PENERAPAN ALGORITMA DEMINA-KUDRYASHOV DALAM MENENTUKAN SOLUSI MEROMORFIK PERSAMAAN OSTROVSKY

Abstrak Persamaan Ostrovsky merupakan persamaan diferensial parsial nonlinear yang dapat ditemukan dalam fenomena fisis seperti tsunami. Persamaan ini telah memiliki banyak solusi khusus analitik terutama untuk menggambarkan penjalaran gelombang soliton. Salah satu solusi khusus yang terkenal berupa solusi tanh kuadrat atau dapat juga dinyatakan dalam sech kuadrat. Paper ini mengkaji solusi meromorfik persamaan Ostrovsky dengan menggunakan algoritma Demina-Kudryashov. Mula-mula, persamaan Ostrovsky ditransformasi ke dalam bentuk persamaan diferensial biasa nonlinear menggunakan model penjalaran gelombang dan selanjutnya diterapkan algoritma tersebut untuk memperoleh solusi meromorfik berdasarkan pada uraian Laurentnya. Solusi yang diperoleh berupa solusi periode tunggal, solusi periode ganda (solusi eliptik), dan solusi rasional, di mana solusi-solusi ini bersifat umum. Pada akhirnya, ditampilkan suatu solusi khusus berupa solusi tanh kuadrat sebagai pemilihan keadaan khusus berdasarkan salah satu solusi meromorfik.  Kata kunci : solusi meromorfik; algoritma Demina-Kudryashov; persamaan Ostrovsky.   Abstract Ostrovsky equation is a nonlinear partial diferential equation which we found in many problems of physics such as tsunami. This equation has many special analytical solutions especially for describing the travelling of soliton. One of the famous special solution is containing  quadratic tanh term or we can express it in sech term. In this paper, the meromorphic solutions of Ostrovsky equation have analyzed by using Demina-Kudryashov algorithm. Firstly, this equation was transformed to nonlinear ordinary differential equation by using travelling wave model and then by using this algorithm and based on Laurent series, the meromorphic solutions can be contructed. Finally, the general solutions was found. These solutions take form in three types, such as simply periodic, doubly periodic (elliptic solutions), and rational solution. And then, the special solution of this equation was showed by choosing a special condition.    Keywords : meromorphic solutions; Demina-Kudryashov algorithm; Ostrovsky equation

A note on "new travelling wave solutions to the Ostrovsky equation"

In a recent paper by Yaşar [E. Yaşar, New travelling wave solutions to the Ostrovsky equation, Appl. Math. Comput. 216 (2010), 3191-3194], 'new' travelling-wave solutions to the transformed reduced Ostrovsky equation are presented. In this note it is shown that some of these solutions are disguised versions of known solutions

Periodic and Solitary Travelling-Wave Solutions of an Extended Reduced Ostrovsky Equation

Periodic and solitary travelling-wave solutions of an extended reduced Ostrovsky equation are investigated. Attention is restricted to solutions that, for the appropriate choice of certain constant parameters, reduce to solutions of the reduced Ostrovsky equation. It is shown how the nature of the waves may be categorized in a simple way by considering the value of a certain single combination of constant parameters. The periodic waves may be smooth humps, cuspons, loops or parabolic corner waves. The latter are shown to be the maximum-amplitude limit of a one-parameter family of periodic smooth-hump waves. The solitary waves may be a smooth hump, a cuspon, a loop or a parabolic wave with compact support. All the solutions are expressed in parametric form. Only in one circumstance can the variable parameter be eliminated to give a solution in explicit form. In this case the resulting waves are either a solitary parabolic wave with compact support or the corresponding periodic corner waves

A note on solitary travelling-wave solutions to the transformed reduced Ostrovsky equation

Two recent papers are considered in which solitary travelling-wave solutions to the transformed reduced Ostrovsky equation are presented. It is shown that these solutions are disguised versions of previously known solutions

Orbital stability of periodic waves in the class of reduced Ostrovsky equations

Periodic travelling waves are considered in the class of reduced Ostrovsky equations that describe low-frequency internal waves in the presence of rotation. The reduced Ostrovsky equations with either quadratic or cubic nonlinearities can be transformed to integrable equations of the Klein--Gordon type by means of a change of coordinates. By using the conserved momentum and energy as well as an additional conserved quantity due to integrability, we prove that small-amplitude periodic waves are orbitally stable with respect to subharmonic perturbations, with period equal to an integer multiple of the period of the wave. The proof is based on construction of a Lyapunov functional, which is convex at the periodic wave and is conserved in the time evolution. We also show numerically that convexity of the Lyapunov functional holds for periodic waves of arbitrary amplitudes.Comment: 34 page

Comment on “Application of (G′/G)-expansion method to travelling-wave solutions of three nonlinear evolution equation" [Comput Fluids 2010;39;1957-63]

In a recent paper [Abazari R. Application of (G′ G )-expansion method to travelling wave solutions of three nonlinear evolution equation. Computers & Fluids 2010;39:1957–1963], the (G′/G)-expansion method was used to find travelling-wave solutions to three nonlinear evolution equations that arise in the mathematical modelling of fluids. The author claimed that the method delivers more general forms of solution than other methods. In this note we point out that not only is this claim false but that the delivered solutions are cumbersome and misleading. The extended tanh-function expansion method, for example, is not only entirely equivalent to the (G′/G)-expansion method but is more efficient and user-friendly, and delivers solutions in a compact and elegant form

Solutions Classification to the Extended Reduced Ostrovsky Equation

An alternative to the Parkes' approach [SIGMA 4 (2008) 053, arXiv:0806.3155] is suggested for the solutions categorization to the extended reduced Ostrovsky equation (the exROE in Parkes' terminology). The approach is based on the application of the qualitative theory of differential equations which includes a mechanical analogy with the point particle motion in a potential field, the phase plane method, analysis of homoclinic trajectories and the like. Such an approach is seemed more vivid and free of some restrictions contained in the above mentioned Parkes' paper.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Meromorphic solutions of nonlinear ordinary differential equations

Exact solutions of some popular nonlinear ordinary differential equations are analyzed taking their Laurent series into account. Using the Laurent series for solutions of nonlinear ordinary differential equations we discuss the nature of many methods for finding exact solutions. We show that most of these methods are conceptually identical to one another and they allow us to have only the same solutions of nonlinear ordinary differential equations