Resonant Scattering and Generation of Waves

1. Front Matter

   Pages i-xx

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2. The Mathematical Model

   • Lutz Angermann, Vasyl V. Yatsyk

   Pages 1-43

3. Existence and Uniqueness of a Weak Solution of the Nonlinear Boundary Value Problem

   • Lutz Angermann, Vasyl V. Yatsyk

   Pages 45-53

4. The Equivalent System of Nonlinear Integral Equations

   • Lutz Angermann, Vasyl V. Yatsyk

   Pages 55-76

5. Spectral Analysis

   • Lutz Angermann, Vasyl V. Yatsyk

   Pages 77-100

6. Numerical Solution of the Nonlinear Boundary Value Problem

   • Lutz Angermann, Vasyl V. Yatsyk

   Pages 101-113

7. Numerical Treatment of the System of Integral Equations

   • Lutz Angermann, Vasyl V. Yatsyk

   Pages 115-122

8. Numerical Experiments

   • Lutz Angermann, Vasyl V. Yatsyk

   Pages 123-186

9. Conclusion and Outlook

   • Lutz Angermann, Vasyl V. Yatsyk

   Pages 187-196

10. Back Matter

    Pages 197-208

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This monograph deals with theoretical aspects and numerical simulations of the interaction of electromagnetic fields with nonlinear materials. It focuses in particular on media with nonlinear polarization properties. It addresses the direct problem of nonlinear Electrodynamics, that is to understand the nonlinear behavior in the induced polarization and to analyze or even to control its impact on the propagation of electromagnetic fields in the matter. 
The book gives a comprehensive presentation of the results obtained by the authors during the last decade and put those findings in a broader, unified context and extends them in several directions.
It is divided into eight chapters and three appendices. Chapter 1 starts from the Maxwell’s equations and develops a wave propagation theory in plate-like media with nonlinear polarizability. In chapter 2 a theoretical framework in terms of weak solutions is given in order to prove the existence and uniqueness of a solution of the semilinear boundary-value problem derived in the first chapter. Chapter 3 presents a different approach to the solvability theory of the reduced frequency-domain model. Here the boundary-value problem is reduced to finding solutions of a system of one-dimensional nonlinear Hammerstein integral equations. Chapter 4 describes an approach to the spectral analysis of the linearized system of integral equations. Chapters 5 and 6 are devoted to the numerical approximation of the solutions of the corresponding mathematical models. Chapter 7 contains detailed descriptions, discussions and evaluations of the numerical experiments. Finally, chapter 8 gives a summary of the results and an outlook for future work.

• third-harmonic generation
• finite element methods
• Q-factor analysis
• nonlinear boundary value problem
• cubic susceptibility
• frequency tripling
• Maxwell's equations
• nonlinear polarizability
• nonlinear integral equations
• Sturm-Liouville boundary value problems
• wave propagation
• frequency domain model
• Hemmerstein integral equation
• Kerr nonlinearity
• cubic polarization
• trace inequality
• spectral analysis
• spectral energy relationships
• solvability theory
• numerical spectral analysis

“The book is a useful reference work, not only for professional theoreticians dealing with problems of nonlinear electrodynamics, but also for graduate students who can widely benefit from it.” (Vladimir Čadež, zbMATH 1414.78001, 2019)

• Institut für Mathematik, Technische Universität Clausthal, Clausthal-Zellerfeld, Germany

  Lutz Angermann

• O.Ya. Usikov Insitute for Radiophysics and Electronics, National Academy of Sciences of Ukraine, Kharkiv, Ukraine

  Vasyl V. Yatsyk

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