Abstract
In this paper, theory of dynamical systems is employed to investigate periodic waves of a singular integrable equation. These periodic waves contain smooth periodic waves, periodic cusp waves and periodic cusp loop waves. Under fixed parameter conditions, their exact parametric expressions are given.
Highlights
Introduction and Main ResultsQiao [4] presented a completely integrable water wave equation: ut − uxxt + 3u2ux − u3x = (4u − 2uxx)uxuxx + (u2 − u2x)uxxx, (1.1)where u is the fluid velocity and subscripts denote the partial derivatives
Where u is the fluid velocity and subscripts denote the partial derivatives. This equation can be derived from the two-dimensional Euler equation by using the approximation procedure
It is well known that the closed orbit of the travelling system gives a periodic wave solution of the corresponding nonlinear wave equation
Summary
Taking special wave speed and using integral method, Qiao [4] showed a W-shape-peak explicit solution as follows: u(χ) = 2 − 3 cosh χ + (cosh χ + 1/3) 3(3 cosh χ + 1)(cosh χ − 1), (1.2). Using bifurcation method of dynamical systems, Li and Zhang [3] showed that there exist smooth solitary solutions and periodic waves of Eq (1.1) when some parameter conditions are satisfied. They explained why the so-called W-shape-peaks and. We draw bifurcation curves and bifurcation phase portraits of the travelling wave system By using these closed orbits, the exact periodic wave solutions of Eq (1.1) are obtained. −=φ. 0u5.=p5l4a3−n8e19.3w59,e4w6dreaanwhdapvzee4ri=ho. 0d3i1c=.c7u94s33p151.w016a62v85e. ,gSzrua1bps=ht.itau−st1iFn9igg.5.t82h7e0(s0be5).d63a,taz2int=o (z13.4=.T),a5ko.8nin5g7ξ8−φ070u2=9p1laφann2ed, wwze4ed=.hraa3wv1e.8ph9e07ri2o=8.d2i5c474c.1u.Ss1pu1b8lso4to8itp6u7wt,ianzvg1et=ghreaszpe2hd=a.atsa−Fi1nig8t..o827(17(.c54)9).2, 4o9n, ξ − u plane we draw cusp loop solitary wave graph as Fig. 2 (d)
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