Asymptotics of solutions of higher-dimensional integrable equations and their perturbations

Kiselev, O. M.

doi:10.1007/s10958-006-0347-8

Cited by 9 publications

(12 citation statements)

References 72 publications

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“…A very precise asymptotics as t → ∞ is given in [114] (see also [113]) for a specific class of scattering data. It differs according to different domains in the (x, y,t) space, expressed in terms of the variables ξ = x/t and η = y/t.…”

Section: The Kp II Equationmentioning

confidence: 99%

“…Concerning the Cauchy problem, the global existence and uniqueness of a solution ψ ∈ C(R; S (R 2 )) of DS I for data ψ 0 ∈ S (R 2 ), φ 1 , φ 2 ∈ C(R; S (R)) is proven in [64]. Under a smallness condition, the solutions with trivial boundary conditions φ 1 = φ 2 = 0 disperse as 1/t (Kiselev [115], see also [113]). A precise asymptotics is also given.…”

Section: Ds I By Ist Comparison With Elliptic-hyperbolic Dsmentioning

confidence: 99%

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IST Versus PDE: A Comparative Study

Klein

Saut

2015

Hamiltonian Partial Differential Equations and Applications

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Section: The Kp II Equationmentioning

confidence: 99%

Section: Ds I By Ist Comparison With Elliptic-hyperbolic Dsmentioning

confidence: 99%

IST Versus PDE: A Comparative Study

Klein

Saut

2015

Hamiltonian Partial Differential Equations and Applications

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“…Perry [41] established the same large time asymptotic behaviour in the L ∞ norm, for initial data in H 1,1 ∩ L 1 . Kiselev ( [29], [30]) had similar results under more restrictive assumptions.…”

Section: Introductionmentioning

confidence: 75%

A nonlinear Plancherel theorem with applications to global well-posedness for the defocusing Davey–Stewartson equation and to the inverse boundary value problem of Calderón

2019

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We prove a Plancherel theorem for a nonlinear Fourier transform in two dimensions arising in the Inverse-Scattering method for the defocusing Davey-Stewartson II equation. We then use it to prove global well-posedness and scattering in L 2 for defocusing DSII. This Plancherel theorem also implies global uniqueness in the inverse boundary value problem of Calderón in dimension 2, for conductivities σ > 0 with log σ ∈Ḣ 1 . The proof of the nonlinear Plancherel theorem includes new estimates on fractional integrals in Sobolev spaces, as well as a new result on L 2 -boundedness of pseudo-differential equations with non-smooth symbols, valid in all dimensions. Estimates on Fractional Integrals and Pseudo-differential OperatorsThis section is devoted to the proofs of new boundedness theorems on fractional integrals, pointwise multipliers in negative Besov spaces and pseudo-differential operators with nonsmooth symbols. These results will be crucial in the rest of the paper. The proofs in this section are valid in all dimensions.

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“…In the case of negative energy, N V − is in some sense reminiscent of the KPII equation (see, for example, [17,16] for results on N V − and [5,6,4,22] for related results on KPII). KPII was proved to be globally well-posed in L 2 (R 2 ) by Bourgain in [7].…”

Section: The Nv Equationmentioning

confidence: 99%

Dispersive estimates for rational symbols and local well-posedness of the nonzero energy NV equation

Kazeykina

Muñoz

2016

Journal of Functional Analysis

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We continue our study on the Cauchy problem for the two-dimensional Novikov-Veselov (NV) equation, integrable via the inverse scattering transform for the two dimensional Schrödinger operator at a fixed energy parameter. This work is concerned with the case of positive energy. For the solution of the linearized equation we derive smoothing and Strichartz estimates by combining two different frequency regimes. At non-low frequencies we also derive improved smoothing estimates with gain of almost one derivative. We combine the linear estimates with the Fourier decomposition method and X s,b spaces to obtain local well-posedness of NV at positive energy in H s , s > 1 2 . Our result implies, in particular, that at least for s > 1 2 , NV does not change its behavior from semilinear to quasilinear as energy changes sign, in contrast to the closely related Kadomtsev-Petviashvili equations.As a supplement, we provide some new explicit solutions of NV at zero energy which exhibit an interesting behaviour at large times.

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Asymptotics of solutions of higher-dimensional integrable equations and their perturbations

Cited by 9 publications

References 72 publications

IST Versus PDE: A Comparative Study

IST Versus PDE: A Comparative Study

A nonlinear Plancherel theorem with applications to global well-posedness for the defocusing Davey–Stewartson equation and to the inverse boundary value problem of Calderón

Dispersive estimates for rational symbols and local well-posedness of the nonzero energy NV equation

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