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Volume 49, Issue 3
Singular Solutions of a Boussinesq System for Water Waves

Jerry L. Bona & Min Chen

DOI: 10.4208/jms.v49n3.16.01

J. Math. Study, 49 (2016), pp. 205-220.

Published online: 2016-09

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  • Abstract

Studied here is the Boussinesq system $$η_t+u_x+(ηu)_x+au_{xxx}-bη_{xxt}=0,$$ $$u_t+η_x+\frac{1}{2}(u²)_x+cη_{xxx}-du_{xxt}=0,$$of partial differential equations. This system has been used in theory and practice as a model for small-amplitude, long-crested water waves. The issue addressed is whether or not the initial-value problem for this system of equations is globally well posed.
The investigation proceeds by way of numerical simulations using a computer code based on a a semi-implicit, pseudo-spectral code. It turns out that larger amplitudes or velocities do seem to lead to singularity formation in finite time, indicating that the problem is not globally well posed.

  • Keywords

Boussinesq systems, global well-posedness, singular solutions, Fourier spectral method, nonlinear water wave.

  • AMS Subject Headings

35Q02, 35E02, 76B02, 65B02

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

jbona@uic.edu (Jerry L. Bona)

chen45@purdue.edu (Min Chen)

  • BibTex
  • RIS
  • TXT
@Article{JMS-49-205, author = {Bona , Jerry L. and Chen , Min}, title = {Singular Solutions of a Boussinesq System for Water Waves}, journal = {Journal of Mathematical Study}, year = {2016}, volume = {49}, number = {3}, pages = {205--220}, abstract = {

Studied here is the Boussinesq system $$η_t+u_x+(ηu)_x+au_{xxx}-bη_{xxt}=0,$$ $$u_t+η_x+\frac{1}{2}(u²)_x+cη_{xxx}-du_{xxt}=0,$$of partial differential equations. This system has been used in theory and practice as a model for small-amplitude, long-crested water waves. The issue addressed is whether or not the initial-value problem for this system of equations is globally well posed.
The investigation proceeds by way of numerical simulations using a computer code based on a a semi-implicit, pseudo-spectral code. It turns out that larger amplitudes or velocities do seem to lead to singularity formation in finite time, indicating that the problem is not globally well posed.

}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v49n3.16.01}, url = {http://global-sci.org/intro/article_detail/jms/999.html} }
TY - JOUR T1 - Singular Solutions of a Boussinesq System for Water Waves AU - Bona , Jerry L. AU - Chen , Min JO - Journal of Mathematical Study VL - 3 SP - 205 EP - 220 PY - 2016 DA - 2016/09 SN - 49 DO - http://doi.org/10.4208/jms.v49n3.16.01 UR - https://global-sci.org/intro/article_detail/jms/999.html KW - Boussinesq systems, global well-posedness, singular solutions, Fourier spectral method, nonlinear water wave. AB -

Studied here is the Boussinesq system $$η_t+u_x+(ηu)_x+au_{xxx}-bη_{xxt}=0,$$ $$u_t+η_x+\frac{1}{2}(u²)_x+cη_{xxx}-du_{xxt}=0,$$of partial differential equations. This system has been used in theory and practice as a model for small-amplitude, long-crested water waves. The issue addressed is whether or not the initial-value problem for this system of equations is globally well posed.
The investigation proceeds by way of numerical simulations using a computer code based on a a semi-implicit, pseudo-spectral code. It turns out that larger amplitudes or velocities do seem to lead to singularity formation in finite time, indicating that the problem is not globally well posed.

Bona , Jerry L. and Chen , Min. (2016). Singular Solutions of a Boussinesq System for Water Waves. Journal of Mathematical Study. 49 (3). 205-220. doi:10.4208/jms.v49n3.16.01
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