Funding: This work is supported by NSFC (Grant No. 11901067), the Fundamental Research Funds for the Central Universities (Grant No. 2024CDJXY018, 2022CDJJCLK002), and Key Laboratory of Nonlinear Analysis and its Applications (Chongqing University), Ministry of Education, the National Natural Science Foundation of Chongqing (Grant No. CSTB2024NSCQ-MSX1059 and CSTB2024NSCQ-MSX1227), Chongqing Municipal Education Commission Science and Technology Research Program (Grant No. KJQN202401529 and KJQN202401528), Chongqing University of Science and Technology Talent Induction Project (Grant No. ckrc20240624).

Read the full text

About

PDF

Tools

Share a link

Email
Wechat
Bluesky

ABSTRACT

We consider the global well-posedness in modulation spaces and scaling limit modulation spaces under smallness assumption on the initial data for the coupled generalized KdV-KdV (gKdV) system. Moreover, we prove a ${H}&#x0005E;1$ scattering criterion for the system.

Open Research

Data Availability Statement

Data sharing not applicable to this article as no data sets were generated or analysed during the current study.

References

1M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, “Nonlinear-Evolution Equations of Physical Significance,” Physical Review Letters 31, no. 2 (1973): 125–127.
10.1103/PhysRevLett.31.125
Web of Science® Google Scholar
2J. Gear and R. Grimshaw, “Weak and Strong Interactions Between Internal Solitary Waves,” Studies in Applied Mathematics 70 (1984): 235–258.
10.1002/sapm1984703235
Web of Science® Google Scholar
3C. E. Kenig, G. Ponce, and L. Vega, “Oscillatory Integrals and Regularity of Dispersive Equations,” Indiana University Mathematics Journal 40 (1991): 33–69.
10.1512/iumj.1991.40.40003
Web of Science® Google Scholar
4L. Molinet and F. Ribaud, “On the Cauchy Problem for the Generalized Korteweg-de Vries Equation,” Communications in Partial Differential Equations 28 (2003): 2065–2091.
10.1081/PDE-120025496
Web of Science® Google Scholar
5H. Koch and J. L. Marzuola, “Small Data Scattering and Soliton Stability in ${\dot{H}}&#x0005E;{-\frac{1}{6}}$ for the Quartic KdV Equation,” Analysis PDE 5 (2012): 145–198.
10.2140/apde.2012.5.145
Google Scholar
6C. E. Kenig, G. Ponce, and L. Vega, “Well-Posedness and Scattering Results for the Generalized Korteweg-de Vries Equation via the Contraction Principle,” Communications on Pure and Applied Mathematics 46 (1993): 527–620.
10.1002/cpa.3160460405
Web of Science® Google Scholar
7G. Ponce and L. Vega, “Nonlinear Small Data Scattering for the Generalized Korteweg-de Vries Equation,” Journal of Functional Analysis 90 (1990): 445–457.
10.1016/0022-1236(90)90092-Y
Web of Science® Google Scholar
8L. G. Farah, F. Linares, A. Pastor, and N. Visciglia, “Large Data Scattering for the Defocusing Supercritical Generalized KdV Equation,” Communications in Partial Differential Equations 43 (2018): 118–157.
10.1080/03605302.2018.1439063
Google Scholar
9E. Alarcon, J. Angulo, and J. F. Montenegro, “Stability and Instability of Solitary Waves for a Nonlinear Dispersive System,” Nonlinear Analysis: Theory, Methods & Applications 36, no. 8 (1999): 1015–1035.
10.1016/S0362-546X(97)00724-4
Google Scholar
10Montenegro, J. F., 1995. “ Sistemas de equações não-lineares; Estudo local, global e estabilidade de ondas solitarias,” Ph. D. Thesis, IMPA, Rio de Janeiro.
Google Scholar
11M. Panthee and M. Scialom, “On the Cauchy Problem for a Coupled System of KdV Equations: Critical Case,” Advances in Differential Equations 13 (2008): 1–26.
