Khokhlov Zabolotskaya Equation Research Articles

Cartan’s Structure Theory of Symmetry Pseudo-Groups, Coverings and Multi-Valued Solutions for the Khokhlov–Zabolotskaya Equation

We derive two non-equivalent coverings for the modified Khokhlov–Zabolotskaya equation from Maurer–Cartan forms of its symmetry pseudo-group. Also we find Backlund transformations between the obtained covering equations. We apply these results to constructing multi-valued solutions for the Khokhlov–Zabolotskaya equation.

Coverings of Differential Equations and Cartan’s Structure Theory of Lie Pseudo-Groups

We establish relations between Maurer–Cartan forms of symmetry pseudo-groups and coverings of differential equations. Examples include Liouville’s equation, the Khokhlov–Zabolotskaya equation, and the Boyer–Finley equation.

Nonlinear focusing of acoustic shock waves at a caustic cusp

The present study investigates the focusing of acoustical weak shock waves incoming on a cusped caustic. The theoretical model is based on the Khokhlov-Zabolotskaya equation and its specific boundary conditions. Based on the so-called Guiraud's similitude law for a step shock, a new explanation about the wavefront unfolding due to nonlinear self-refraction is proposed. This effect is shown to be associated not only to nonlinearities, as expected by previous authors, but also to the nonlocal geometry of the wavefront. Numerical simulations confirm the sensitivity of the process to wavefront geometry. Theoretical modeling and numerical simulations are substantiated by an original experiment. This one is carried out in two steps. First, the canonical Pearcey function is synthesized in linear regime by the inverse filter technique. In the second step, the same wavefront is emitted but with a high amplitude to generate shock waves during the propagation. The experimental results are compared with remarkable agreement to the numerical ones. Finally, applications to sonic boom are briefly discussed.

Long waves in ferromagnetic media, Khokhlov–Zabolotskaya equation

We are interested here in small perturbations of electromagnetic waves in a saturated ferromagnetic media. By means of an asymptotic expansion we prove that the solution remains close on long times of the one of the Khokhlov–Zabolotskaya equation.

A modified Khokhlov-Zabolotskaya equation governing shear waves in a prestrained hyperelastic solid.

Weakly nonlinear, weakly diffracting, two-dimensional shear waves propagating in a prestrained hyperelastic solid are examined. A modification of the classical Khokhlov-Zabolotskaya equation is derived using a systematic perturbation scheme. Both dissipative and nondissipative materials were considered. The principal effect of the prestrain was seen to be the inclusion of a quadratic nonlinearity to the cubic nonlinearity found in the case of zero prestrain. Further new results include the shock jump relations and the prediction of shocks having a speed which is identical to the nonlinear wave speed ahead of or behind the shock. Explicit expressions for the nonlinearity coefficients for the special case of a Blatz-Ko material were provided.

Waveguide propagation of sound beams in a nonlinear medium

A possibility of a waveguide propagation of sound beams in the case of compensation of the diffraction divergence by the nonlinear refraction is demonstrated theoretically. A stationary (with respect to the longitudinal coordinate) solution is obtained to the nonlinear equation for a sound beam (the Khokhlov—Zabolotskaya equation); the solution describes the characteristic bow-shaped profile of the beam and the self-localized (with respect to the transverse coordinate) distribution of the peak values of this profile. The physical and mathematical features of this phenomenon belonging to nonlinear acoustics are discussed and compared with those of the well-known analog from nonlinear optics. A scheme of an experimental realization of the waveguide propagation of acoustic beams is proposed.

Oscillations in harmonics generated by the interaction of acoustic beams

A numerical model of nonlinear propagation is used to investigate two cases of monochromatic ultrasonic beams interacting at small angles in a nonlinear medium. Two finite Young’s slits are seen to produce fringes at harmonic frequencies of the source in places where the source frequency is absent, which can be seen as a combination of harmonic generation near the source, and in the beam. Two intersecting beams with shaded edges are seen to produce similar fringes in the near field, with an oscillatory structure. Algebraic solutions to a simplified model, using the weak-field Khokhlov–Zabolotskaya equation, are invoked to illustrate the origin of the oscillations, and of the far-field directivity, providing an alternative view of the fringes due to Young’s slits. It is seen that two weakly interacting beams can produce fringes of second harmonic where the source frequency has low amplitude, if the beams coincide at the point of observation, or if a boundary condition is imposed on the second harmonic where the beams coincide.

