Soliton Resolution for the Energy-Critical Nonlinear Wave Equation in the Radial Case

Jendrej, Jacek; Lawrie, Andrew

doi:10.1007/s40818-023-00159-4

Ann. PDE

2023

DOI: 10.1007/s40818-023-00159-4

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Soliton Resolution for the Energy-Critical Nonlinear Wave Equation in the Radial Case

Jacek Jendrej,

Andrew Lawrie

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Cited by 3 publications

References 75 publications

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Dynamics of the Collision of Two Nearly Equal Solitary Waves for the Zakharov–Kuznetsov Equation

Pilod,

Valet

2024

Commun. Math. Phys.

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We study the dynamics of the collision of two solitary waves for the 2 and 3-dimensional Zakharov–Kuznetsov equation, a high-dimensional non-integrable version of the Korteweg-de Vries equation that appears as an asymptotic model in plasma physics. We describe the evolution of the solution behaving as a sum of 2-solitary waves of nearly equal speeds at time $$t=-\infty $$ t = - ∞ up to time $$t=+\infty $$ t = + ∞ . We show that this solution behaves as the sum of two modulated solitary waves and an error term which is small in $$H^1$$ H 1 for all time $$t \in {\mathbb {R}}$$ t ∈ R . Finally, we also prove the stability of this solution for large times around the collision. The proofs are a non-trivial extension of the ones of Martel and Merle for the quartic generalized Korteweg-de Vries equation to higher dimensions. First, despite the non-explicit nature of the solitary wave, we construct an approximate solution in an intrinsic way by canceling the error to the equation only in the natural directions of scaling and translation. Then, to control the difference between a solution and the approximate solution, we use a modified energy functional and a refined modulation estimate in the transverse variable. Moreover, we rely on the Hamiltonian structure of the ODE governing the distance between the waves, which cannot be approximated by explicit solutions, to close the bootstrap estimates on the parameters. We hope that the techniques introduced here are robust and will prove useful in studying the collision phenomena for other focusing non-linear dispersive equations with non-explicit solitary waves.

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Dynamics of the Collision of Two Nearly Equal Solitary Waves for the Zakharov–Kuznetsov Equation

Pilod,

Valet

2024

Commun. Math. Phys.

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Long-time asymptotics of the damped nonlinear Klein–Gordon equation with a delta potential

Ishizuka

2025

Nonlinear Analysis

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Stable blowup for focusing semilinear wave equations in all dimensions

Ostermann

2024

Trans. Amer. Math. Soc.

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We consider the wave equation with focusing power nonlinearity. The associated ODE in time gives rise to a self-similar solution known as the ODE blowup. We prove the nonlinear asymptotic stability of this blowup mechanism outside of radial symmetry in all space dimensions and for all superlinear powers. This result covers for the first time the whole energy-supercritical range without symmetry restrictions.

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Soliton Resolution for the Energy-Critical Nonlinear Wave Equation in the Radial Case

Cited by 3 publications

References 75 publications

Dynamics of the Collision of Two Nearly Equal Solitary Waves for the Zakharov–Kuznetsov Equation

Dynamics of the Collision of Two Nearly Equal Solitary Waves for the Zakharov–Kuznetsov Equation

Long-time asymptotics of the damped nonlinear Klein–Gordon equation with a delta potential

Stable blowup for focusing semilinear wave equations in all dimensions

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