[
{
"id": "authors:k6fwn-27z53",
"collection": "authors",
"collection_id": "k6fwn-27z53",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:FOKjmp80",
"type": "article",
"title": "A symmetry approach to exactly solvable evolution equations",
"author": [
{
"family_name": "Fokas",
"given_name": "A. S.",
"clpid": "Fokas-A-S"
}
],
"abstract": "A method is developed for establishing the exact solvability of nonlinear evolution equations in one space dimension which are linear with constant coefficient in the highest-order derivative. The method, based on the symmetry structure of the equations, is applied to second-order equations and then to third-order equations which do not contain a second-order derivative. In those cases the most general exactly solvable nonlinear equations turn out to be the Burgers equation and a new third-order evolution equation which contains the Korteweg-de Vries (KdV) equation and the modified KdV equation as particular cases.",
"doi": "10.1063/1.524581",
"issn": "0022-2488",
"publisher": "American Institute of Physics",
"publication": "Journal of Mathematical Physics",
"publication_date": "1980-06",
"series_number": "6",
"volume": "21",
"issue": "6",
"pages": "1318-1325"
},
{
"id": "authors:craz1-78y47",
"collection": "authors",
"collection_id": "craz1-78y47",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:COHsiamjam78c",
"type": "article",
"title": "Proof of some asymptotic results for a model equation for low Reynolds number flow",
"author": [
{
"family_name": "Cohen",
"given_name": "D. S.",
"clpid": "Cohen-D-S"
},
{
"family_name": "Fokas",
"given_name": "A.",
"clpid": "Fokas-A-S"
},
{
"family_name": "Lagerstrom",
"given_name": "P. A.",
"clpid": "Lagerstrom-P-A"
}
],
"abstract": "A two-point boundary value problem in the interval [\u03b5, \u221e], \u03b5 > 0 is studied. The problem contains additional parameters \u03b1 \u2265 0, \u03b2 \u2265 0, 0 \u2264 U < \u221e, k real. It was originally proposed by Lagerstrom as a model for viscous flow at low Reynolds numbers. A related initial value problem is transformed into an integral equation which is shown to have a unique solution by a pincer method. The integral representation is used for a simple proof of the existence of a solution of the boundary value problem for a \u03b1 > 0; for \u03b1 = 0 an explicit construction shows that no solution exists unless k > 1. A special method is used to show uniqueness. For \u03b5 \u2193 0, k \u2265 1, various results had previously been obtained by the method of matched asymptotic expansions. Examples of these results are verified rigorously using the integral representation. For k < 1, the problem is shown not to be a layer-type problem, a fact previously demonstrated explicitly for k = 0. If k is an integer \u2265 0 the intuitive understanding of the problem is aided by regarding it as spherically symmetric in k + 1 dimensions. In the present study, however, k may be any real number, even negative.",
"doi": "10.1137/0135015",
"issn": "0036-1399",
"publisher": "SIAM",
"publication": "SIAM Journal on Applied Mathematics",
"publication_date": "1978-07",
"series_number": "1",
"volume": "35",
"issue": "1",
"pages": "187-207"
}
]