August  2022, 21(8): 2615-2641. doi: 10.3934/cpaa.2022063

Well-posedness for a coupled system of Kawahara/KdV type equations with polynomials nonlinearities

Departamento de Matemática, Universidade Federal de São Carlos, São Carlos, SP 13565-905, Brazil

*Corresponding author

Received  September 2021 Published  August 2022 Early access  March 2022

Fund Project: The first author is supported by CAPES - finance code 001

We consider the initial value problem (IVP) associated to a coupled system of modified Kawahara/KdV type equations with polynomials nonlinearities. For the model in question, the Cauchy problem is of interest, and is shown to be well-posed for given data in a Gevrey spaces. Our results make use of techniques presented in Grujić and Kalisch, who studied the Gevrey regularity for a class of water-wave models and the well-posedness of a IVP associated to a general equation. The proof relies on estimates in space-time norms adapted to the linear part of the equations. In particular, estimates in Bourgain spaces are proven for the linear and nonlinear terms of the system and the main result is obtained by a contraction principle. The class of system in view generalizes the system of modified Kawahara/KdV type equations studied by Kondo and Pes, which contains a number of systems arising in the modeling of waves in fluids, stability and instability of solitary waves and models for wave propagation in physical systems where both nonlinear and dispersive effects are important.

Citation: Cezar Kondo, Ronaldo Pes. Well-posedness for a coupled system of Kawahara/KdV type equations with polynomials nonlinearities. Communications on Pure and Applied Analysis, 2022, 21 (8) : 2615-2641. doi: 10.3934/cpaa.2022063
References:
[1]

E. AlarconJ. Angulo and J. F. Montenegro, Stability and instability of solitary waves for a nonlinear dispersive system, Nonlinear Anal., 36 (1999), 1015-1035.  doi: 10.1016/S0362-546X(97)00724-4.

[2]

A. Benedek and R. Panzone, The spaces $L^p$, with mixed norm, Duke Math. J., 28 (1961), 301-324. 

[3]

J. L. BonaJ. Cohen and G. Wang, Global well-posedness for a system of KDV-type Equations with coupled quadratic nonlinearities, Nagoya Math. J., 215 (2014), 67-149.  doi: 10.1215/00277630-2691901.

[4]

J. L. BonaG. PonceJ. Saut and M. M. Tom, A model system for strong interaction between internal solitary waves, Commun. Math. phys., 143 (1992), 287-313. 

[5]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations I. Schröndiger equations, Geometr. Funct. Anal., 3 (1993), 107-156.  doi: 10.1007/BF01896020.

[6]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations II. The KdV-equation, Geometr. Funct. Anal., 3 (1993), 209-262.  doi: 10.1007/BF01895688.

[7]

X. Carvajal and M. Panthee, Sharp well-posedness for a coupled system of mKdV type equations, J. Evol. Equ., 19 (2019), 1167-1197.  doi: 10.1007/s00028-019-00508-6.

[8]

A. Corcho and M. Panthee, Global well-posedness for a coupled modified KdV system, Bullet. Braz. Math. Soc., 43 (2012), 27-57.  doi: 10.1007/s00574-012-0004-4.

[9]

L. C. Evans, Partial Differential Equations, 2$^{nd}$ edition, American Mathematical Society, 1998. doi: 978-0821807729.

[10]

J. Gear and R. Grimshaw, Weak and strong interactions between internal solitary waves equation, Stud. Appl. Math., 70 (1984), 235-258.  doi: 10.1002/sapm1984703235.

[11]

Z. Grujić and H. Kalisch, Gevrey regularity for a class of water-wave models, Nonlinear Anal., 71 (2009), 1160-1170.  doi: 10.1016/j.na.2008.11.047.

[12]

J. Ginibre, Y. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system, J. Funct. Anal., $2^{nd}$, 151 (1997), 384–436. doi: 10.1006/jfan.1997.3148.

[13]

T. Kawahara, Oscillatory solitary waves in dispersive media, J. Phys. Soc. Japan, 33 (1972), 260-264. 

[14]

C. E. KenigG. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univer. Math. J., 40 (1991), 33-69.  doi: 10.1512/iumj.1991.40.40003.

