Normal form: Difference between revisions
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The method of '''normal forms''' transforms the Hamiltonian of an equation via a canonical transformation to remove (or attenuate) non-resonant portions of the nonlinearity, replacing them with more tractable terms. For instance, normal forms can replace a quadratic nonlinearity with a cubic one. They are particularly useful in [[wave equations|nonlinear wave equations]]. | The method of '''normal forms''' transforms the Hamiltonian of an equation via a canonical transformation to remove (or attenuate) non-resonant portions of the nonlinearity, replacing them with more tractable terms. For instance, normal forms can replace a quadratic nonlinearity with a cubic one. They are particularly useful in [[wave equations|nonlinear wave equations]]. | ||
Normal forms should not be confused with the unrelated concept of a [[null form]]. | In [[Bo-p2]] the method of normal forms was shown to be compatible with the [[I-method]], and used to improve the low-regularity global regularity theory for certain [[NLS|nonlinear Schrodinger equations]]. | ||
Normal forms should not be confused with the unrelated concept of a [[null form]]. They achieve a similar effect as [[gauge transform|gauge transformations]], although the latter arise from the differential geometry of connections and bundles rather than from the structure of the Hamiltonian. | |||
[[Category:Transforms]] | [[Category:Transforms]] | ||
Revision as of 06:47, 31 July 2006
The method of normal forms transforms the Hamiltonian of an equation via a canonical transformation to remove (or attenuate) non-resonant portions of the nonlinearity, replacing them with more tractable terms. For instance, normal forms can replace a quadratic nonlinearity with a cubic one. They are particularly useful in nonlinear wave equations.
In Bo-p2 the method of normal forms was shown to be compatible with the I-method, and used to improve the low-regularity global regularity theory for certain nonlinear Schrodinger equations.
Normal forms should not be confused with the unrelated concept of a null form. They achieve a similar effect as gauge transformations, although the latter arise from the differential geometry of connections and bundles rather than from the structure of the Hamiltonian.
