Nonlinear Nonlocal Equations in the Theory of Waves

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Translations of Mathematical Monographs 133

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Nonlinear Nonlocal Equations in theTheory of Waves

® Copyright 1994 by the American Mathematical Society
     ISBN 0-8218-4573-X
     QC157.N3813 1994 532\'.593\'01515353-dc20
     LCCN 9308452

Contents

Introduction
     §1. Physical problems leading to nonlinear nonlocal equations
     §2. Brief review of the content of this book

CHAPTER 1  Simplest Properties of Solutions of Nonlinear Nonlocal Equations
     §1. Conservation laws. Solitary waves
     §2. Wave peaking
     §3. Breaking of waves in the case of a monotone kernel

CHAPTER 2  The Cauchy Problem for the Whitham Equation
     §1. Introduction
     §2. The existence of a classical solution for the Cauchy problem on a finite time-interval
          2.1. Regular operator
          2.2. Dissipative operator
          2.3. Antidissipative operator
     §3. The existence of a global in time solution
          3.1. Strongly dissipative operator
          3.2. Strongly dissipative operator
     §4. Smoothing of solutions
          4.1. Smoothing for short time intervals
          4.2. The existence of smoothed global solution
     §5. Breaking of waves for a conservative or dissipative operator of order less than 3/5
     §6. Breaking of waves for arbitrary operators of order less than 2/3
     §7. Proof of Theorem 10

CHAPTER 3  The Periodic Problem
     §1. Introduction
     §2. Breaking of waves for a conservative or dissipative operator ]K of order a < 3/5
          2.1. Local (in time) existence of a solution of problem (1.1
     §3. On the existence of a global solution of the Cauchy problem
     §4. Smoothing of solutions of the Cauchy problem
          4.1. Global existence of a solution of the Cauchy problem for smooth initial conditions
          4.2. Smoothing
     §5. The periodic problem with a weak interaction

CHAPTER 4  The System of Equations of Surface Waves
     §1. Conservation laws
     §2. The Cauchy problem for the system of equations of surface waves with a regular operator
     §3. The Cauchy problem for the system of equations of surface waves with a dissipative or conservative operator
     §4. Breaking of waves
          4.1. The theorem on breaking of wave
          4.2. Technical lemmas
          4.3. The second theorem on breaking of waves
     §5. Existence of a global solution of the Cauchy problem
     §6. Smoothing of the initial perturbations
     §7. Smoothing of initial perturbations from L2
     §8. The Cauchy problem for the system of equations for surface waves with weak nonlocal interaction

CHAPTER 5  Generalized Solutions
     §1. Introduction
     §2. The dissipative Whitham equation
     §3. The conservative Whitham equation
     §4. The shallow water equation
     §5. Nonlinear nonlocal Schrodinger equation
     §6. The system of surface waves

CHAPTER 6  The Asymptotics as t -00 of Solutions of the Generalized Kolmogorov-Petrovskii-Piskunov Equation
     §1. Introduction
     §2. Proof of the theorem
     §3. Computation of the functions I(p)
          3.1. Perturbation theory
          3.2. The first formula for Qy
          3.3. A second formula for Q
          3.4. A formula for I (p).
          3.5. Bounds for cly(p).

CHAPTER 7  Asymptotics of Solutions of the Whitham Equation for Large Times
     §1. Introduction
     §2. Technical lemmas
     §3. Proof of the theorem
          3.1. Perturbation theory
          3.2. The estimation of the integrals
          3.3. The leading term of the asymptotics
          3.4. Estimation of the Holder term
          3.5. The asymptotics
     §4. Computation of the numbers
          4.1. Perturbation theor
          4.2. First formula for the symbol Qy
          4.3. Second formula for Q,,.
          4.4. Formula for
     §5. Asymptotics of solutions of the KDV equation

CHAPTER 8  Asymptotics as t - 00 of Solutions of the Nonlinear Nonlocal Schrodinger Equation
     §1. Introduction
     §2. Technical lemmas
     §3. Proof of Theorem 1
          3.1. Perturbation theory
          3.2. Estimates of the integrals
          3.3. Estimation of the Holder term J
          3.4. The asymptotics
     §4. Computation of the numbers
          4.1. Perturbation theory
          4.2. First formula for Q,.
          4.3. Second formula(*) for Q
     §5. Computation of the asymptotics for the Landau-Ginzburg equation
     §6. Asymptotics of solutions for periodic problem of the nonlinear Schrodinger equation for large times

CHAPTER 9  Asymptotics of Solutions for a System of Equations of Surface Waves for Large Times
     §1. Introduction
     §2. Lemmas
     §3. Proof of the theorem
          3.1. Perturbation theory
          3.2. Estimation of the function v,,
          3.3. Estimates of the integrals J1, J2i J3
          3.4. Estimation of the integral J4
          3.5. Estimation of the Holder term J5
          3.6. The asymptotics
     §4. Computation of the vectors
          4.1. Perturbation theory
          4.2. First formula for Q
          4.3. Second formula for Q
          4.4. A formula for

CHAPTER 10  The Step-Decaying Problem for the Korteweg-de Vries-Burgers Equation
     §1. Introduction
     §2. First theorem
     §3. Second theorem
     §4. A lemma
     §5. The step-decaying problem for the Kuramoto-Sivashinsky equation

References

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