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PreprintCC BY 4.0HAL

A nonhomogeneous boundary-value problem for the nonlinear KdV equation on star graphs

  • Roberto de Almeida Capistrano Filho
  • Hugo Parada
  • Jandeilson Santos da Silva
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Abstract

This paper investigates a boundary-value problem for the Korteweg-de Vries (KdV) equation on a finite equilateral star-graph. We use the notion of s-compatibility, which extends classical compatibility conditions to the coupled boundary conditions arising at the central vertex, inspired by the work of Bona, Sun, and Zhang [14]. By combining analytical techniques, fixed-point argument, and nonlinear interpolation on suitable augmented spaces, we establish global well-posedness for the nonlinear problem in Hs 0≤s≤3. At the critical indices s = 1/2,3/2,5/2, the admissible data are defined through constrained interpolation; away from these indices, they are characterized by the corresponding pointwise compatibility conditions. The nonlinear compatibility condition at the central vertex is handled by introducing an auxiliary variable, which allows the interpolation argument to be carried out on a pair of linear spaces.

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Cite

BibTeXRIS
BibTeX
@misc{filho2026nonhomogeneous,
  title = {{A nonhomogeneous boundary-value problem for the nonlinear KdV equation on star graphs}},
  author = {Roberto de Almeida Capistrano Filho and Hugo Parada and Jandeilson Santos da Silva},
  howpublished = {HAL},
  year = {2026},
  month = sep,
  url = {https://hal.science/hal-05421349v2},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A nonhomogeneous boundary-value problem for the nonlinear KdV equation on star graphs
AU  - Roberto de Almeida Capistrano Filho
AU  - Hugo Parada
AU  - Jandeilson Santos da Silva
PY  - 2026
DA  - 2026/09/24
PB  - HAL
UR  - https://hal.science/hal-05421349v2
AB  - This paper investigates a boundary-value problem for the Korteweg-de Vries (KdV) equation on a finite equilateral star-graph. We use the notion of s-compatibility, which extends classical compatibility conditions to the coupled boundary conditions arising at the central vertex, inspired by the work of Bona, Sun, and Zhang [14]. By combining analytical techniques, fixed-point argument, and nonlinear interpolation on suitable augmented spaces, we establish global well-posedness for the nonlinear problem in Hs 0≤s≤3. At the critical indices s = 1/2,3/2,5/2, the admissible data are defined through constrained interpolation; away from these indices, they are characterized by the corresponding pointwise compatibility conditions. The nonlinear compatibility condition at the central vertex is handled by introducing an auxiliary variable, which allows the interpolation argument to be carried out on a pair of linear spaces.
ER  -