − 2 2 −( ′ + 10) ′ − αϕ 2 t α ϕ α t ϕ+β t β = 0.

7 ◦ . “Two-dimensional” solution:

w = U (r, z), z = x + βy + λt, r = y + µt,

where β, λ, and µ are arbitrary constants, and the function U = U (r, z) is determined by the differential equation

With λ=β 2 and µ = 2β, we obtain an equation of the form 5.1.5.1.

8 ◦ . “Two-dimensional” solution:

where the function

V = V (p, q) is determined by the differential equation

3 p(3V p + 1) 2 + (4 q

V + 1)

+2 q(6pV + 1)

−1 −1 w = u(r)x 2 y , r = (At + B) x y ,

where

A and B are arbitrary constants, and the function u = u(r) is determined by the ordinary differential equation

r ( u − Ar + 4)u rr + r ( u ′ r ) − r(6u − Ar + 6)u ′ r + 6( u + 1)u = 0.

References for equation 7.1.2.1: Y. Kodama (1988), Y. Kodama and J. Gibbons (1989), N. H. Ibragimov (1994, pp. 299–

300; 1995, pp. 447–450), A. M. Vinogradov and I. S. Krasil’shchik (1997), A. D. Polyanin and V. F. Zaitsev (2002).

∂x∂t ∂x

∂x

∂y 2

The transformation

b w(x, y, t) = u(x, y, τ ), τ = −bt

leads to an equation of the form 7.1.2.1:

– g(t)

Generalized Khokhlov–Zabolotskaya equation .

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

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

2 = w(ξ, η, t) + ϕ(t), ξ = x + λy + [ f (t)ϕ(t) + λ g(t)] dt, η = y + 2λ g(t) dt, where C 1 , C 2 , C 3 , and λ are arbitrary constants and ϕ = ϕ(t) is an arbitrary function, are also

solutions of the equation.

2 ◦ . Solutions:

w(x, y, t) = −x

f dt + C

+ ϕy + ψ,

t −2 fϕ w(x, y, t) = 2ϕx + 2 y + ψy + χ,

w(x, y, t) = (ϕy + ψ)x −

where ϕ = ϕ(t), ψ = ψ(t), χ = χ(t), and θ = θ(t) are arbitrary functions; C is an arbitrary constant;

f = f (t) and g = g(t); the prime denotes a derivative with respect to t.

3 ◦ . “Two-dimensional” solution:

w(x, y, t) = U (z, t) + ϕ(t), z = x + λy,

where the function ϕ(t) is an arbitrary function, λ is an arbitrary constant, and the function U = U (z, t) is determined by the first-order partial differential equation [ ψ(t) is an arbitrary function]

− f (t)U

−[ f (t)ϕ(t) + λ g(t)]

A complete integral of this equation is sought in the form U = A(t)z + B(t), which allows obtaining the general solution (see Polyanin, Zaitsev, and Moussiaux, 2002).

4 ◦ . “Two-dimensional” generalized separable solution quadratic in x:

w = ϕ(y, t)x 2 + ψ(y, t)x + χ(y, t),

where the function ϕ = ϕ(y, t), ψ = ψ(y, t), and χ = χ(y, t) are determined by the system of differential equations

gϕ 2 yy = −6 fϕ , gψ yy = −6 f ϕψ + 2ϕ t ,

gχ 2 yy =− f (2ϕχ + ψ )+ ψ t .

The subscripts ✠✂✡ y and t denote the corresponding partial derivatives, f = f (t) and g = g(t).

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

∂x∂t ∂x

∂x 2 ∂y 2

∂z 2

Three-dimensional Khokhlov–Zabolotskaya equation .

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

2 w 2 = w(x + λy + µz + ϕ(t), y + 2λt, z + 2µt, t) + ϕ t ( t) − λ − µ ,

w 3 = w(x, y cos β + z sin β, −y sin β + z cos β, t), where C 1 , ...,C 6 , λ, µ, and β are arbitrary constants, and ϕ = ϕ(t) is an arbitrary function, are also

solutions of the equation.

2 ◦ . Solutions:

1 2 w(x, y, z, t) = 2α 2 x + (α ′ 1 −2 α 1 − α 2 ) 2 y + α 3 y+α 2 z + βz + γ,

C 4 tx − y 2 − 2 z

w(x, y, z, t) =

t 3 /2

where α 1 , α 2 , α 3 , β, γ are arbitrary functions of t, and C is an arbitrary constant.

3 ◦ . “Three-dimensional” solution:

w = u(x, ξ, t), ξ = y sin β + z cos β,

where β is an arbitrary constant and the function u = u(x, ξ, t) is determined by the Khokhlov– Zabolotskaya equation of the form 7.1.2.1:

4 ◦ . “Three-dimensional” generalized separable solution linear in x:

w = f (y, z, t)x + g(y, z, t), where the functions f = f (y, z, t) and g = g(y, z, t) are determined by the differential equations

f yy + f zz = 0,

g 2 yy + g zz = f t − f .

