part i foundations of the theory of waves in nondispersive
media

1 nonlinear equations of the first order

1.1 simple wave equation

1.1.1 the canonical form of the equation

1.1.2 particle flow

1.1.3 discussion of the riemann solution

1.1.4 compressions and expansions of the particle flow

1.1.5 continuity equation

1.1.6 construction of the density field

1.1.7 momentum-conservation law

1.1.8 fourier transforms of density and velocity

1.2 line-growth equation

1.2.1 forest-fire propagation

1.2.2 anisotropic surface growth

1.2.3 solution of the surface-growth equation

1.3 one-dimensional laws of gravitation

1.3.1 lagrangian description of one-dimensional gravitation

1.3.2 eulerian description of one-dimensional gravitation

1.3.3 collapse of a one-dimensional universe

1.4 problems to chapter 1

references

2 generalized solutions of nonlinear equations

2.1 standard equations

2.1.1 particle-flow equations

2.1.2 line growth in the small angle approximation

2.1.3 nonlinear acoustics equation

2.2 multistream solutions

2.2.1 interval of single-stream motion

2.2.2 appearance of multistreamness

2.2.3 gradient catastrophe

2.3 sum of streams

2.3.1 total particle flow

2.3.2 summation of streams by inverse fourier transform

2.3.3 algebraic sum of the velocity field

2.3.4 density of a "warm" particle flow

2.4 weak solutions of nonlinear equations of the first order

2.4.1 forest fire

2.4.2 the lax-oleinik absolute minimum principle

2.4.3 geometric construction of weak solutions

2.4.4 convex hull

2.4.5 maxwell's rule

2.5 the e-rykov-sinai global principle

2.5.1 flow of inelasfically coalescing particles

2.5.2 inelastic collisions of particles

2.5.3 formulation of the global principle

2.5.4 mechanical meaning of the global principle

2.5.5 condition of physical realizability

2.5.6 geometry of the global principle

2.5.7 solutions of the continuity equation

2.6 line-growth geometry

2.6.1 parametric equations of a line

2.6.2 contour in polar coordinates

2.6.3 contour envelopes

2.7 problems to chapter 2

references

3 nonlinear equations of the second order

3.1 regularization of nonlinear equations

3.1.1 the kardar-parisi-zhang equation

3.1.2 the burgers equation

3.2 properties of the burgers equation

3.2.1 galilean invariance

3.2.2 reynolds number

3.2.3 hubble expansion

3.2.4 stationary wave

3.2.5 khokhlov's solution

3.2.6 rudenko's solution

3.3 general solution of the burgers equation

3.3.1 the hopf-cole substitution

3.3.2 general solution of the burgers equation

3.3.3 averaged lagrangian coordinate

3.3.4 solution of the burgers equation with vanishing
viscosity

3.4 model equations of gas dynamics

3.4.1 one-dimensional model of a polytropic gas

3.4.2 discussion of physical properties of a model gas

3.5 problems to chapter 3

references

4 field evolution within the framework of the burgers
equation

4.1 evolution of one-dimonsional signals

4.1.1 self-similar solution, once more

4.1.2 approach to the linear stage

4.1.3 n-wave and u-wave

4.1.4 sawtooth waves

4.1.5 periodic waves

4.2 evolution of complex signals

4.2.1 quasiperiodic complex signals

4.2.2 evolution of fractal signals

4.2.3 evolution of multi-scale signals - a dynamic turbulence
model

4.3 problems to chapter 4

references

5 evolution of a noise field within the framework of the burgers
equation

5.1 burgers turbulence - acoustic turbulence

5.2 the burgers turbulence at the initial stage of evolution

5.2.1 one-point probability density of a random eulerian velocity
field

5.2.2 properties of the probability density of a random velocity
field

5.2.3 spectra of a velocity field

5.3 turbulence evolution at the stage of developed
discontinuities

5.3.1 phenomenology of the burgers turbulence

5.3.2 evolution of the burgers turbulence: statistically
homogeneous potential and velocity (n ] 1 and n [ -3)

5.3.3 exact self-similarity (n ] 2)

5.3.4 violation of self-similarity (1 [ n [ 2)

5.3.5 evolution of turbulence: statistically inhomogeneous
potential (-3 [ n [ 1)

5.3.6 statistically homogeneous velocity and inhomogeneous
potential (-1 [ n [ 1)

5.3.7 statistically inhomogeneous velocity and in_homogeneous
potential (-3 [ n [ -1)

5.3.8 evolution of intense acoustic noise

references

6 multidimensional nonlinear equations

6.1 nonlinear equations of the first order

6.1.1 main equations of three-dimensional flows

6.1.2 lagrangian and eulerian description of a three-dimentional
low

6.1.3 jacobian matrix for the transformation from lagrangian to
eulerian coordinates

