MATHEMATICS AND IMACS COMPUTERS IN SIMULATION ELSEVIER Mathematics and Computers in Simulation 37... more MATHEMATICS AND IMACS COMPUTERS IN SIMULATION ELSEVIER Mathematics and Computers in Simulation 37 (1994) 265-277 On the asymptotic behavior of solutions to nonlinear, dispersive, dissipative wave equations Jerry Bona ab *, Keith Promislow a, Gene Wayne ...
The renormalization techniques for determining the long-time asymp-totics of nonlinear parabolic ... more The renormalization techniques for determining the long-time asymp-totics of nonlinear parabolic equations pioneered by Bricmont, Kupiainen and Lin are shown to be eeective in analyzing nonlinear wave equations featuring both dissi-pation and dispersion. These methods allow us to recover recent results of Dix in a way which is both transparent and has interesting prospects for generalization. This study is focussed on the long-time behavior of solutions of the damped, non-linear wave equation u t ? Mu + u xxx + u p u x = 0; (x 2 R; t > 1); (1.1) with initial data u(; 1) = f 0 (); (x 2 R): (1.2) Here M is a Fourier-multiplier operator, which in Fourier transformed variables has the form d Mu(k) = ?jkj 2 b u, where 1 2 < 1, and p 2. The initial-value problem (1.1){(1.2) is always locally well posed, but for larger values of p and large initial data, it appears that it may not be globally well posed (see Bona et al. 1993). If attention is restricted to small initial data, then (1...
Transactions of the American Mathematical Society, 1989
We study the group of germs of contact diffeomorphisms at a fixed point. We prove that the abelia... more We study the group of germs of contact diffeomorphisms at a fixed point. We prove that the abelianization of this group is isomorphic to the multiplicative group of real positive numbers. The principal ingredient in this proof is a version of the Sternberg linearization theorem in which the conjugating diffeomorphism preserves the contact structure.
MATHEMATICS AND IMACS COMPUTERS IN SIMULATION ELSEVIER Mathematics and Computers in Simulation 37... more MATHEMATICS AND IMACS COMPUTERS IN SIMULATION ELSEVIER Mathematics and Computers in Simulation 37 (1994) 265-277 On the asymptotic behavior of solutions to nonlinear, dispersive, dissipative wave equations Jerry Bona ab *, Keith Promislow a, Gene Wayne ...
MATHEMATICS AND IMACS COMPUTERS IN SIMULATION ELSEVIER Mathematics and Computers in Simulation 37... more MATHEMATICS AND IMACS COMPUTERS IN SIMULATION ELSEVIER Mathematics and Computers in Simulation 37 (1994) 265-277 On the asymptotic behavior of solutions to nonlinear, dispersive, dissipative wave equations Jerry Bona ab *, Keith Promislow a, Gene Wayne ...
The renormalization techniques for determining the long-time asymp-totics of nonlinear parabolic ... more The renormalization techniques for determining the long-time asymp-totics of nonlinear parabolic equations pioneered by Bricmont, Kupiainen and Lin are shown to be eeective in analyzing nonlinear wave equations featuring both dissi-pation and dispersion. These methods allow us to recover recent results of Dix in a way which is both transparent and has interesting prospects for generalization. This study is focussed on the long-time behavior of solutions of the damped, non-linear wave equation u t ? Mu + u xxx + u p u x = 0; (x 2 R; t > 1); (1.1) with initial data u(; 1) = f 0 (); (x 2 R): (1.2) Here M is a Fourier-multiplier operator, which in Fourier transformed variables has the form d Mu(k) = ?jkj 2 b u, where 1 2 < 1, and p 2. The initial-value problem (1.1){(1.2) is always locally well posed, but for larger values of p and large initial data, it appears that it may not be globally well posed (see Bona et al. 1993). If attention is restricted to small initial data, then (1...
Transactions of the American Mathematical Society, 1989
We study the group of germs of contact diffeomorphisms at a fixed point. We prove that the abelia... more We study the group of germs of contact diffeomorphisms at a fixed point. We prove that the abelianization of this group is isomorphic to the multiplicative group of real positive numbers. The principal ingredient in this proof is a version of the Sternberg linearization theorem in which the conjugating diffeomorphism preserves the contact structure.
MATHEMATICS AND IMACS COMPUTERS IN SIMULATION ELSEVIER Mathematics and Computers in Simulation 37... more MATHEMATICS AND IMACS COMPUTERS IN SIMULATION ELSEVIER Mathematics and Computers in Simulation 37 (1994) 265-277 On the asymptotic behavior of solutions to nonlinear, dispersive, dissipative wave equations Jerry Bona ab *, Keith Promislow a, Gene Wayne ...
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