On the linearized log-KdV equation

Pelinovsky, Dmitry E.

doi:10.4310/cms.2017.v15.n3.a13

Cited by 6 publications

(2 citation statements)

References 11 publications

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“…Similar questions arise in the context of granular chains and involve the logarithmic versions of the Burgers and Korteweg-de Vries equations [10,11]. The logarithmic nonlinearity is more singular than the modular nonlinearity, hence questions of well-posedness and stability of nonlinear waves remain open for some time [4,20,24].…”

Section: Introductionmentioning

confidence: 93%

Asymptotic stability of viscous shocks in the modular Burgers equation

Pelinovsky

Poullet

2021

Nonlinearity

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Dynamics of viscous shocks is considered in the modular Burgers equation, where the time evolution becomes complicated due to singularities produced by the modular nonlinearity. We prove that the viscous shocks are asymptotically stable under odd and general perturbations. For the odd perturbations, the proof relies on the reduction of the modular Burgers equation to a linear diffusion equation on a half-line. For the general perturbations, the proof is developed by converting the time-evolution problem to a system of linear equations coupled with a nonlinear equation for the interface position. Exponential weights in space are imposed on the initial data of general perturbations in order to gain the asymptotic decay of perturbations in time. We give numerical illustrations of asymptotic stability of the viscous shocks under general perturbations.

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Section: Introductionmentioning

confidence: 93%

Asymptotic stability of viscous shocks in the modular Burgers equation

Pelinovsky

Poullet

2021

Nonlinearity

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“…Natali et al [10] established the orbital stability of periodic waves related to the logarithmic-KdV equation. Pelinovsky [11] addressed the properties of Gaussian solitary waves solutions to the linearized logarithmic-KdV equation. Inc. et al [12] analyzed the logarithmic-KdV equation involving the new fractional operator called Atangana-Baleanu fractional derivative with Mittag-Leffler type kernel.…”

Section: Introductionmentioning

confidence: 99%

Structure-preserving analysis on Gaussian solitary wave solution of logarithmic-KdV equation

Hu¹,

Hu²,

Fan³

et al. 2021

Phys. Scr.

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The existence of the Gaussian solitary wave solution in the logarithmic-KdV equation has aroused considerable interests recently. Focusing on the defects of the reported multi-symplectic analysis on the Gaussian solitary wave solution of the logarithmic-KdV equation and considering the dynamic symmetry breaking of the logarithmic-KdV equation, new approximate multi-symplectic formulations for the logarithmic-KdV equation are deduced and the associated structure-preserving scheme is constructed to simulate the evolution of the Gaussian solitary wave solution. In the structure-preserving simulation process of the Gaussian solitary wave solution, the residuals of three conservation laws are recorded in each step. Comparing with the reported numerical results, it can be found that the Gaussian solitary wave solution can be simulated with tiny numerical errors and three conservation laws are preserved perfectly in the simulation process by the structure-preserving scheme presented in this paper, which implies that the proposed structure-preserving scheme improved the precision as well as the structure-preserving properties of the reported multi-symplectic approach.

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