Preface ...................................................... vii
1 Some auxiliary results from the differential geometry ....... 1
2 Cauchy method and envelope method for integration of
first order fully nonlinear partial differential
equations .................................................. 21
2.1 On the Cauchy problem for the Burgers-Hopf equation ... 21
2.2 First integrals of systems of ordinary differential
equations ............................................. 25
2.3 Fully nonlinear PDE of first order. Cauchy method ..... 26
2.4 Complete, general and singular integrals of first
order nonlinear PDE. Applications to the Cauchy
problem. Method of the envelopes ...................... 44
2.5 Lagrange-Charpit method for finding out of complete
integrals ............................................. 51
3 Some applications of first order nonlinear PDE to
mechanics and geometry ..................................... 67
3.1 Canonical transformations and equations of the
Hamilton-Jacobi type .................................. 67
3.2 Complete integrals and envelopes in the
multidimensional case. Solvability of the Cauchy
problem ............................................... 70
3.3 On the integrability of the canonical Hamilton
systems ............................................... 73
3.4 An application of first order nonlinear PDE to
the geodesies in the differential geometry ............ 80
4 Cauchy problem for Monge-Ampere type partial differential
equations .................................................. 85
4.1 Characteristics for second order nonlinear PDE ........ 85
4.2 Characteristics for Monge-Ampere type PDE ............. 87
4.3 Cauchy problem for the hyperbolic equations of
Monge-Ampere type. Geometrical approach ............... 90
4.4 Proof of Theorem 4.1 .................................. 94
5 Characteristics of quasilinear hyperbolic systems in
the hodograph plane and applications to mechanics ......... 101
5.1 Introduction ......................................... 101
5.2 Epicycloids and their properties ..................... 101
5.3 First order ODE satisfied by the arcs of the
epicycloid ........................................... 104
5.4 Characteristics of some classes of quasilinear
first order hyperbolic systems in the hodograph
plane. Epicycloids and applications to mechanics ..... 108
6 Examples of anomalous singularities of the solutions to
some classes of weakly hyperbolic semilinear systems in
the plane ................................................. 115
6.1 Introduction ......................................... 115
6.2 Formulation and proof of the main results ............ 115
7 Lorentz transformations and creation of logarithmic
singularities to the solutions of some nonstrictly
hyperbolic semilinear systems with two space variables .... 125
7.1 Introduction ......................................... 125
7.2 Statement of the problem and formulation of the
main results ......................................... 126
7.3 Some preliminary notes ............................... 127
7.4 Lorentz transformations applied to some hyperbolic
equations ............................................ 129
7.5 Proof of the main Theorem 7.1 ........................ 137
Appendix ..................................................... 143
References ................................................... 153
Index ........................................................ 157
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