scholarly journals Local Analyticity in the Time and Space Variables and the Smoothing Effect for the Fifth-Order KdV-Type Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-39
Author(s):  
Kyoko Tomoeda
Keyword(s):  
Kdv Equation ◽  
Type Equation ◽  
Smoothing Effect ◽  
Initial Value ◽  
Time And Space ◽  
Third Order ◽  
Bootstrap Argument ◽  
Real Analytic ◽  
Fifth Order ◽  
Bourgain Space

We consider the initial value problem for the reduced fifth-order KdV-type equation: , , , . This equation is obtained by removing the nonlinear term from the fifth-order KdV equation. We show the existence of the local solution which is real analytic in both time and space variables if the initial data satisfies the condition , for some constant . Moreover, the smoothing effect for this equation is obtained. The proof of our main result is based on the contraction principle and the bootstrap argument used in the third-order KdV equation (K. Kato and Ogawa 2000). The key of the proof is to obtain the estimate of on the Bourgain space, which is accomplished by improving Kenig et al.'s method used in (Kenig et al. 1996).

Solutions of time-fractional third- and fifth-order Korteweg–de-Vries equations using homotopy perturbation transform method

2019 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Perumandla Karunakar ◽  
Snehashish Chakraverty
Keyword(s):  
Kdv Equation ◽  
Convergent Series ◽  
Third Order ◽  
Transform Method ◽  
Content Type ◽  
Homotopy Perturbation ◽  
De Vries ◽  
Series Type ◽  
Fifth Order ◽  
Wave Phenomena

Purpose This study aims to find the solution of time-fractional Korteweg–de-Vries (tfKdV) equations which may be used for modeling various wave phenomena using homotopy perturbation transform method (HPTM). Design/methodology/approach HPTM, which consists of mainly two parts, the first part is the application of Laplace transform to the differential equation and the second part is finding the convergent series-type solution using homotopy perturbation method (HPM), based on He’s polynomials. Findings The study obtained the solution of tfKdV equations. An existing result “as the fractional order of KdV equation given in the first example decreases the wave bifurcates into two peaks” is confirmed with present results by HPTM. A worth mentioning point may be noted from the results is that the number of terms required for acquiring the convergent solution may not be the same for different time-fractional orders. Originality/value Although third-order tfKdV and mKdV equations have already been solved by ADM and HPM, respectively, the fifth-order tfKdV equation has not been solved yet. Accordingly, here HPTM is applied to two tfKdV equations of order three and five which are used for modeling various wave phenomena. The results of third-order KdV and KdV equations are compared with existing results.

Integrable Models

2018 ◽  
Author(s):  
S. G. Rajeev
Keyword(s):  
Fluid Mechanics ◽  
Soliton Solution ◽  
Water Waves ◽  
Heisenberg Model ◽  
Kdv Equation ◽  
Short Review ◽  
Lax Pair ◽  
Shallow Water Waves ◽  
Initial Value ◽  
The Kdv Equation

Some exceptional situations in fluid mechanics can be modeled by equations that are analytically solvable. The most famous example is the Korteweg–de Vries (KdV) equation for shallow water waves in a channel. The exact soliton solution of this equation is derived. The Lax pair formalism for solving the general initial value problem is outlined. Two hamiltonian formalisms for the KdV equation (Fadeev–Zakharov and Magri) are explained. Then a short review of the geometry of curves (Frenet–Serret equations) is given. They are used to derive a remarkably simple equation for the propagation of a kink along a vortex filament. This equation of Hasimoto has surprising connections to the nonlinear Schrödinger equation and to the Heisenberg model of ferromagnetism. An exact soliton solution is found.

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