American Institute of Mathematical Sciences

Advanced
July  2018, 17(4): 1407-1448. doi: 10.3934/cpaa.2018069

Asymptotics for the modified witham equation

Nakao Hayashi 1,, , Pavel I. Naumkin 2, and  Isahi Sánchez-Suárez 2,

1. 

Department of Mathematics, Graduate School of Science, Osaka University, Osaka, Toyonaka 560-0043, Japan
2. 

Centro de Ciencias Matemáticas, UNAM Campus Morelia, AP 61-3 (Xangari), Morelia CP 58089, Michoacán, Mexico

* Corresponding author

Received  August 2016 Revised  May 2017 Published  April 2018

We consider the modified Witham equation
${{\partial }_{t}}v+{{\partial }_{x}}\sqrt{{{a}^{2}}-\partial _{x}^{2}v}={{\partial }_{x}}\left( {{v}^{3}} \right),\ \ \left( t,x \right)\in \mathbb{R}\times \mathbb{R},$
where
$\sqrt{a^{2}-\partial _{x}^{2}}$
 means the dispersion relation which correspond to nonlinear Kelvin and continental-shelf waves. We develop the factorization technique to study the large time asymptotics of solutions.
Keywords: Asymptotics of solutions, modified Witham equation.
Mathematics Subject Classification: Primary: 35Q55; Secondary: 35B40.
Citation: Nakao Hayashi, Pavel I. Naumkin, Isahi Sánchez-Suárez. Asymptotics for the modified witham equation. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1407-1448. doi: 10.3934/cpaa.2018069
References:
[1]

M. V. Fedoryuk, Asymptotic Methods in Analysis, in Analysis. I. Integral representations and asymptotic methods, Encyclopaedia of Mathematical Sciences, 13. Springer-Verlag, Berlin, 1989. vi+238 pp. Google Scholar

[2]

N. Hayashi and E. Kaikina, Asymptotics for the third-order nonlinear Schrödinger equation in the critical case, to appear in MMAS. Google Scholar

[3]

N. Hayashi, J. Mendez-Navarro and P. Naumkin, Asymptotics for the fourth-order nonlinear Schrödinger equation in the critical case, submitted to JDE, 2016. Google Scholar

[4]

N. Hayashi and P. Naumkin, Factorization technique for the fourth-order nonlinear Schrödinger equation, Z. Angew. Math. Phys., 66 (2015), 2343-2377.   Google Scholar

[5]

N. Hayashi and P. I. Naumkin, Global existence and asymptotic behavior of solutions to the fourth-order nonlinear Schrödinger equation in the critical case, Nonlinear Analysis, 116 (2015), 112-131.   Google Scholar

[6]

N. Hayashi and P. I. Naumkin, Large time asymptotics of solutions for the modified KdV equation with a fifth order dispersive term, submitted to ARMA, 2015. Google Scholar

[7]

R. S. Smith, Nonlinear Kelvin and continental-shelf waves, J. Fluid Mech., 52 (1972), 379-391.   Google Scholar

[8]

E. M. Stein and R. Shakarchi, Functional Analysis. Introduction to Further Topics in Analysis, Princeton Lectures in Analysis, 4. Princeton University Press, Princeton, NJ, 2011. ⅹⅷ+423 pp. Google Scholar

[9]

G. B. Whitham, Variational methods and applications to water waves, Hyperbolic Equations and Waves (Rencontres, Battelle Res. Inst., Seattle, WA, 1968), Springer, Berlin, 1970, pp. 153-172. Google Scholar

[10]

G. B. Whitham, Linear and Nonlinear Waves, Pure Appl. Math., Wiley, New York, 1974. Google Scholar

show all references

References:
[1]

M. V. Fedoryuk, Asymptotic Methods in Analysis, in Analysis. I. Integral representations and asymptotic methods, Encyclopaedia of Mathematical Sciences, 13. Springer-Verlag, Berlin, 1989. vi+238 pp. Google Scholar

[2]

N. Hayashi and E. Kaikina, Asymptotics for the third-order nonlinear Schrödinger equation in the critical case, to appear in MMAS. Google Scholar

[3]

N. Hayashi, J. Mendez-Navarro and P. Naumkin, Asymptotics for the fourth-order nonlinear Schrödinger equation in the critical case, submitted to JDE, 2016. Google Scholar

[4]

N. Hayashi and P. Naumkin, Factorization technique for the fourth-order nonlinear Schrödinger equation, Z. Angew. Math. Phys., 66 (2015), 2343-2377.   Google Scholar

[5]

N. Hayashi and P. I. Naumkin, Global existence and asymptotic behavior of solutions to the fourth-order nonlinear Schrödinger equation in the critical case, Nonlinear Analysis, 116 (2015), 112-131.   Google Scholar

[6]