The subscripts y, z, and t denote the corresponding partial derivatives. The first equation represents the Laplace equation and the second one is a Poisson equation (for g). For solutions of these linear equations, see, for example, Tikhonov and Samarskii (1990) and Polyanin (2002).

5 ◦ . “Three-dimensional” generalized separable solution quadratic in x:

w = f (y, z, t)x 2 + g(y, z, t)x + h(y, z, t),

where the functions f = f (y, z, t), g = g(y, z, t), and h = h(y, z, t) are determined by the system of differential equations

f 2 yy + f zz = −6 f ,

g yy + g zz = −6 f g + 2f t ,

h 2 yy + h zz = −2 fh+g t − g .

2 w(x, y, z, t) = u(ξ)t 2 , ξ=t (4 xt − y − z ), where λ is an arbitrary constant, and the function u = u(ξ) is determined by the ordinary differential

equation

[4 2 u + (1 − λ)ξ]u

ξξ ′′ + 4( u ξ ′ ) = 0.

λ u, and the reduction of order with p ′ u = 1− η(p) result in the first-order equation pηη p ′ − η + 1 = 0. Integrating yields ( η − 1)e η

For 4 λ ≠ 1, the passage to the inverse ξ = ξ(u), the change of variable ξ(u) = p(u) −

= C 1 p.

For λ = 1, we have u(ξ) = ☛

C 1 ξ+C 2 .

7 ◦ . Solution:

w(x, y, z, t) =

2 U (ζ), ζ=

xt

where the function U = U (ζ) is determined by the ordinary differential equation

4 ′ 2 ζ 2 ( ζ U − ζ + 4)U ζζ + ζ ( U ζ ) + ζ(2ζ U − 3ζ + 12)U ζ ′ +4 U = 0.

w(x, y, z, t) =

V (q), q=

where the function

V = V (q) is determined by the ordinary differential equation

References for equation 7.1.2.4: A. M. Vinogradov, I. S. Krasil’shchik, and V. V. Lychagin (1986), N. H. Ibragimov (1994, 1995).

∂t∂x ∂x

∂x

∂y 2 ∂z 2

1 ◦ . For a < 0, b < 0, and c < 0, the passage to the new independent variables according to

x = ¯x − a, y = ¯y − b, z = ¯z − a, t = ¯t/ − a leads to the three-dimensional Khokhlov–Zabolotskaya equation 7.1.2.4.

2 ◦ . Suppose w(x, y, z, t) is a solution of the equation in question. Then the functions w 1 −2 = 2 C 1 C 2 w(C 1 x+C 3 , C 2 y+C 4 , C 2 z+C 5 −1 , 2 C 1 C 2 t+C 6 ),

2 1 2 bλ + cµ

w 2 = w(x + λy + µz + ϕ(t), y − 2bλt, z − 2cµt, t) − ϕ t ′ ( t) −

a a where C 1 , ...,C 6 , λ, µ, and β are arbitrary constants and ϕ = ϕ(t) is an arbitrary function, are also

solutions of the equation.

3 ◦ . Solutions:

w(x, y, z, t) = αy + βz +

x + γ,

at + C

2 2 w(x, y, z, t) = α ln(cy 2 + bz )−( β t +4 abcβ )( cy + bz )+4 bcβx + γ,

where α = α(t), β = β(t), and γ = γ(t) are arbitrary functions and C is an arbitrary constant.

4 ◦ . “Three-dimensional” generalized separable solution linear in x:

w = f (y, z, t)x + g(y, z, t), where the functions f = f (y, z, t) and g = g(y, z, t) are determined by the differential equations

bf yy + cf zz = 0,

bg 2 yy + cg zz =− f t − af .

Remark. The above remains true if the coefficients

a, b, and c are functions of y, z, and t.

5 ◦ . “Three-dimensional” generalized separable solution quadratic in x:

w = f (y, z, t)x 2 + g(y, z, t)x + h(y, z, t),

where the functions f = f (y, z, t), g = g(y, z, t), and h = h(y, z, t) are determined by the system of differential equations

Remark. This remains true if the coefficients

a, b, and c are functions of y, z, and t.

6 ◦ . There are “three-dimensional” solutions of the following forms:

2 w(x, y, z, t) = u(x, t, ξ), 2 ξ = cy + bz ; w(x, y, z, t) = v(p, q, r)x k+2 , p = tx k+1 , q = yx k/2 , r = zx k/2 ,

where k is an arbitrary constant.

7 ◦ . “Two-dimensional” solution:

2 2 w(x, y, z, t) = xU (η, t), −1 η = (cy + bz ) x ,

where the function U = U (η, t) is determined by the differential equation

η(aηU + 4bc) 2

− 2( aηU − 2bc)

8 ◦ . “Two-dimensional” solution:

w(x, y, z, t) = V (ζ, t) − t ′ −4 bc 2 x, 2 ζ = cy + bz + ϕx,

aϕ

where ϕ = ϕ(t) is an arbitrary function, and V = V (ζ, t) is determined by the differential equation

V + 4bcζ)

References : P. Kucharczyk (1967), S. V. Sukhinin (1978), N. H. Ibragimov (1994).