Exact Cnoidal Solutions of the Extended KdV Equation

E. Infelda, A. Karczewskab, G. Rowlandsc, P. Rozmejd aNational Centre for Nuclear Research, Hoża 69, 00-681 Warszawa, Poland bFaculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Z. Szafrana 4a, 65-246 Zielona Góra, Poland cDepartment of Physics, University of Warwick, Coventry, CV4 7AL, UK dInstitute of Physics, Faculty of Physics and Astronomy, University of Zielona Góra, Z. Szafrana 4a, 65-246 Zielona Góra, Poland

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The KdV equation can be derived within the shallow water limit of the Euler equations. Over the last few decades, this equation has been extended to include both higher order effects (KdV2) and an uneven river bottom. Although this equation is not integrable and has only one conservation law, exact periodic and solitonic solutions exist for the even bottom case. The method used to find them assumes the same functional forms as for KdV solutions. The KdV2 equation imposes more constraints on the parameters of solutions. Quite unexpectedly, we found two regions in m parameter space for periodic solutions. For the range of m close to one the cnoidal waves are upright as expected, but are inverted in the m region close to zero which is a completely new feature. The properties of exact solutions for KdV and KdV2 are compared. Numerical evolution of all the discussed exact solutions to KdV2 is stable and confirms the properties of the analytic solutions.

DOI:10.12693/APhysPolA.133.1191 topics: shallow water waves, extended KdV equation, analytic solutions, inverted cnoidal waves