6.1.4 density of a multidimensional flow

6.1.5 weak solution of the surface-growth equation

6.1.6 flows of locally interacting particles and a singular
density field

6.2 multidimensional nonlinear equations of the second order

6.2.1 the two-dimensional kpz equation

6.2.2 the three-dimensional burgers equation

6.2.3 model density field

6.2.4 concentration field

6.3 evolution of the main perturbation types in the kpz equation
and

in the multidimensional burgers equation

6.3.1 asymptotic solutions of the multidimensional burgers
equation and local self-similarity

6.3.2 evolution of simple localized perturbations

6.3.3 evolution of periodic structures under infinite reynolds
numbers

6.3.4 evolution of the anisotropic burgers turbulence

6.3.5 evolution of perturbations with complex internal
structure

6.3.6 asymptotic long-time behavior of a localized
perturbation

6.3.7 appendix to section 6.3. statistical properties of maxima
of inhomogeneous random gaussian fields

6.4 model description of evolution of the large-scale structure of
the universe

6.4.1 gravitational instability in an expanding universe

6.4.2 from the vlasov~poisson equation to the zeldovich
approximation and adhesion model

references

part ii mathematical models and physical phenomena in nonlinear
acoustics

7 model equations and methods of finding their exact
solutions

7.1 introduction

7.1.1 facts from the linear theory

7.1.2 how to add nonlinear terms to simplified equations

7.1.3 more general evolution equations

7.1.4 two types of evolution equations

7.2 lie groups and some exact solutions

7.2.1 exact solutions of the burgers equation

7.2.2 finding exact solutions of the burgers equation by using
the group-theory methods

7.2.3 some methods of finding exact solutions

7.3 the a priori symmetry method

references

8 types of acoustic nonlinearities and methods of nonlinear
acoustic diagnostics

8.1 introduction

8.1.1 physical and geometric nonlinearities

8.2 classification of types of acoustic nonlinearity

8.2.1 boundary nonlinearities

8.3 some mechanisms of bulk structural nonlinearity

8.3.1 nonlinearity of media with strongly compressible
inclusions

8.3.2 nonlinearity of solid structurally inhomogeneous
media

8.4 nonlinear diagnostics

8.4.1 inverse problems of nonlinear diagnostics

8.4.2 peculiarities of nonlinear diagnostics problems

8.5 applications of nonlinear diagnostics methods

8.5.1 detection of bubbles in a liquid and cracks in a
solid

8.5.2 measurements based on the use of radiation pressure

8.5.3 nonlinear acoustic diagnostics in construction
industry

8.6 non-typical nonlinear phenomena in structurally inhomogeneous
media

references

9 nonlinear sawtooth waves

9.1 sawtooth waves

9.2 field and spectral approaches in the theory of nonlinear
waves

9.2.1 general remarks

9.2.2 generation of harmonics

9.2.3 degenerate parametric interaction

9.3 diffracting beams of sawtooth waves

9.4 waves in inhomogeneous media and nonlinear geometric
acoustics

9.5 the focusing of discontinuous waves

9.6 nonlinear absorption and saturation

9.7 kinetics of sawtooth waves

9.8 interaction of waves containing shock fronts

references

10 self-action of spatially bounded waves containing shock
fronts

10.1 introduction

10.2 self-action of sawtooth ultrasonic wave beams due to the
heating of a medium and acoustic wind formation

10.3 self-refraction of weak shock waves in a quardatically
nonlinear medium

10.4 non-inertial self-action in a cubically nonlinear
medium

10.5 symmetries and conservation laws for an evolution equation
describing beam propagation in a nonlinear medium

10.6 conclusions

references

11 nonlinear standing waves, resonance phenomena and frequency
characteristics of distributed systems

11.1 introduction

11.2 methods of evaluation of the characteristics of nonlinear
resonators

11.3 standing waves and the q-factor of a resonator filled with a
dissipating medium

11.4 frequency responses of a quadratically nonlinear
resonator

11.5 q-factor increase under introduction of losses

11.6 geometric nonlinearity due to boundary motion

11.7 resonator filled with a cubically nonlinear medium

references

appendix fundamental properties of generalized functions

a.1 definition of generalized functions

a.2 fundamental sequences

a.3 derivatives of generalized functions

a.4 the leibniz formula

a.5 derivatives of discontinuous functions

a.6 generalized functions of a composite argument

a.7 multidimensional generalized functions

a.8 continuity equation

a.8.1 singular solution

a.8.2 green's function

a.8.3 lagrangian and eulerian coordinates

a.9 method of characteristics inde