Simple Waves and their Interactions in Quasilinear Hyperbolic Systems

This volume presents a new method of construction of solutions for nonlinear and nonelliptic systems of partial differential equations and especially nonhomogeneous ones. A generalization of Riemann invariants method to the case of nonhomogeneous systems has been formulated. 

CONTENTS

Abstract, ...3
1. Introduction, ...4
2. Homogeneous and nonhomogeneous simple integral elements, ...6
3. Simple waves (Riemann waves), ...8
4. Simple states, ...12
5. The examples of applications for the nonlinear nonhomogeneous equations of the second order, ...20
    5.1 The field equation, ...22
    5.2 The Liouville equation, ...25
    5.3 The Sine-Gordon equation, ...27
    5.4 The cosh and sinh - d Alembert equations, ...27
    5.5 The Korteweg de Vries equation, ...28
6. The notion of involution system, ...30
    6.1 The integral manifolds of Pfaff s forms system, ...30
    6.2 The notion of integral element, ...32
    6.3 Quasicharacters, ...33
7. Nonlinear superposition of simple waves in nonhomogeneous systems. Conditions of involution for the case of many
     independent variables, ...36
8. Propagation of a simple wave on a simple state, ...51
    8.1 The case when α1 ≠ 0, ...53
    8.2 The case when α1 = 0, ...57
9. Formulation of the Cauchy problem for the case of the propagation of a simple wave on a simple state, ...59
    9.1 The case when α1 ≠ 0, ...63
    9.2 The case when α1 = 0, ...63
10. Basic notions and definitions for simple waves interaction on a simple state, ...65
11. Nonlinear superposition of simple waves on a simple state. Conditions of involution for the case of two independent
    variables, ...72
12. Introduction. Equations of magnetohydrodynamics, ...83
13. Propagation of one simple wave on a simple state for the case of rotating fluid, ...85
14. Interactions of many simple waves on a simple state, ...87
    14.1 Subsystem F_, F+, B, ...90
    14.2 Subsystem F_, E, F+, B, ...93
    14.3 Subsystem F_, A_, A+, F+, B, ...95
Appendix A1. Simple integral elements for undetermined system, ...96
Appendix A2. The results of the analysis of the homogeneous equations of magnetohydrodynamics, ...97
References, ...100
Streszczenie, ...105