Remarks on Existence/Nonexistence of Analytic Solutions to Higher Order KdV Equations |
| A. Karczewskaa, P. Rozmejb
aFaculty of Mathematics, Computer Science and Econometrics University of Zielona Góra, Z. Szafrana 4a, 65-246 Zielona Góra, Poland bInstitute of Physics, Faculty of Physics and Astronomy University of Zielona Góra, Z. Szafrana 4a, 65-246 Zielona Góra, Poland |
| Received: June 5, 2019; in final form October 15, 2019 |
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| In this note, we discuss the existence of analytic solutions to the nonlinear wave equations of the higher order than the ubiquitous Korteweg-de Vries equation. First, we recall our recent results which show that the extended Korteweg-de Vries equation, that is, the equation obtained within second-order perturbation approach possesses three kinds of analytic solutions. These solutions have the same functional form as the corresponding Korteweg-de Vries solutions. We show, however, that the most intriguing multi-soliton solutions, known for the Korteweg-de Vries equation, do not exist for extended Korteweg-de Vries equation. Moreover, we show that for the equations obtained in the third order perturbation approach (and then in any higher order) analytic solutions in the forms known from Korteweg-de Vries theory do not exist. |
DOI:10.12693/APhysPolA.136.910 topics: shallow water waves, extended KdV equations, analytic solutions |