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Integrability of a Generalized Short Pulse Equation Revisited

Integrability of a Generalized Short Pulse Equation Revisited We further generalize the generalized short pulse equation studied recently in [Commun. Nonlinear Sci. Numer. Simulat. 39 (2016) 21–28; arXiv:1510.08822], and find in this way two new integrable nonlinear wave equations which are transformable to linear Klein–Gordon equations. 1 Introduction In this paper, we study the integrability of the nonlinear wave equation 2 2 u = au u + buu (1) xt xx containing two arbitrary parameters, a and b, not equal zero simultaneously. Actually, there is only one essential parameter in (1), the ratio a/b or b/a, which is invariant under the scale transformations of u, x and t, while the values of a and b are not invariant. We show that this equation (1) is integrable in two (and, most probably, only two) distinct cases, namely, when a/b = 1/2 and a/b = 1, which correspond via scale transformations of variables to the equations u = u (2) xt xx and u = u u , (3) xt xx respectively. There is the following reason to study the nonlinear equation (1). Recently, in [1], we studied the integrability of the generalized short pulse equation 2 2 u = u + au u + buu (4) xt xx http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Research in Applied Mathematics Unpaywall

Integrability of a Generalized Short Pulse Equation Revisited

Research in Applied MathematicsJan 1, 2018
12 pages

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Publisher
Unpaywall
ISSN
2357-0482
DOI
10.11131/2018/101272
Publisher site
See Article on Publisher Site

Abstract

We further generalize the generalized short pulse equation studied recently in [Commun. Nonlinear Sci. Numer. Simulat. 39 (2016) 21–28; arXiv:1510.08822], and find in this way two new integrable nonlinear wave equations which are transformable to linear Klein–Gordon equations. 1 Introduction In this paper, we study the integrability of the nonlinear wave equation 2 2 u = au u + buu (1) xt xx containing two arbitrary parameters, a and b, not equal zero simultaneously. Actually, there is only one essential parameter in (1), the ratio a/b or b/a, which is invariant under the scale transformations of u, x and t, while the values of a and b are not invariant. We show that this equation (1) is integrable in two (and, most probably, only two) distinct cases, namely, when a/b = 1/2 and a/b = 1, which correspond via scale transformations of variables to the equations u = u (2) xt xx and u = u u , (3) xt xx respectively. There is the following reason to study the nonlinear equation (1). Recently, in [1], we studied the integrability of the generalized short pulse equation 2 2 u = u + au u + buu (4) xt xx

Journal

Research in Applied MathematicsUnpaywall

Published: Jan 1, 2018

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