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

Jerry L. Bona & Min Chen

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 wellposedness 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 KW - global wellposedness KW - singular solutions KW - Fourier spectral method KW - 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.

Jerry L. Bona & Min Chen. (2020). 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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