Fifth-order Shallow Water Research Articles

Local well-posedness and time regularity for a fifth-order shallow water equations in analytic Gevrey–Bourgain spaces

This paper studies a Cauchy problems for fifth-order shallow water equations with nonlinear terms in (1). With data in analytic Gevrey spaces on the line, we prove that the problem is well defined. We also treat the regularity in time which belongs to $$G^{5\sigma }$$ near zero for every x on the line. The proof is based mainly on bilinear and trilinear estimates in the analytic Gevrey–Bourgain spaces, relies on the contraction mapping theorem to improve the results in Jia and Huo (J Differ Equ 246:2448–2467, 2009).

Well-posedness of stochastic modified Kawahara equation

In this paper we consider the Cauchy problem for the stochastic modified Kawahara equation, which is a fifth-order shallow water wave equation. We prove local well-posedness for data in H^{s}(mathbb{R}), sgeq -1/4. Moreover, we get the global existence for L^{2}( mathbb{R}) solutions. Due to the non-zero singularity of the phase function, a fixed point argument and the Fourier restriction method are proposed.

The well-posedness of stochastic Kawahara equation: fixed point argument and Fourier restriction method

In this paper, we investigate the Cauchy problem for the stochastic Kawahara equation, which is a fifth-order shallow water wave equation. We prove local well-posedness for data in H^{s}(mathbb {R}), s>−7/4. Moreover, we get global existence for L^{2}(mathbb {R}) solutions. Due to the non-zero singularity of the phase function, a fixed point argument and Fourier restriction method are proposed.

Global well-posedness for a fifth-order shallow water equation on the circle

The periodic initial value problem of a fifth-order shallow water equation ∂ t u − ∂ x 2 ∂ t u + ∂ x 3 u − ∂ x 5 u + 3 u ∂ x u − 2 ∂ x u ∂ x 2 u − u ∂ x 2 u = 0 is shown to be globally well-posed in Sobolev spaces H ˙ s ( T ) for s > 2/3 by I-method. For this equation lacks scaling invariance, we first reconsider the local result and pay special attention to the relationship between the lifespan of the local solution and the initial data, and then prove the almost conservation law, and finally obtain the global well-posedness by an iteration process.

Sharp local well-posedness for a fifth-order shallow water wave equation

In this paper we prove that the already-established local well-posedness in the range s > − 5 / 4 of the Cauchy problem with an initial H s ( R ) data for a fifth-order shallow water wave equation is extendable to s = − 5 / 4 by using the F ¯ s space. This is sharp in the sense that the ill-posedness in the range s < − 5 / 4 of this initial value problem is already known.

Global well-posedness for a fifth-order shallow water equation in Sobolev spaces

The Cauchy problem of a fifth-order shallow water equation ∂ t u − ∂ x 2 ∂ t u + ∂ x 3 u + 3 u ∂ x u − 2 ∂ x u ∂ x 2 u − u ∂ x 3 u − ∂ x 5 u = 0 is shown to be globally well-posed in Sobolev spaces H s ( R ) for s > ( 6 10 − 17 ) / 4 . The proof relies on the I-method developed by Colliander, Keel, Staffilani, Takaoka and Tao. For this equation lacks scaling invariance, we reconsider the local result and pay special attention to the relationship between the lifespan of the local solution and the initial data. We prove the almost conservation law, and combine it with the local result to obtain the global well-posedness.

The Cauchy problem for Kawahara equation in Sobolev spaces with low regularity

This paper is devoted to studying the initial-value problem of the Kawahara equation. By establishing some crucial bilinear estimates related to the Bourgain spaces Xs, b(R2) introduced by Bourgain and homogeneous Bourgain spaces, which is defined in this paper and using I-method as well as L2 conservation law, we show that this fifth-order shallow water wave equation is globally well-posed for the initial data in the Sobolev spaces Hs(R) with \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}$s{>}-\frac{63}{58}$\end{document}. Copyright © 2010 John Wiley & Sons, Ltd.

Well-posedness and ill-posedness for a fifth-order shallow water wave equation

In this paper we establish a new bilinear estimate in suitable Bourgain spaces by using a fundamental estimate on dyadic blocks for the Kawahara equation which was obtained by the [ k ; Z ] multiplier norm method of Tao (2001) [2]; then the local well-posedness of the Cauchy problem for a fifth-order shallow water wave equation in H s ( R ) with s > − 5 4 is obtained by the Fourier restriction norm method. And some ill-posedness in H s ( R ) with s < − 5 4 is derived from a general principle of Bejenaru and Tao.

Well-posedness for the fifth-order shallow water equations

The Cauchy problems for some kind of fifth-order shallow water equations ∂ t u + α ∂ x 5 u + β ∂ x 3 u + γ ∂ x u + F ( u , ∂ x u , ∂ x 2 u ) = 0 , x , t ∈ R × R , are considered by the Fourier restriction norm method, where nonlinear terms F ( u , ∂ x u , ∂ x 2 u ) are μ ∂ x ( u k ) , k = 2 , 3 , μ u ∂ x 2 u or μ ∂ x u ∂ x 2 u respectively. The local well-posedness is established for data in H s ( R ) with s > − 7 4 for the Kawahara equation ( F = μ ∂ x ( u 2 ) ) and is established for data in H s ( R ) with s ⩾ − 1 4 for the modified Kawahara equation ( F = μ ∂ x ( u 3 ) ), respectively. Moreover, the local result is established for data in H s ( R ) with s > 0 if F = μ u ∂ x 2 u and is established for data in H s ( R ) with s > − 1 4 if F = μ ∂ x u ∂ x 2 u , respectively.

Global well-posedness of the Cauchy problem of the fifth-order shallow water equation

By using the I-method, we prove that the Cauchy problem of the fifth-order shallow water equation ∂ t u − ∂ x 2 ∂ t u + ∂ x 3 u + 3 u ∂ x u − 2 ∂ x u ∂ x 2 u − u ∂ x 3 u − ∂ x 5 u = 0 is globally well-posed in the Sobolev space H s ( R ) provided s > ( 5 7 − 10 ) / 4 .

Dynamics of Combined Solitary-waves in the General Shallow Water Wave Models

We find new analytic solitary-wave solutions, having a nonzero background at infinity, of the general fifth-order shallow water wave models using the hyperbolic function ansatz method. We study the dynamical properties of the solutions in the combined form of a bright and a dark solitary-wave by using numerical simulations. It is shown that the solitary-waves can be stable or marginally stable, depending on the coefficients of the model.We study the interaction dynamics by using the combined solitary-waves as the initial profiles to show the formation of sech2-type solitary-waves in the presence of a strong nonlinear dispersion term. - PACS: 03.40.Kf, 02.30.Jr, 47.20.Ky, 52.35.Mw

SOLITONIC INVESTIGATION ON THE GENERAL FIFTH-ORDER SHALLOW WATER WAVE MODELS

For the general fifth-order shallow water wave models, we perform computerized symbolic computation to obtain two auto-Bäcklund transformations and four families of the bell-shaped solitonic solutions, which are exact analytic.