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...generalizations of the Fourier transform (such as scattering transforms or Fourier integral operators) can be employed.

| title = Restriction of Fourier Transform to Quadratic Surfaces and Decay of Solutions of Wave Equations

...e = Schrödinger Maximal Function and Restriction Properties of the Fourier transform

| title = Fourier transform restriction phenomena for certain lattice subsets and applications to nonli

| title = Fourier transform restriction phenomena for certain lattice subsets and applications to nonli

| title = The inverse scattering transform-Fourier analysis for nonlinear problems

...is drift term can be eliminated by a <span class="SpellE">Gallilean</span> transform, but this is not available on the half-line). ...rm. One approach is to use the Fourier-<span class="SpellE">Laplace</span> transform instead.

...frequency outputs, and '''non-resonant''' otherwise. Using the [[Fourier transform]], one can often usefully decompose a given nonlinearity into resonant and One can also attempt to use [[transform]]s to eliminate (or at least attenuate) nonresonant portions of the nonline

...eStTkTa-p2]], via the <span class="SpellE">KdV</span> theory and the Miura transform, for both the <span class="SpellE">focussing</span> and <span class="SpellE ...o below H^{1/2} provided that one has a decay like O(|k|^{-1+\eps}) on the Fourier coefficients (which is indeed the case almost surely).

...er transform [[VaVe2001]]. For instance one has LWP for data whose Fourier transform decays like <math>|x|^{-1/6-}\,</math>. Ideally one would like to replace t ...allilean spaces in [[VaVe2001]]. For instance one has GWP when the Fourier transform of the data decays like <math>|x|^{-5/12-}\,</math>. Ideally one would like

...s="SpellE">H^s</span> for s >= -3/4 [[CtCoTa-p]], using a modified [[Miura transform]] and the [[mKdV on R|mKdV theory]]. This is despite the failure of the key ...>KdV</span> this has also been established in [[CtCoTa-p]], by the [[Miura transform]] and the [[mKdV on R| corresponding result for mKdV]].

...near equation one can always write down a solution using Fourier series or transform. These solutions represent superpositions of traveling waves.

...into a solution that blows up in finite time (cf. how the pseudoconformal transform is used to achieve a similar effect without the potential). ** For potentials which decay like <math><x>^{-7-}\,</math> and whose Fourier transform is in <math>L^1\,,</math> a version of this estimate is in [[JouSfSo1991]]

...ify all computations. (If ''L'' is constant coefficient, then the Fourier transform can usually be used to justify everything so long as one works in the categ

...ms are more likely to disappear in energy estimates). One can often "gauge transform" the equation (in a manner dependent on the solution <math>u\,</math>) so t

M. Ablowitz, D. Kaup, A. Newell, H. Segur, ''The inverse scattering transform-Fourier analysis for nonlinear problems'', Studies in Appl. Math. '''53''' (1974), M. Ablowitz, A. Fokas, ''The inverse scattering transform for the Benjamin-Ono equation, a pivot for multidimensional problems'', Stu