Asymptotically unified exact solutions of the Khokhlov–Zabolotskaya equation

For the large distance traversed by the sonic beam the exact analytical and physically realistic solutions of the Khokhlov–Zabolotskaya (KZ) equation were found for the first time. The procedure of finding these solutions as the self-similar traveling waves identifies their asymptotical generality with respect to the initial arbitrary wave profiles. The resulting solutions consist of the periodically repeating parts of the second-order curvers such as hyperbolas, ellipses, and, in particular, parabolas. The boundaries between these parts are defined with the use of the preserved integrals of the KZ equation. The influence of nonlinearity, diffraction, which is seen in many-dimensional cases, and initial curvature of the phase front of the wave beam on the forming of the typical form of a specific asymptotical wave profile (N shaped or U shaped) was discussed. A comparison was made between analytical solutions with the numerical results for the KZ equation and the experimental data and a full agreement was discovered. These solutions play the same role among all the variations of solutions of the KZ equation, such as the well-known sawtoothlike wave for the equation of the simple waves (Riemann’s equation). [Work supported by CRDF.]

Direct Similarity Solution Method and Comparison with the Classical Lie Symmetry Solutions

Abstract We study the general applicability of the Clarkson–Kruskal’s direct method, which is known to be related to symmetry reduction methods, for the similarity solutions of nonlinear evolution equations (NEEs). We give a theorem that will, when satisfied, immediately simplify the reduction procedure or ansatz before performing any explicit reduction expansions. We shall apply the method to both scalar and vector NEEs in either 1+1 or 2+1 dimensions, including in particular, a variable coefficient KdV equation and the 2+1 dimensional Khokhlov–Zabolotskaya equation. Explicit solutions that are beyond the classical Lie symmetry method are obtained, with comparison discussed in this connection.

Similarity Reductions for the Khokhlov–Zabolotskaya Equation

In this paper, we first give the Lie point symmetries of the Khokhlov–Zabolotskaya equation by a simple direct method developed by LOU for some -dimensional evolution equations, and then discuss all their similarity reductions from these symmetries, finally obtain their similarity analysis by using direct method introduced by Clarkson and Kruskal.

Comment on 'Towards the conservation laws and Lie symmetries for the Khokhlov-Zabolotskaya equation in three dimensions'

The authors comment on the Lie point symmetries for the Khokhlov-Zabolotskaya equation as calculated by Roy Chowdhury and Nasker (1986), and demonstrate that their result for the coefficients of the vector field is correct but incomplete.

An analytical approximation of strong nonlinear effects in bounded sound beams

The propagation of large nonlinear bounded sound beams in inviscid fluids is studied by means of the method of renormalization. Starting from a nonuniform quasilinear expansion of a solution of the Khokhlov–Zabolotskaya equation for a Gaussian source, a straining of the retarded time is introduced, which leads to a uniform approximation. The main point is the choice of a nonlinear phase shift, which yields a single smooth continuous representation for the wave, both before and beyond the shock formation point. A computation of the harmonics shows the asymmetrical distortion of the wave profile due to the coupling between nonlinearity and diffraction. A comparison with the results of a finite-difference numerical algorithm turns out favorable. The method is then extended to general plane sources.

Symmetries of the Khokhlov-Zabolotskaya equation

The group of Lie symmetries of the Khokhlov-Zabolotskaya equation in two and three space dimensions is given.

Quasilongitudinal propagation of narrow beams of nonlinear magnetohydrodynamic waves

An equation, analogous to the Khokhlov-Zabolotskaya equation, is derived for narrow beams of quasitransverse waves propagating at small angles to a magnetic field. The effect of diffraction on wave propagation is investigated in the linear approximation.

Towards the conservation laws and Lie symmetries for the Khokhlov-Zabolotskaya equation in three dimensions

The authors have obtained the explicit structure for the generating function of Lie symmetries for the Khokhlov-Zabolotskaya equation describing the propagation of a sound beam in a non-linear medium. In the absence of a canonical framework they have derived the set of conservation laws through the technique of differential forms and prolongation. Such conservation laws are of utmost importance for the analysis of a sound beam in the medium.