[15]

C. E. KenigG. Ponce and L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J., 71 (1993), 1-21.  doi: 10.1215/S0012-7094-93-07101-3.

[16]

C. I. Kondo and R. B. Pes, Well posedness for a coupled system of Kawahara-KdV type equations, Appl. Mathe. Optim., 84 (2021), 2985-3024.  doi: 10.1007/s00245-020-09737-5.

[17]

D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Phil. Magaz., 39 (1895), 422-443. 

[18]

F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, $2^{nd}$ Ed., Springer-Verlag, NewYork, 2015. doi: 10.1007/978-1-4939-2181-2.

[19]

T. Oh, Diophantine conditions in global well-posedness for coupled KdV-type systems, Electron. J. Differe. Equ., 52 (2009), 3516-3556.  doi: 10.1093/imrn/rnp063.

show all references

References:
[1]

E. AlarconJ. Angulo and J. F. Montenegro, Stability and instability of solitary waves for a nonlinear dispersive system, Nonlinear Anal., 36 (1999), 1015-1035.  doi: 10.1016/S0362-546X(97)00724-4.

[2]

A. Benedek and R. Panzone, The spaces $L^p$, with mixed norm, Duke Math. J., 28 (1961), 301-324. 

[3]

J. L. BonaJ. Cohen and G. Wang, Global well-posedness for a system of KDV-type Equations with coupled quadratic nonlinearities, Nagoya Math. J., 215 (2014), 67-149.  doi: 10.1215/00277630-2691901.

[4]

J. L. BonaG. PonceJ. Saut and M. M. Tom, A model system for strong interaction between internal solitary waves, Commun. Math. phys., 143 (1992), 287-313. 

[5]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations I. Schröndiger equations, Geometr. Funct. Anal., 3 (1993), 107-156.  doi: 10.1007/BF01896020.

[6]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations II. The KdV-equation, Geometr. Funct. Anal., 3 (1993), 209-262.  doi: 10.1007/BF01895688.

[7]

X. Carvajal and M. Panthee, Sharp well-posedness for a coupled system of mKdV type equations, J. Evol. Equ., 19 (2019), 1167-1197.  doi: 10.1007/s00028-019-00508-6.

[8]

A. Corcho and M. Panthee, Global well-posedness for a coupled modified KdV system, Bullet. Braz. Math. Soc., 43 (2012), 27-57.  doi: 10.1007/s00574-012-0004-4.

[9]

L. C. Evans, Partial Differential Equations, 2$^{nd}$ edition, American Mathematical Society, 1998. doi: 978-0821807729.

[10]

J. Gear and R. Grimshaw, Weak and strong interactions between internal solitary waves equation, Stud. Appl. Math., 70 (1984), 235-258.  doi: 10.1002/sapm1984703235.

[11]

Z. Grujić and H. Kalisch, Gevrey regularity for a class of water-wave models, Nonlinear Anal., 71 (2009), 1160-1170.  doi: 10.1016/j.na.2008.11.047.

[12]

J. Ginibre, Y. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system, J. Funct. Anal., $2^{nd}$, 151 (1997), 384–436. doi: 10.1006/jfan.1997.3148.

[13]

T. Kawahara, Oscillatory solitary waves in dispersive media, J. Phys. Soc. Japan, 33 (1972), 260-264. 

[14]

C. E. KenigG. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univer. Math. J., 40 (1991), 33-69.  doi: 10.1512/iumj.1991.40.40003.

[15]

C. E. KenigG. Ponce and L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J., 71 (1993), 1-21.  doi: 10.1215/S0012-7094-93-07101-3.

[16]

C. I. Kondo and R. B. Pes, Well posedness for a coupled system of Kawahara-KdV type equations, Appl. Mathe. Optim., 84 (2021), 2985-3024.  doi: 10.1007/s00245-020-09737-5.

[17]

D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Phil. Magaz., 39 (1895), 422-443. 

[18]

F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, $2^{nd}$ Ed., Springer-Verlag, NewYork, 2015. doi: 10.1007/978-1-4939-2181-2.

[19]

T. Oh, Diophantine conditions in global well-posedness for coupled KdV-type systems, Electron. J. Differe. Equ., 52 (2009), 3516-3556.  doi: 10.1093/imrn/rnp063.

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