part i foundations of the theory of waves in nondispersivemedia
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
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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
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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 vanishingviscosity
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
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4 field evolution within the framework of the burgersequation
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 turbulencemodel
4.3 problems to chapter 4
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5 evolution of a noise field within the framework of the burgersequation
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 velocityfield
5.2.2 properties of the probability density of a random velocityfield
5.2.3 spectra of a velocity field
5.3 turbulence evolution at the stage of developeddiscontinuities
5.3.1 phenomenology of the burgers turbulence
5.3.2 evolution of the burgers turbulence: statisticallyhomogeneous 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 inhomogeneouspotential (-3 [ n [ 1)
5.3.6 statistically homogeneous velocity and inhomogeneouspotential (-1 [ n [ 1)
5.3.7 statistically inhomogeneous velocity and in_homogeneouspotential (-3 [ n [ -1)
5.3.8 evolution of intense acoustic noise
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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-dimentionallow
6.1.3 jacobian matrix for the transformation from lagrangian toeulerian 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 singulardensity 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 equationand
in the multidimensional burgers equation
6.3.1 asymptotic solutions of the multidimensional burgersequation and local self-similarity
6.3.2 evolution of simple localized perturbations
6.3.3 evolution of periodic structures under infinite reynoldsnumbers
6.3.4 evolution of the anisotropic burgers turbulence
6.3.5 evolution of perturbations with complex internalstructure
6.3.6 asymptotic long-time behavior of a localizedperturbation
6.3.7 appendix to section 6.3. statistical properties of maximaof inhomogeneous random gaussian fields
6.4 model description of evolution of the large-scale structure ofthe universe
6.4.1 gravitational instability in an expanding universe
6.4.2 from the vlasov~poisson equation to the zeldovichapproximation and adhesion model
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part ii mathematical models and physical phenomena in nonlinearacoustics
7 model equations and methods of finding their exactsolutions
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 usingthe 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 nonlinearacoustic 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 compressibleinclusions
8.3.2 nonlinearity of solid structurally inhomogeneousmedia
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 asolid
8.5.2 measurements based on the use of radiation pressure
8.5.3 nonlinear acoustic diagnostics in constructionindustry
8.6 non-typical nonlinear phenomena in structurally inhomogeneousmedia
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9 nonlinear sawtooth waves
9.1 sawtooth waves
9.2 field and spectral approaches in the theory of nonlinearwaves
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 geometricacoustics
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
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10 self-action of spatially bounded waves containing shockfronts
10.1 introduction
10.2 self-action of sawtooth ultrasonic wave beams due to theheating of a medium and acoustic wind formation
10.3 self-refraction of weak shock waves in a quardaticallynonlinear medium
10.4 non-inertial self-action in a cubically nonlinearmedium
10.5 symmetries and conservation laws for an evolution equationdescribing beam propagation in a nonlinear medium
10.6 conclusions
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11 nonlinear standing waves, resonance phenomena and frequencycharacteristics of distributed systems
11.1 introduction
11.2 methods of evaluation of the characteristics of nonlinearresonators
11.3 standing waves and the q-factor of a resonator filled with adissipating medium
11.4 frequency responses of a quadratically nonlinearresonator
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
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