Abstract
Abstract
We consider two types of the generalized Korteweg–de Vries equation, where the nonlinearity is given with or without absolute values, and, in particular, including the low powers of nonlinearity, an example of which is the Schamel equation. We first prove the local well-posedness of both equations in a weighted subspace of H
1 that includes functions with polynomial decay, extending the result of Linares et al (2019 Commun. Contemp. Math.
21 1850056) to fractional weights. We then investigate solutions numerically, confirming the well-posedness and extending it to a wider class of functions that includes exponential decay. We include a comparison of solutions to both types of equations, in particular, we investigate soliton resolution for the positive and negative data with different decay rates. Finally, we study the interaction of various solitary waves in both models, showing the formation of solitons, dispersive radiation and even breathers, all of which are easier to track in nonlinearities with lower power.
Funder
Division of Mathematical Sciences
Subject
Applied Mathematics,General Physics and Astronomy,Mathematical Physics,Statistical and Nonlinear Physics
Reference65 articles.
1. Nonlocal models for nonlinear, dispersive waves;Abdelouhab;Physica D,1989
2. Concentration compactness and the stability of solitary-wave solutions to nonlocal equations;Albert,1999
3. Well-posedness in weighted spaces for the generalized Hartree equation with p < 2;Arora;Commun. Contemp. Math.,2022
4. The stability of solitary waves;Benjamin;Proc. R. Soc. A,1972
5. Model equations for long waves in nonlinear dispersive systems;Benjamin;Phil. Trans. R. Soc. A,1972
Cited by
2 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献