Khokhlov Zabolotskaya Equation Research Articles

Dispersive nonlinear acoustic waves

The main phenomena and mathematical models of the theory of nonlinear dispersive acoustic waves are described. For the physical dispersion of the relaxation type, the form of the kernels of the integro-differential equation for several relaxation times and the continuous spectrum of such times is indicated. The possibility of obtaining simple equations for media with a finite “memory time” is indicated. As a medium with resonant dispersion, where the kernels have an oscillatory character, a liquid with gas bubbles is considered. The dispersion curves are analyzed. It is argued that the introduction of resonant elements into a homogeneous matrix is a special case of creating metamaterials with unusual nonlinear-dispersion properties. As an example of the geometric dispersion, important problems of nonlinear focusing are considered based on the Ostrovsky–Vakhnenko equation, which is interpreted here as the projection of the Khokhlov–Zabolotskaya equation onto the acoustic beam axis. When analyzing the nonlinear dispersion, it was indicated that in acoustics the dependence of the wave propagation velocity on the magnitude of the perturbation is a common phenomenon. It leads to a distortion of the wave profile in time, as well as to a deformation of the spatial shape of the beam — to effects like self-refraction. In acoustics, nonlinearities of odd degrees are encountered much less frequently than in optics; cubic nonlinearity is possessed, for example, by shear waves in solids. Nevertheless, it leads to original effects — the formation of trapezoidal sawtooth waves and unusual self-focusing. The article systematically presents both known and new results.

Instantaneous blow-up of solutions to the Cauchy problem for the fractional Khokhlov-Zabolotskaya equation

Abstract In this paper, we consider the Cauchy problem for a second-order nonlinear equation with mixed fractional derivatives related to the fractional Khokhlov-Zabolotskaya equation. We prove the nonexistence of a classical local in time solution. The obtained instantaneous blow-up result is proved via the nonlinear capacity method.

Localized Wave Structures Determined by Exact Solutions of the Khokhlov–Zabolotskaya Equation

A brief description of physically meaningful exact solutions of nonlinear partial differential equations that describe concrete physical objects, processes, etc., is presented. New exact solutions of the Khokhlov–Zabolotskaya equation that have a physical meaning (including a variant of the stationary equation with respect to the direction of propagation of a wave structure) are presented that correspond to the description of localized wave structures either only with spatial localization (beams in the self-trapped propagation mode), or with spatiotemporal localization (pulse-type structures, which are sometimes referred to as wave (acoustic) bullets). The solutions and the localized structures that are described by them are not obvious for nonlinear acoustics or predictable from “physical considerations” (as, for example, for the self-trapped propagation of beams in nonlinear optics), since there are no explicit conditions for the balance of the “necessary” effects in the dispersion-free state.

Nonlinear acoustics at Moscow State University

The rapid development of nonlinear acoustics, which began in the second half of the 20th century, had several distinct centers in different countries, and one of them was M. V. Lomonosov Moscow State University (MSU) in Russia. From MSU came one of the basic equations in nonlinear acoustics—the Khokhlov-Zabolotskaya (KZ) equation, the 50th anniversary of which is celebrated this year [Sov. Phys. Acoust. 15(1), 35–40 (1969)]. By the 1960–1970s, the work conducted by Academician Rem Khokhlov and his colleagues made MSU a leading international center for nonlinear acoustics and, even more influentially, for nonlinear optics. Nonlinear optics at that time underwent rapid growth related to the invention of lasers, and many phenomena in optics had analogs in acoustics. One of them was a parabolic approximation for wave beams, which led to formulation of the KZ equation after translation to acoustics with consideration of the details of nonlinear effects in weakly dispersive media. Along with the development of theoretical foundations, initial experiments on nonlinear acoustic phenomena were carried out at MSU. The current paper highlights these studies, as well as contemporary work and achievements on nonlinear acoustics at Moscow State University.

Three-component nonlocal conservation laws for Lax-integrable 3D partial differential equations

We found two three-component nonlocal conservation laws for the 3D rdDym equation, the universal hierarchy equation, the modified Veronese web equation, and the Veronese web equation. The aforementioned equations together with Pavlov’s equation are related via Bäcklund transformations. We study correspondences between the nonlocal conservation laws yielded by the Bäcklund transformations. In particular, we prove that the nonlocal conservation laws that depend on one pseudopotential are generated from a local conservation law of the Veronese web equation via appropriate superpositions of the Bäcklund transformations. Also, we prove nontriviality of the conservation laws found in the paper as well as the ones found in Makridin and Pavlov (2017) for the Khokhlov–Zabolotskaya equation and Pavlov’s equation.

Solution Blowup for Nonlinear Equations of the Khokhlov–Zabolotskaya Type

We consider several nonlinear evolution equations sharing a nonlinearity of the form ∂2u2/∂t2. Such a nonlinearity is present in the Khokhlov–Zabolotskaya equation, in other equations in the theory of nonlinear waves in a fluid, and also in equations in the theory of electromagnetic waves and ion–sound waves in a plasma. We consider sufficient conditions for a blowup regime to arise and find initial functions for which a solution understood in the classical sense is totally absent, even locally in time, i.e., we study the problem of an instantaneous blowup of classical solutions.

