Certain Special Self-Similar Solutions of Khokhlov - Zabolotskaya - Kuznetsov Equation, Generalized Burgers Equation in (1 + 2)-dimensions

Certain special forms of self-similar solutions which satisfy Khokhlov-Zabolotskaya-Kuznetsov equations or a generalized Burgers equation in (1 + 2)-dimensions are reported.


Introduction
The Khokhlov -Zabolotskaya-Kuznetsov (KZK) equation [1,2] is describes a plethora of phenomena such as • the quasi-one-dimensional propagation of a signal in a nonlinear, homogeneous, isentropic medium • the nonlinear propagation of a finite-amplitude sound beam pulse in the thermo-viscous medium • the long waves in ferromagnetic media In addition, the KZK equation is used in lithotripsy in which kidney stones are broken with the help of ultrasound waves.
The scheme of this paper is as follows: In Section 2, some special exact, analytic travelling wave solutions of the Khokhlov -Zabolotskaya -Kuznetsov equation (1) is reported; one of these solutions tends to a constant limit as t → ∞.

A Special Travelling Waves of Khokhlov -Zabolotskaya -Kuznetsov Equation
We shall proceed to obtain a special travelling wave solutions [3] of (1). For if an ansatz where c 1 and c 2 are wave speeds, is inserted into the KZK equation (1) then the partial differential equation (PDE) for v(r, s) may be written in the conservation form We shall solve (3) by the method of equation-splitting [4]; for, we write Integrating (4) with respect to r, we get where A 2 (s) is the function of integration. With A(s) = A 2 (s) = 0, (6) simplifies to a Riccati equation It is easily verified that with B ( s) a function of s solves (7). Now differentiating (8) with respect to r and s, and inserting into (5), with A 1 (r) = 0, we have The general solution of (9) is where c is an arbitrary constant. Substituting (10) in (8), we thus find that v(r, s) = 1 Equations (2) and (11) lead to the following travelling wave solution of the KZK equation (2): It is evident from (12) that u(x, y, t) tends to 'twice the wave speed' 2c 1 as t → ∞.
The very form of the solution (12) of KHZ equation (2) suggests that we seek an ansatz in the form u = φ(z) z = ax + by − ct.
Inserting (13), equation (14) becomes Integrating equation (15) once with respect to z, we get where p is an integration constant. Now we shall proceed to find some special solution of equation (16).

Case -1: p=0
It is easy to ascertain that where q is a free constant, is a solution of (16) with p = 0, namely In order to obtain a solution which is more general than (17), we write Substitution of (19) in to (18) leads to (qB − Hz + Bw(z))w (z) + Aw (z) = 0.
An integration of (20) with respect to z yields where c * is an arbitrary constant. Thus (19) becomes

Case -2: H=0
In this case, equation (16) reduces to The general solution of (23) is After integration, with l a free constant, (25) yields a Riccati equation with the solution In (27), C is an arbitrary constant.

Khokhlov -Zabolotskaya-Kuznetsov Equation with A Variable Coefficient
Now we consider Khokhlov -Zabolotskaya-Kuznetsov (KZK) equation with a variable coeffient If we seek solutions in the form then the PDE governing v(r, s) is 2v r + rv rr + sv rs + δv rrr − 2v rr v − 2v r 2 − 2 0 v ss = 0.
Thus, in turn, the solution of (1) is