Large time asymptotics of solutions to the short-pulse equation

Okamoto, Mamoru

doi:10.1007/s00030-017-0464-8

Cited by 6 publications

(5 citation statements)

References 17 publications

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“…And there are others aspects of the SP equation have been addressed in the literature, including integrable semi-discrete and full-discrete analogues [18], well-posedness of the Cauchy problem [19,20] and Riemann-Hilbert approach [21]. Using the method of testing by wave packets, Okamoto prove the unique global existence of small solutions to the SP equation (1.1) when the small initial data u 0 [22].…”

Section: Introductionmentioning

confidence: 99%

Soliton Resolution for the Short-pluse Equation

Yang,

Fan

2020

Preprint

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In this paper, we apply ∂ steepest descent method to study the Cauchy problem for the focusing nonlinear short-pluse equationwhere H 1,1 (R) is a weighted Sobolev space. We construct the solution of the shortpluse equation via a the solution of Riemann-Hilbert problem in the new scale (y, t).In any fixed space-time cone of the new scale (y, t) which stratify that v1 ≤ v1 ∈ R − and ξ = y t < 0,we compute the long time asymptotic expansion of the solution u(x, t), which prove soliton resolution conjecture consisting of three terms: the leading order term can be characterized with an N (I)-soliton whose parameters are modulated by a sum of localized soliton-soliton interactions as one moves through the cone; the second t −1/2 order term coming from soliton-radiation interactions on continuous spectrum up to an residual error order O(|t| −1 ) from a ∂ equation. Our results also show that soliton solutions of short-pluse equation are asymptotically stable.

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Section: Introductionmentioning

confidence: 99%

Soliton Resolution for the Short-pluse Equation

Yang,

Fan

2020

Preprint

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“…provided that a ≥ 0 and a + c ≥ 0. Hence, the second inequality follows from the same argument as in Lemma 3.1 in [23].…”

Section: Decay Estimatesmentioning

confidence: 80%

“…To obtain (1.5) in the self-similar region X 0 (t) or the oscillatory region X − (t), we use the method of testing by wave packets as in [6,7,12,23]. In §5, we observe that the wave packet is a good approximate solution to the linear fifth-order mKdV equation.…”

Section: Introductionmentioning

confidence: 99%

Long-time behavior of solutions to the fifth-order modified KdV-type equation

Okamoto¹

2017

Preprint

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“…This is a consequence of the following lemma. See Lemma 3.1 in [16] and Lemma 4.1 [17] for the proof.…”

Section: Hyperbolic and Elliptic Parts Of Umentioning

confidence: 99%

Long-time behavior of solutions to a fourth-order nonlinear Schrödinger equation with critical nonlinearity

Okamoto

Uriya

2021

J. Evol. Equ.

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We consider the long-time behavior of solutions to a fourth-order nonlinear Schrödinger (NLS) equation with a derivative nonlinearity. By using the method of testing by wave packets, we construct an approximate solution and show that the solution for the fourth-order NLS has the same decay estimate for linear solutions. We prove that the self-similar solution is the leading part of the asymptotic behavior.

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Large time asymptotics of solutions to the short-pulse equation

Abstract: Abstract. We consider the long-time behavior of solutions to the short-pulse equation. Using the method of testing by wave packets, we prove small data global existence and modified scattering.

Cited by 6 publications

References 17 publications

Soliton Resolution for the Short-pluse Equation

Soliton Resolution for the Short-pluse Equation

Long-time behavior of solutions to the fifth-order modified KdV-type equation

Long-time behavior of solutions to a fourth-order nonlinear Schrödinger equation with critical nonlinearity

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