Invariant solutions to the Khokhlov-Zabolotskaya singular manifold equation and their application

We find the Lie algebra of point symmetries for the singular manifold equation associated with the Khokhlov–Zabolotskaya equation (KhZ), classify one-dimensional subalgebras of this algebra, and find corresponding invariant solutions . Then we use the Backlund transformation between the singular manifold equation and KhZ to find new exact solutions to the last equation.

Projection of the Khokhlov–Zabolotskaya equation on the axis of wave beam as a model of nonlinear diffraction

An equation is obtained that describes the nonlinear diffraction of a focused wave in a half-space starting from the wave source, then through the focus region up to the far zone, where the wave be ...

Instantaneous blow-up of classical solutions to the Cauchy problem for the Khokhlov–Zabolotskaya equation

The Cauchy problem for a second-order nonlinear equation with mixed derivatives is considered. It is proved that its classical local-in-time solution does not exist. The blow-up of the solution is proved by applying S.I. Pohozaev and E.L. Mitidieri’s nonlinear capacity method.

Deformed cohomologies of symmetry pseudo-groups and coverings of differential equations

Abstract The work establishes a relation between deformed cohomologies of symmetry pseudo-groups and coverings of differential equations. Examples include the potential Khokhlov–Zabolotskaya equation and the Boyer–Finley equation.

Setting boundary conditions on the Khokhlov–Zabolotskaya equation for modeling ultrasound fields generated by strongly focused transducers

An equivalent source model is developed for setting boundary conditions on the parabolic diffraction equation in order to simulate ultrasound fields radiated by strongly focused medical transducers. The equivalent source is defined in a plane; corresponding boundary conditions for pressure amplitude, aperture, and focal distance are chosen so that the axial solution to the parabolic model in the focal region of the beam matches the solution to the full diffraction model (Rayleigh integral) for a spherically curved uniformly vibrating source. It is shown that the proposed approach to transferring the boundary condition from a spherical surface to a plane makes it possible to match the solutions over an interval of several diffraction maxima around the focus even for focused sources with F-numbers less than unity. This method can be used to accurately simulate nonlinear effects in the fields of strongly focused therapeutic transducers using the parabolic Khokhlov–Zabolotskaya equation.

Effect of the angular aperture of medical ultrasound transducers on the parameters of nonlinear ultrasound field with shocks at the focus

Certain modern applications of high-intensity focused ultrasound (HIFU) in medicine use the nonlinear effect of shock front formation in the focal waveform. However, an important problem remains unsolved: determination of transducer parameters that provide the given pressure levels of the shock wave field at the focus required for a specific application. In this paper, simulations based on the Khokhlov-Zabolotskaya equation are performed to test and confirm the hypothesis that angular aperture of the transducer is the main parameter that determines the characteristic amplitude of the shock front and corresponding values for the peak positive and negative pressures at the focus. A criterion for formation of a developed shock in the acoustic waveform, as well as a method for determining its amplitude is proposed. Quantitative dependences of the amplitude of the developed shock and the peak pressures in the wave profile on the angular aperture of the transducer are calculated. The effects of saturation and the range of changes of the shock waveform parameters at the focus are analyzed for a typical HIFU transducer.

An equivalent source model for simulating high intensity focused ultrasound fields using a nonlinear parabolic equation

The nonlinear parabolic Khokhlov-Zabolotskaya (KZ) equation is commonly used for simulation of nonlinear focused fields generated by medical ultrasound transducers. Axially symmetric codes can be easily implemented on personal computers with simulation times on the order of tens of minutes even for the most extensive simulations of shock formation conditions. However, model accuracy can be limited for high intensity focused ultrasound sources with large convergence angles. In previous studies, it was shown that nonlinear parabolic modeling can provide good matching with measurements and full diffraction modeling in the focal region for both single element piston sources and multi-element arrays. Boundary conditions for the modeling were adjusted by varying the aperture of an equivalent planar source with a uniform pressure magnitude and a phase distribution for parabolic focusing. With this approach, there were slight discrepancies in the lateral pressure distributions in the focal plane. Here, it is proposed that an additional variation in the position of the equivalent source enables matching both axial and lateral distributions of the real HIFU sources with higher accuracy. Several examples are presented to compare modeling and measurements of the fields generated by laboratory and clinical HIFU transducers. [Work supported by RSF 14-12-00974 and NIH EB007643.]

Diffraction of high-intensity field in focal region as dynamics of nonlinear system with low-frequency dispersion

The stationary profile in the focal region of a focused nonlinear acoustic wave is described. Three models following from the Khokhlov-Zabolotskaya (KZ) equation with three independent variables are used: (i) the simplified one-dimensional Ostrovsky-Vakhnenko equation, (ii) the system of equations for paraxial series expansion of the acoustic field in powers of transverse coordinates, and (iii) the KZ equation reduced to two independent variables. The structure of the last equation is analogous to the Westervelt equation. Linearization through the Legendre transformation and reduction to the well-studied Euler-Tricomi equation is shown. At high intensities the stationary profiles are periodic sequences of arc sections having singularities of derivative in their matching points. The occurrence of arc profiles was pointed out by Makov. These appear in different nonlinear systems with low-frequency dispersion. Profiles containing discontinuities (shock fronts) change their form while passing through the focal region and are non-stationary waves. The numerical estimations of maximum pressure and intensity in the focus agree with computer calculations and experimental measurements.

