Hamiltonian Dynamical Systems and Applications pp 179-212 | Cite as
Four lectures on KAM for the non-linear Schrödinger equation
Conference paper
We discuss the KAM-theory for lower-dimensional tori for the non-linear Schrödinger equation with periodic boundary conditions and a convolution potential in dimension d. Central in this theory is the homological equation and a condition on the small divisors often known as the second Melnikov condition. The difficulties related to this condition are substantial when d≥ 2.
We discuss this difficulty, and we show that a block decomposition and a Töplitz- Lipschitz-property, present for non-linear Schrödinger equation, permit to overcome this difficuly. A detailed proof is given in [EK06].
Keywords
Normal Form Lipschitz Domain Homological Equation Block Decomposition Small Divisor
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