The linear stability problem for solitary wave states of the Kawahara—or fifthorder KdV-type—equation and its generalizations is considered. A new formulation of the stability problem in terms of the symplectic Evans matrix is presented. The formulation is based on a multisymplectification of the Kawahara equation, and leads to a new characterization of the basic solitary wave, including change...

The paper concerns capillary-gravity surface waves propagating on a two-dimensional incompressible and inviscid fluid with or without an obstacle placed at flat horizontal bottom. Formally second-order correct model equations, analogous to the first-order KdV equation, are derived under classical assumptions of long wavelength small amplitude waves. equations fifth-order type where bottom intro...

We introduce a new solution for Kawahara-KdV equations. The Lie group analysis is used to carry out the integration of this equations. The similarity reductions and exact solutions are obtained based on the optimal system method.

Abstract The initialand boundary-value problem for the Kawahara equation, a fifthorder KdV type equation, is studied in weighted Sobolev spaces. This functional framework is based on the dual-Petrov–Galerkin algorithm, a numerical method proposed by Shen (2003 SIAM J. Numer. Anal. 41 1595–619) to solve third and higher odd-order partial differential equations. The theory presented here includes...

In this manuscript, we consider the internal control problem for fifth-order KdV type equation, commonly called Kawahara on unbounded domains. Precisely, under certain hypotheses over initial and boundary data, can prove that there exists an input such solutions of equation satisfy integral overdetermination condition. This condition is satisfied when domain posed in real line, left half-line, ...

In this paper, we study the computability of the solution operator of the initial problem for the nonlinear Kawahara equation, which is based on the Type-2 Turing machines. We will prove that in Sobolev space 2 ( ) s H R , for 0 s ≥ , the solution operator: 2 : ( ) s R K H R → ( ; C R 2 ( )) s H R is ( ,[ s H δ ρ → ]) s H δ -computable. The conclusion enriches the theory of computability. Key w...

In this article, we investigate the generalized Kawahara–KdV system. A new topological approach is applied to prove existence of at least one classical solution and two non-negative solutions. The arguments are based upon recent theoretical results.

We propose a novel algorithm, based on physics-informed neural networks (PINNs) to efficiently approximate solutions of nonlinear dispersive PDEs such as the KdV-Kawahara, Camassa-Holm andBenjamin-Ono equations. The stability these is leveraged prove rigorous bounds resulting error. present several numerical experiments demonstrate thatPINNs can very accurately

Dissipationless hydrodynamics regularized by dispersion describe a number of physical media including water waves, nonlinear optics, and Bose–Einstein condensates. As in the classical theory of hyperbolic equations where a nonconvex flux leads to nonclassical solution structures, a nonconvex linear dispersion relation provides an intriguing dispersive hydrodynamic analogue. Here, the fifth orde...

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.