Nonlinear nonlocal equations in the theory of waves Book

	Introduction	1
1	Physical problems leading to nonlinear nonlocal equations	1
2	Brief review of the content of this book	6
Ch. 1	Simplest Properties of Solutions of Nonlinear Nonlocal Equations	11
1	Conservation laws. Solitary waves	11
2	Wave peaking	14
3	Breaking of waves in the case of a monotone kernel	18
Ch. 2	The Cauchy Problem for the Whitham Equation	29
2	The existence of a classical solution for the Cauchy problem on a finite time-interval	31
3	The existence of a global in time solution	40
4	Smoothing of solutions	43
5	Breaking of waves for a conservative or dissipative operator of order less than 3/5	47
6	Breaking of waves for arbitrary operators of order less than 2/3	51
7	Proof of Theorem 10	60
Ch. 3	The Periodic Problem	65
2	Breaking of waves for a conservative or dissipative operator K of order [alpha] < 3/5	65
3	On the existence of a global solution of the Cauchy problem	74
4	Smoothing of solutions of the Cauchy problem	76
5	The periodic problem with a weak interaction	84
Ch. 4	The System of Equations of Surface Waves	89
1	Conservation laws	89
2	The Cauchy problem for the system of equations of surface waves with a regular operator	91
3	The Cauchy problem for the system of equations of surface waves with a dissipative or conservative operator	97
4	Breaking of waves	101
5	Existence of a global solution of the Cauchy problem	125
6	Smoothing of the initial perturbations	126
7	Smoothing of initial perturbations from L[subscript 2]	133
8	The Cauchy problem for the system of equations for surface waves with weak nonlocal interaction	138
Ch. 5	Generalized Solutions	141
2	The dissipative Whitham equation	143
3	The conservative Whitham equation	145
4	The shallow water equation	147
5	Nonlinear nonlocal Schrodinger equation	150
6	The system of surface waves	153
Ch. 6	The Asymptotics as [actual symbol not reproducible] of Solutions of the Generalized Kolmogorov-Petrovskii-Piskunov Equation	159
2	Proof of the theorem	160
3	Computation of the functions [actual symbol not reproducible]	164
Ch. 7	Asymptotics of Solutions of the Whitham Equation for Large Times	179
2	Technical lemmas	180
3	Proof of the theorem	183
4	Computation of the numbers [actual symbol not reproducible]	190
5	Asymptotics of solutions of the KdV equation	194
Ch. 8	Asymptotics as [actual symbol not reproducible] of Solutions of the Nonlinear Nonlocal Schrodinger Equation	209
2	Technical lemmas	210
3	Proof of Theorem I	212
4	Computation of the numbers [actual symbol not reproducible]	217
5	Computation of the asymptotics for the Landau-Ginzburg equation	227
6	Asymptotics of solutions for periodic problem of the nonlinear Schrodinger equation for large times	229
Ch. 9	Asymptotics of Solutions for a System of Equations of Surface Waves for Large Times	235
2	Lemmas	238
3	Proof of the theorem	239
4	Computation of the vectors [actual symbol not reproducible]	252
Ch. 10	The Step-Decaying Problem for the Korteweg-de Vries-Burgers Equation	261
2	First theorem	262
3	Second theorem	267
4	A lemma	271
5	The step-decaying problem for the Kuramoto-Sivashinsky equation	276
	References	281
	Supplementary References	288