New Exact Solutions for a Higher-Order Wave Equation of KdV Type Using the Multiple Simplest Equation Method

Zhao, Yun-Mei

doi:10.1155/2014/848069

Cited by 27 publications

(16 citation statements)

References 23 publications

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“…Several papers [7,17,18,19,20,21,22,23,24,25] claim existence of higher invariants and integrability of second order KdV type equations. In particular Benjamin and Olver [7] discussed Hamiltonian structure, symmetries and conservation laws for water waves.…”

Section: Introductionmentioning

confidence: 99%

Adiabatic invariants of the extended KdV equation

et al. 2017

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When the Euler equations for shallow water are taken to the next order, beyond KdV, momentum and energy are no longer exact invariants. (The only one is mass.) However, adiabatic invariants (AI) can be found. When the KdV expansion parameters are zero, exact invariants are recovered. Existence of adiabatic invariants results from general theory of near-identity transformations (NIT) which allow us to transform higher order nonintegrable equations to asymptotically equivalent (when small parameters tend to zero) integrable form. Here we present the direct method of calculations of adiabatic invariants. It does not need a transformation to a moving reference frame nor performing a near-identity transformation. Numerical tests show that deviations of AI from almost constant values are indeed small.

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Section: Introductionmentioning

confidence: 99%

Adiabatic invariants of the extended KdV equation

et al. 2017

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“…By the general expressions for ( )/ ( ) in [12], one can obtain the following expressions for ( )/ ( ):…”

Section: Description Of the Extended Fractional ( / )-Expansion Methodsmentioning

confidence: 99%

The Extended Fractional (DξαG/G)‐Expansion Method and Its Applications to a Space‐Time Fractional Fokas Equation

Zhao

2017

Mathematical Problems in Engineering

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Based on a fractional subequation and the properties of the modified Riemann-Liouville fractional derivative, we propose a new analytical method named extended fractional ( / )-expansion method for seeking traveling wave solutions of fractional partial differential equations. To illustrate the effectiveness of the method, we discuss a space-time fractional Fokas equation, many types of exact analytical solutions are obtained, and the solutions include hyperbolic function and trigonometric and negative exponential solutions.

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“…Recently, Bilige et al [15,16], extended and improved this method which is called the extended simplest equation method. After wards, several researchers applied this method to obtain new exact solutions for nonlinear PDEs [15][16][17][18][19][20].…”

Section: Introductionmentioning

confidence: 99%