We determine the existence of cosymmetries for the scalar evolution equations ut = uk + f(u, · · · , uk−1) by using the symbolic method and emphasize the role played by the permutation group in answering divisibility questions. In some special cases, we conclude that there is only one possible cosymmetry. The method can also be used to classify the evolution equations by generating functions. A...

We introduce a new solution for Kawahara-KdV equations. The Lie group analysis is used to carry out the integration of this equations. The similarity reductions and exact solutions are obtained based on the optimal system method.

The Cauchy problem for the higher order equations in the mKdV hierarchy is investigated with data in the spaces b H s (R) defined by the norm ‖v0‖ b Hr s (R) := ‖〈ξ〉 b v0‖Lr′ ξ , 〈ξ〉 = (1 + ξ) 1 2 , 1 r + 1 r = 1. Local well-posedness for the jth equation is shown in the parameter range 2 ≥ r > 1, s ≥ 2j−1 2r . The proof uses an appropriate variant of the Fourier restriction norm method. A coun...

We show that a nonlinear equation that represents third-order approximation of long wavelength, small amplitude waves of inviscid and incompressible fluids is integrable for a particular choice of its parameters, since in this case it is equivalent with an integrable equation which has recently appeared in the literature. We also discuss the integrability of both secondand third-order approxima...

We study here the water waves problem for uneven bottoms in a highly nonlinear regime where the small amplitude assumption of the Korteweg-de Vries (KdV) equation is enforced. It is known that, for such regimes, a generalization of the KdV equation (somehow linked to the CamassaHolm equation) can be derived and justified [Constantin and Lannes, Arch. Ration. Mech. Anal. 192 (2009) 165–186] when...

The main purpose of this research is to propose a new methodology observe class time-fractional generalized fifth-order Korteweg–de Vries equations. Laplace transform along with homotopy perturbation algorithm utilized for the solution and analysis in current study. This extended technique provides improved convergent series solutions through symbolic computation. proposed applied Sawada–Kotera...

We give a systematic account of isospectral deformations for Sturm-Liouville and Dirac-type operators and associated hierarchies of nonlinear evolution equations. In particular, we study generalized KdV and modified KdV-hierarchies and their reduction to the standard (m)KdV-hierarchy. As an example we study the Harry Dym equation in some detail and relate its solutions to KdV-solutions and to H...