Group analysis of evolutionary integro-differential equations describing nonlinear waves: the general model

The paper deals with an evolutionary integro-differential equation describing nonlinear waves. A particular choice of the kernel in the integral leads to well-known equations such as the Khokhlov–Zabolotskaya equation, the Kadomtsev–Petviashvili equation and others. Since the solutions of these equations describe many physical phenomena, the analysis of the general model studied in this paper is important. One of the methods for obtaining solutions of differential equations is provided by the Lie group analysis. However, this method is not applicable to integro-differential equations. Therefore, we discuss new approaches developed in modern group analysis and apply them to the general model considered in this paper. Reduced equations and exact solutions are also presented.

The 40th anniversary of the Khokhlov-Zabolotskaya equation

The 40th anniversary of the Khokhlov-Zabolotskaya equation was marked by a special session of the 158th Meeting of the Acoustical Society of America (October 2009, San Antonio, Texas, United States). A response on the part of Acoustical Physics to this date is also quite appropriate, all the more so because Russian scientists were the main players involved in formulating and using this equation during the period of time between the middle 1960s and middle 1980s. In this article, the author—a participant and witness of those events—presents his view of the dramatic history of the formulation of this equation and related models in the context of earlier and independent work in aerodynamics and nonlinear wave theory. The main problems and physical phenomena described by these mathematical models are briefly considered.

Historical aspects of the Khokhlov–Zabolotskaya equation and its generalizations.

Derivation of the Khokhlov–Zabolotskaya (KZ) equation provided a new approach to describing the combined effects of nonlinear propagation and diffraction in sound beams. In this paper, historical aspects of the KZ equation and its generalizations are presented. The interest in nonlinear acoustic beams of Academician Khokhlov and his colleagues at Moscow State University was inspired in the 1960s by emerging developments in laser physics and the corresponding models of nonlinear optical beams. The two cases, acoustical and optical, represent two limiting cases of nonlinear beams in weakly and strongly dispersive media, respectively, which required different theoretical approaches. The KZ equation and analogous nonlinear evolution equations of nonlinear wave physics are reviewed. It is illustrated how theoretical studies combined with numerical modeling resulted in predictions of new physical phenomena in nonlinear acoustic beams. Concurrently, newer applications of nonlinear acoustics such as parametric arrays, sonic booms, and medical acoustics stimulated the derivation of generalized KZ-type equations together with analytical and numerical methods to solve them. Modern applications and corresponding generalized KZ-type models that include effects such as frequency-dependent absorption, weak dispersion, scalar and vectorial inhomogeneities of the propagation medium, different orders of nonlinearity, and more accurate description of diffraction are presented.

Minkowski metrics on solutions of the Khokhlov-Zabolotskaya equation

In this paper, we show that every solution of the Khokhlov-Zabolotskaya equation possesses a Minkowski metric. This metric is responsible for transport of singularities of KZ-solutions. We find explicit solutions for which the metrics are either locally-flat or Ricci-flat or conformally-flat.

Group analysis of the Khokhlov–Zabolotskaya type equations.

Khokhlov–Zabolotskaya (KZ) equation is the basic equation of the theory of acoustic beams propagating in quadratically nonlinear media. Being generalized for the case of a medium with general-type nonlinearity and expressed in dimensionless notation, it has the form [uz+P(u)uτ]τ=uxx+uyy, where u represents waveform, τ is retarded time, z is axial coordinate, x and y are lateral coordinates, and P(u) characterizes a nonlinear addition to the linear wave velocity: quadratic nonlinearity corresponds to P(u)=u, cubic one to P(u)=u2, etc. No general analytical solution has been obtained for this nonlinear KZ-type equation, so numerical methods or asymptotic approximations are usually employed. Additional powerful mathematical tool is Lie group analysis that enables to find general symmetries of differential equations. These symmetries help to generalize known analytical and numerical solutions, derive new solutions, and obtain conservation laws. Results of group analysis of KZ and KZK equations are discussed for various nonlinearities P(u). Examples of obtaining of new solutions and deriving of reduced equations are presented. It is also shown that the generalized KZ equation can be written in Euler–Lagrange form, which makes it possible to apply Nöther theorem and derive new conservation laws for nonlinear acoustic beams.

Stable-Range Approach to Short Wave and Khokhlov-Zabolotskaya Equations

Short wave equations were introduced in connection with the nonlinear reflection of weak shock waves. They also relate to the modulation of a gas-fluid mixture. Khokhlov-Zabolotskaya equation is used to describe the propagation of a diffraction sound beam in a nonlinear medium. We give a new algebraic method of solving these equations by using certain finite-dimensional stable range of the nonlinear terms and obtain large families of new explicit exact solutions parameterized by several functions for them. These parameter functions enable one to find the solutions of some related practical models and boundary value problems.