This paper is concerned with the Cauchy problem of the modified Kawahara equation. By using the Fourier restriction norm method introduced by Bourgain, and using the I-method as well as the L 2 conservation law, we prove that the modified Kawahara equation is globally well-posed for the initial data in the Sobolev space H s (R) with s > − 3 22 .

In this work we apply an extended hyperbolic function method to solve the nonlinear family of third order Korteweg de-Vries (KdV) equations, namely, the KdV equation, the modified KdV (mKdV) equation, the potential KdV (pKdV) equation, the generalized KdV (gKdV) equation and gKdV with two power nonlinearities equation. New exact travelling wave solutions are obtained for the KdV, mKdV and pKdV ...

Exact solutions of the Kawahara equation by Assas [L.M.B. Assas, J. Com. Appl. Math. 233 (2009) 97–102] are analyzed. It is shown that all solutions do not satisfy the Kawahara equation and consequently all nontrivial solutions by Assas are wrong. Recently Assas in [1] looked for exact solutions of the Kawahara equation using the Exp function method. This equation can be written as ut + α uux +...

Soliton interactions for the extended Korteweg-de Vries (KdV) equation are examined. It is shown that the extended KdV equation can be transformed (to its order of approximation) to a higher-order member of the KdV hierarchy of integrable equations. This transformation is used to derive the higher-order, two-soliton solution for the extended KdV equation. Hence it follows that the higher-order ...

  In this paper, we establish exact solutions for the time-fractional Klein-Gordon equation, and the time-fractional Hirota-Satsuma coupled KdV system. The He’s semi-inverse and the Kudryashov methods are used to construct exact solutions of these equations. We apply He’s semi-inverse method to establish a variational theory for the time-fractional Klein-Gordon equation, and the time-fractiona...

In this paper, we study a class of distributed parameter control system described by the Kawahara equation posed on a periodic domain T (a unit circle in the plane) with an internal control acting on an arbitrary small nonempty subdomain of T. Aided by the Bourgain smoothing property of the Kawahara equation on a periodic domain, we show that the system is locally exactly controllable and expon...

We investigate simulations of exact solutions of the stochastic Dr. Herman Numerical Realizations of Solutions f the Stochastic KdV Equation Exact Solution of Stochastic KdV Wadati 1983 The One Soliton Solution Under Noise Statistical Averages The Exact Solution for < u(x, t) > via the Diffusion Equation Solving the Diffusion Equation The Damped Stochastic KdV Asymptotics Numerical Simulation o...

The connection between supersymmetric quantummechanics and the Kortewegde Vries (KdV) equation is discussed, with particular emphasis on the KdV conservation laws. It is shown that supersymmetric quantum mechanics aids in the derivation of the conservation laws, and gives some insight into the Miura transformation that converts the KdV equation into the modified KdV equation. The construction o...

The Kawahara equation is a higher-order Korteweg-de Vries equation with an additional fifth order derivative term. It was derived by Hasimoto as a model of capillary-gravity waves in an infinitely long canal over a flat bottom in a long wave regime when the Bond number is nearly one third. In this paper, we give a mathematically rigorous justification of this modeling and show that the solution...