10.57262/ade/1355867358
Google Scholar
12Feichtinger H. G., 1983. “ Modulation Spaces on Locally Compact Abelian Group,” Technical report, University Vienna.
Google Scholar
13A. Bény, K. Gröchenig, and K. A. Okoudjou, “Unimodular Fourier Multipliers for Modulation Spaces,” Journal of Functional Analysis 246 (2007): 366–384.
10.1016/j.jfa.2006.12.019
Google Scholar
14B. Wang and H. Hudzik, “The Global Cauchy Problem for the NLS and NLKG With Small Rough Data,” Journal of Differential Equations 232 (2007): 36–73.
10.1016/j.jde.2006.09.004
Web of Science® Google Scholar
15B. Wang and C. Huang, “Frequency-Uniform Decomposition Method for the Generalized BO, KdV and NLS Equations,” Journal of Differential Equations 239 (2007): 213–250.
10.1016/j.jde.2007.04.009
Web of Science® Google Scholar
16T. Kato, “Well-Posedness for the Generalized Zakharov-Kuznetsov Equation on Modulation Spaces,” Journal of Fourier Analysis and Applications 23 (2017): 612–655.
10.1007/s00041-016-9480-z
Web of Science® Google Scholar
17T. Oh and Y. Wang, “On Global Well-Posedness of the Modified KdV Equation in Modulation Spaces,” Discrete and Continuous Dynamical Systems 41 (2021): 2971–2992.
10.3934/dcds.2020393
Web of Science® Google Scholar
18M. Sugimoto and N. Tomita, “The Dilation Property of Modulation Spaces and Their Inclusion Relation With Besov Spaces,” Journal of Functional Analysis 248 (2007): 79–106.
10.1016/j.jfa.2007.03.015
Web of Science® Google Scholar
19Á. Bényi and T. Oh, “Modulation Spaces With Scaling Symmetry,” Applied and Computational Harmonic Analysis 48 (2020): 496–507.
10.1016/j.acha.2019.04.005
Web of Science® Google Scholar
20M. Sugimoto and B. Wang, “Scaling Limit of Modulation Spaces and Their Applications,” Applied and Computational Harmonic Analysis 53 (2021): 54–94.
10.1016/j.acha.2021.01.002
Web of Science® Google Scholar
21Y. Lu, “Well-Posedness of Generalized KdV and One-Dimensional Fourth-Order Derivative Nonlinear Schrödinger Equations for Data With an Infinite ${L}&#x0005E;2$ norm,” Studies in Applied Mathematics 150 (2023): 883–898.
10.1111/sapm.12559
Google Scholar
22L. G. Farah and B. Pigott, “Nonlinear Profile Decomposition and the Concentration Phenomenon for Supercritical Generalized KdV Equations,” Indiana University Mathematics Journal 67 (2018): 1857–1892.
10.1512/iumj.2018.67.7471
Google Scholar
23A. Bényi and K. A. Okoudjou, Modulation Spaces-With Applications to Pseudodifferential Operators and Nonlinear Schrödinger Equations (Appl. Numer. Harmon. Anal. Birkhäuser/Springer, 2020), xvi+169.
Google Scholar
24J. Chen, Y. Lu, and B. Wang, “The Embedding Property of the Scaling Limit of Modulation Spaces,” Journal of Mathematical Analysis and Applications 512 (2022): 126188.
10.1016/j.jmaa.2022.126188
Google Scholar
25K. Gröchenig, Foundations of Time-Frequency Analysis (Appl. Numer. Harmon. Anal. Birkhäuser Boston, Inc., 2001), xvi+359.
10.1007/978-1-4612-0003-1
Google Scholar

Volume48, Issue12

August 2025

Pages 11923-11936

Global Well-Posedness for a Coupled Generalized KdV-KdV System in Modulation Spaces and a Scattering Criterion in a Sobolev Space

ABSTRACT

Open Research

Data Availability Statement

References

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley

Global Well-Posedness for a Coupled Generalized KdV-KdV System in Modulation Spaces and a Scattering Criterion in a Sobolev Space

ABSTRACT

Open Research

Data Availability Statement

References

References

Related

Information