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

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.

35Q02, 35E02, 76B02, 65B02

jbona@uic.edu (Jerry L. Bona)

chen45@purdue.edu (Min Chen)

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@Article{JMS-49-205, author = {Jerry L. and Bona and jbona@uic.edu and 7093 and Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago IL 60607, USA and Jerry L. Bona and Min and Chen and chen45@purdue.edu and 7094 and Department of Mathematics, Purdue University, West Lafayette IN 47907, USA and Min Chen}, 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.

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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