ILL-POSEDNESS OF MODIFIED KAWAHARA EQUATION AND KAUP-KUPERSHMIDT EQUATION

In this article, we consider the Cauchy problems for the modified Kawahara equationδtu＋μδx（u3）＋αδx5u＋βδx3u＋γδxu=0 and the Kaup-Kupershmidt equation δtu＋μuδx2u＋αδx5u＋βδx3u＋βδx3u＋γδxu=0Using the general well-posedness principle introduced by I. Bejenaru and T. Tao, we prove 1 that the modified Kawahara equation is ill-posed for the initial data in H8 （It） with s 〈 - and that the Kaup-Kupershmidt equation is ill-posed for the initial data in HS（It） with s〈0.

On Exact Traveling Wave Solutions for (1 + 1) Dimensional Kaup-Kupershmidt Equation

In this present paper, the Fan sub-equation method is used to construct exact traveling wave solutions of the (1 + 1) dimensional Kaup-Kupershmidt equation. Many exact traveling wave solutions are successfully obtained, which contain solitary wave solutions, trigonometric function solutions, hyperbolic function solutions and Jacobian elliptic function periodic solutions with double periods.

Approximate Solution Method of the Seventh Order KdV Equations by Decomposition Method

In this paper, Adomian decomposition method (ADM) is implemented to approximate the solution of the Korteweg-de Vries (KdV) equations of seventh order, which are Kaup-Kuperschmidt equation and seventh order Kawahara equation. The results obtained by the ADM are compared with the exact solutions. It is found that the ADM is very efficient and convenient and can be applied to a large class of problems. The conservation properties of solution are examined by calculating the first three invariants.

Analysis of the seventh-order Caputo fractional KdV equation:applications to the Sawada-Kotera-Ito and Lax equations

In this study,we investigate the seventh-order nonlinear Caputo time-fractional KdV equation.The suggested model's solutions,which have a series form,are obtained using the hybrid ZZ-transform under the aforementioned fractional operator.The proposed approach combines the homotopy perturbation method(HPM)and the ZZ-transform.We consider two specific examples with suitable initial conditions and find the series solution to test their applicability.To demonstrate the utility of the presented technique,we explore its applications to the fractional Sawada–Kotera–Ito problem and the Lax equation.We observe the impact of a few fractional orders on the wave solution evolution for the problems under consideration.We provide the efficiency and reliability of the ZZHPM by calculating the absolute error between the series solution and the exact solution of both the Sawada–Kotera–Ito and Lax equations.The convergence and uniqueness of the solution are portrayed via fixed-point theory.

AN INSTABILITY THEOREM FOR A CERTAIN SEVENTH-ORDER DIFFERENTIAL EQUATION

The object of the paper is to establish an instability theorem for seventh-order differential equations of the form (1.10). The proof is based on the use of Krasovskii criteria.

Local and Global Existence of Solutions to Initial Value Problems of Nonlinear Kaup-Kupershmidt Equations

This paper is devoted to studying the initial value problems of the nonlinear Kaup Kupershmidt equations δu/δt ＋ α1 uδ^2u/δx^2 ＋ βδ^3u/δx^3 ＋ γδ^5u/δx^5 = 0, （x,t）∈ E R^2, and δu/δt ＋ α2 δu/δx δ^2u/δx^2 ＋ βδ^3u/δx^3 ＋ γδ^5u/δx^5 = 0, （x, t） ∈R^2. Several important Strichartz type estimates for the fundamental solution of the corresponding linear problem are established. Then we apply such estimates to prove the local and global existence of solutions for the initial value problems of the nonlinear Kaup- Kupershmidt equations. The results show that a local solution exists if the initial function u0（x） ∈ H^s（R）, and s ≥ 5/4 for the first equation and s≥301/108 for the second equation.

Conservatory of Kaup-Kupershmidt Equation to the Concept of Fractional Derivative with and without Singular Kernel

Making use of the traditional Caputo derivative and the newly introduced Caputo-Fabrizio deriva- tive with fractional order and no singular kernel, we extent the nonlinear Kaup-Kupershmidt to the span of fractional calculus. In the analysis, different methods of fixed-point theorem together with the concept of piccard L-stability are used, allowing us to present the existence and uniqueness of the exact solution to models with both versions of derivatives. Finally, we present techniques to perform some numerical simulations for both non-linear models and graphical simulations are provided for values of the order α = 1.00; 0.90. Solutions are shown to behave similarly to the standard well-known traveling wave solution of Kaup-Kupershmidt equation.

Local Solvability of Cauchy Problem for Kaup-Kupershmidt Equation

This paper deals with the local solvability of initial value problem for Kaup-Kupershmidt equations. Indeed, using Bourgain method, we prove that the Cauchy problem of Kaup-Kupershmidt equation is local well-posed in H8 whenever s 〉 9/8, which improves the former results in [5].