N. Hayashi and P. I. Naumkin, Large time asymptotics of solutions for the modified KdV equation with a fifth order dispersive term, submitted to ARMA, 2015. Google Scholar

[7]

R. S. Smith, Nonlinear Kelvin and continental-shelf waves, J. Fluid Mech., 52 (1972), 379-391.   Google Scholar

[8]

E. M. Stein and R. Shakarchi, Functional Analysis. Introduction to Further Topics in Analysis, Princeton Lectures in Analysis, 4. Princeton University Press, Princeton, NJ, 2011. ⅹⅷ+423 pp. Google Scholar

[9]

G. B. Whitham, Variational methods and applications to water waves, Hyperbolic Equations and Waves (Rencontres, Battelle Res. Inst., Seattle, WA, 1968), Springer, Berlin, 1970, pp. 153-172. Google Scholar

[10]

G. B. Whitham, Linear and Nonlinear Waves, Pure Appl. Math., Wiley, New York, 1974. Google Scholar

[1]

Yanghong Huang, Andrea Bertozzi. Asymptotics of blowup solutions for the aggregation equation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1309-1331. doi: 10.3934/dcdsb.2012.17.1309
[2]

Josef DiblÍk, Rigoberto Medina. Exact asymptotics of positive solutions to Dickman equation. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 101-121. doi: 10.3934/dcdsb.2018007
[3]

Qian Li. Generalized Wronskian solutions of modified Boussinesq equation. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020272
[4]

Joel Kübler, Tobias Weth. Spectral asymptotics of radial solutions and nonradial bifurcation for the Hénon equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3629-3656. doi: 10.3934/dcds.2020032
[5]

Stephen Anco, Daniel Kraus. Hamiltonian structure of peakons as weak solutions for the modified Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4449-4465. doi: 10.3934/dcds.2018194
[6]

Jerry L. Bona, Stéphane Vento, Fred B. Weissler. Singularity formation and blowup of complex-valued solutions of the modified KdV equation. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 4811-4840. doi: 10.3934/dcds.2013.33.4811
[7]

Yixia Shi, Maoan Han. Existence of generalized homoclinic solutions for a modified Swift-Hohenberg equation. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020114
[8]

Katrin Grunert, Helge Holden, Xavier Raynaud. Global conservative solutions to the Camassa--Holm equation for initial data with nonvanishing asymptotics. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4209-4227. doi: 10.3934/dcds.2012.32.4209
[9]

Gerald Teschl. On the spatial asymptotics of solutions of the Toda lattice. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1233-1239. doi: 10.3934/dcds.2010.27.1233
[10]

Min Zhu, Shuanghu Zhang. Blow-up of solutions to the periodic modified Camassa-Holm equation with varying linear dispersion. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7235-7256. doi: 10.3934/dcds.2016115
[11]

Min Zhu, Ying Wang. Blow-up of solutions to the periodic generalized modified Camassa-Holm equation with varying linear dispersion. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 645-661. doi: 10.3934/dcds.2017027
[12]

Min Zhu, Shuanghu Zhang. On the blow-up of solutions to the periodic modified integrable Camassa--Holm equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2347-2364. doi: 10.3934/dcds.2016.36.2347
[13]

Wen Tan, Bo-Qing Dong, Zhi-Min Chen. Large-time regular solutions to the modified quasi-geostrophic equation in Besov spaces. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3749-3765. doi: 10.3934/dcds.2019152
[14]

Marek Fila, Hannes Stuke. Special asymptotics for a critical fast diffusion equation. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 725-735. doi: 10.3934/dcdss.2014.7.725
[15]

Darya V. Verveyko, Andrey Yu. Verisokin. Application of He's method to the modified Rayleigh equation. Conference Publications, 2011, 2011 (Special) : 1423-1431. doi: 10.3934/proc.2011.2011.1423
[16]

Yongsheng Mi, Chunlai Mu, Pan Zheng. On the Cauchy problem of the modified Hunter-Saxton equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2047-2072. doi: 10.3934/dcdss.2016084
[17]

Yu Gao, Jian-Guo Liu. The modified Camassa-Holm equation in Lagrangian coordinates. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2545-2592. doi: 10.3934/dcdsb.2018067
[18]

Jerry L. Bona, Laihan Luo. Large-time asymptotics of the generalized Benjamin-Ono-Burgers equation. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 15-50. doi: 10.3934/dcdss.2011.4.15
[19]

Dimitra Antonopoulou, Georgia Karali, Georgios T. Kossioris. Asymptotics for a generalized Cahn-Hilliard equation with forcing terms. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1037-1054. doi: 10.3934/dcds.2011.30.1037
[20]

Naoufel Ben Abdallah, Hédia Chaker. Mixed high field and diffusion asymptotics for the fermionic Boltzmann equation. Kinetic & Related Models, 2009, 2 (3) : 403-424. doi: 10.3934/krm.2009.2.403

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (66)
  • HTML views (146)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences