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doi: 10.3934/dcdsb.2022024
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On the solutions for a Benney-Lin type equation

1. 

Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari, via E. Orabona 4, 70125 Bari, Italy

2. 

Dipartimento di Matematica, Università di Bari, via E. Orabona 4, 70125 Bari, Italy

*Corresponding author: Giuseppe Maria Coclite

Received  May 2021 Revised  December 2021 Early access February 2022

Fund Project: GMC is member of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). GMC has been partially supported by the Research Project of National Relevance "Multiscale Innovative Materials and Structures" granted by the Italian Ministry of Education, University and Research (MIUR Prin 2017, project code 2017J4EAYB and the Italian Ministry of Education, University and Research under the Programme Department of Excellence Legge 232/2016 (Grant No. CUP - D94I18000260001)

The Benney-Lin equation describes the evolution of long waves in various problems in fluid dynamics. In this paper, we prove the well-posedness of the Cauchy problem, associated with this equation.

Citation: Giuseppe Maria Coclite, Lorenzo di Ruvo. On the solutions for a Benney-Lin type equation. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022024
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show all references

References:
[1]

R. Adams and S. C. Mancas, Stability of solitary and cnoidal traveling wave solutions for a fifth order Korteweg–de Vries equation, Appl. Math. Comput., 321 (2018), 745-751.  doi: 10.1016/j.amc.2017.11.005.

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E. M. Al-Ali, Traveling wave solutions for a generalized Kawahara and Hunter-Saxton equations, Int. J. Math. Anal. (Ruse), 7 (2013), 1647-1666.  doi: 10.12988/ijma.2013.3483.

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A. Armaou and P. D. Christofides, Feedback control of the Kuramoto-Sivashinsky equation, Phys. D, 137 (2000), 49-61.  doi: 10.1016/S0167-2789(99)00175-X.

[4]

A. Başhan, An efficient approximation to numerical solutions for the Kawahara equation via modified cubic B-spline differential quadrature method, Mediterr. J. Math., 16 (2019), Paper No. 14, 19. doi: 10.1007/s00009-018-1291-9.

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D. J. Benney, Long waves on liquid films, J. Math. and Phys., 45 (1966), 150-155.  doi: 10.1002/sapm1966451150.

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N. G. Berloff and L. N. Howard, Solitary and periodic solutions of nonlinear nonintegrable equations, Stud. Appl. Math., 99 (1997), 1-24.  doi: 10.1111/1467-9590.00054.

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H. A. BiagioniJ. L. BonaR. J. Iório Jr. and M. Scialom, On the Korteweg-de Vries-Kuramoto-Sivashinsky equation, Adv. Differential Equations, 1 (1996), 1-20. 

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H. A. Biagioni and F. Linares, On the Benney-Lin and Kawahara equations, J. Math. Anal. Appl., 211 (1997), 131-152.  doi: 10.1006/jmaa.1997.5438.

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A. Biswas, Solitary wave solution for the generalized Kawahara equation, Appl. Math. Lett., 22 (2009), 208-210.  doi: 10.1016/j.aml.2008.03.011.

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R. d. A. Capistrano-Filho and M. M. d. S. Gomes, Well-posedness and controllability of Kawahara equation in weighted Sobolev spaces, Nonlinear Anal., 207 (2021), Paper No. 112267, 24 pp. doi: 10.1016/j.na.2021.112267.

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M. Cavalcante and C. Kwak, Local well-posedness of the fifth-order KdV-type equations on the half-line, Commun. Pure Appl. Anal., 18 (2019), 2607-2661.  doi: 10.3934/cpaa.2019117.

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