x Contents §8.4. Rays and nonlinear transport 285 Chapter 9. Resonant Interaction and Quasilinear Systems 291 §9.1. Introduction to resonance 291 §9.2. The three wave interaction partial differential equation 294 §9.3. The three wave interaction ordinary differential equation 298 §9.4. Formal asymptotic solutions for resonant quasilinear geometric optics 302 §9.5. Existence for quasiperiodic principal profiles 307 §9.6. Small divisors and correctors 310 §9.7. Stability and accuracy of the approximate solutions 313 §9.8. Semilinear resonant nonlinear geometric optics 314 Chapter 10. Examples of Resonance in One Dimensional Space 317 §10.1. Resonance relations 317 §10.2. Semilinear examples 321 §10.3. Quasilinear examples 327 Chapter 11. Dense Oscillations for the Compressible Euler Equations 333 §11.1. The 2 d isentropic Euler equations 333 §11.2. Homogeneous oscillations and many wave interaction systems 336 §11.3. Linear oscillations for the Euler equations 338 §11.4. Resonance relations 341 §11.5. Interaction coefficients for Euler’s equations 343 §11.6. Dense oscillations for the Euler equations 346 11.6.1. The algebraic/geometric part 346 11.6.2. Construction of the profiles 347 Bibliography 351 Index 359
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