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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.
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This content will become publicly available on April 28, 2026
On the well-posedness of the periodic fractional Schrödinger equation
We consider the periodic fractional nonlinear Schroedinger equation, where the nonlinearity is expressed in two ways either as a derivative of a polynomial (including negative powers, such including logarithmic nonlinearities), or given as a sum of powers, possibly infinite. We obtain the local well-posedness for the Cauchy problem of this equation in weighted Sobolev spaces and with non-vanishing initial data.
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- PAR ID:
- 10598501
- Publisher / Repository:
- Springer
- Date Published:
- Journal Name:
- Partial Differential Equations and Applications
- Volume:
- 6
- Issue:
- 3
- ISSN:
- 2662-2963
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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