Kawahara Equation Research Articles

Global attractor for the weakly damped forced Kawahara equation on the torus

We study the long-time behaviour of solutions for the weakly damped forced Kawahara equation on the torus. More precisely, we prove the existence of a global attractor in L 2 , to which as time passes all solutions draw closer. In fact, we show that the global attractor turns out to lie in a smoother space H 2 and be bounded therein. Further, we give an upper bound of the size of the attractor in H 2 that depends only on the damping parameter and the norm of the forcing term.

A Rapid Numerical Method for Nonlinear Generalized Time-Fractional Kawahara Equations via Domination Polynomials of Complete Graph.

Abstract The key objective of this study is to examine the nonlinear Time-fractional generalized Kawahara equation (N-TF-GKE), a significant nonlinear equation that appears in a variety of physical domains, including ion-acoustic waves in plasmas, shallow water waves, and nonlinear acoustics. This Equation exhibits complex dynamic properties, most notably the appearance of shock waves, solitary waves, and chaotic solutions. We use a graph theoretic polynomial approach to obtain numerical solutions of the N-TF-GKE. We present the dominance polynomials of complete graphs (DPC), a new class of graph polynomials, to develop an efficient operational integration matrix to approximate the N-TF-GKE. The N-TF-GKE transformed into a collection of nonlinear algebraic equations with the standard collocation points. We acquire the numerical solutions for the Domination polynomial using Newton's approach. We validate the efficiency and robustness of our domination polynomial collocation method (DCM) with a set of numerical experiments and comparative analyses.

Computational study of time-fractional non-linear Kawahara equations using Quintic B-spline and Galerkin’s method

This study presents two numerical methods focused on Quintic B-spline (QBS) and Galerkin finite element method (GFEM) for solving time-fractional Kawahara equations. The QBS is utilized as both the basis and test function in the FEM approach. We apply Caputo formula with quadrature rule for evaluation of temporal fractional part. The QBS and GFEM formulation are used to approximate the space functions and their derivatives. Furthermore, a four-point Gauss Legendre quadrature is employed to evaluate the source term in the GFEM. The efficiency and accuracy of the proposed scheme are evaluated using the E2 and E∞ norms. Additionally, Fourier stability analysis is conducted, and it is revealed that the method exhibits unconditional stability. The results, presented in the form of tables and graphs to demonstrate the effectiveness of the scheme.

Floquet theory and stability analysis for Hamiltonian PDEs

Abstract We analyze Floquet theory as it applies to the stability and instability of periodic traveling waves in Hamiltonian PDEs. Our investigation focuses on several examples of such PDEs, including the generalized KdV and BBM equations (third order), the nonlinear Schrödinger and Boussinesq equations (fourth order), and the Kawahara equation (fifth order). Our analysis reveals that the characteristic polynomial of the monodromy matrix inherits symmetry from the underlying PDE, enabling us to determine the essential spectrum along the imaginary axis and bifurcations of the spectrum away from the axis, employing the Floquet discriminant. We present numerical evidence to support our analytical findings.

Stability Analysis Study of Time-Fractional Nonlinear Modified Kawahara Equation Based on the Homotopy Perturbation Sadik Transform

In this manuscript, we survey a numerical algorithm based on the combination of the homotopy perturbation method and the Sadik transform for solving the time-fractional nonlinear modified shallow water waves (called Kawahara equation) within the frame of the Caputo–Prabhakar (CP) operator. The nonlinear terms are handled with the assistance of the homotopy polynomials. The stability analysis of the implemented method is studied by using S-stable mapping and the Banach contraction principle. Also, we use the fixed-point method to determine the existence and uniqueness of solutions in the given suggested model. Finally, some numerical simulations are illustrated to display the accuracy and efficiency of the present numerical method. Moreover, numerical behaviors are captured to validate the reliability and efficiency of the scheme.

Landau quantization effects on damping Kawahara solitons in electron–positron–ion plasma in rotating ionized medium

For the dynamics of three-dimensional electron–positron–ion plasmas, a fluid quantum hydrodynamic model is proposed by considering Landau quantization effects in dense plasma. Ion–neutral collisions in the presence of the Coriolis force are also considered. The application of the reductive perturbation technique produces a wave evolution equation represented by a damped Korteweg–de Vries equation. This equation, however, is insufficient for describing waves in our system at very low dispersion coefficients. As a result, we considered the highest-order perturbation, which resulted in the damped Kawahara equation. The effects of the magnetic field, Landau quantization, the ratio of positron density to electron density, the ratio of positron density to ion density, and the direction cosine on linear dispersion laws as well as soliton and conoidal solutions of the damped Kawahara equation are explored. The understanding from this research can contribute to the broader field of astrophysics and aid in the interpretation of observational data from white dwarfs.

Approximation theorem for the Kawahara operator and its application in the control theory

AbstractControl properties of the Kawahara equation are considered when the equation is posed on an unbounded domain. Precisely, the paper's main results are related to an approximation theorem that ensures the exact (internal) controllability in . Following [23], the problem is reduced to prove an approximate theorem which is achieved thanks to a global Carleman estimate for the Kawahara operator.

Analytical Scheme for Time Fractional Kawahara and Modified Kawahara Problems in Shallow Water Waves

The Kawahara equation exhibits signal dispersion across lines of transmission and the production of unstable waves from the water in the broad wavelength area. This article explores the computational analysis for the approximate series of time fractional Kawahara (TFK) and modified Kawahara (TFMK) problems. We utilize the Shehu homotopy transform method (SHTM), which combines the Shehu transform (ST) with the homotopy perturbation method (HPM). He’s polynomials using HPM effectively handle the nonlinear terms. The derivatives of fractional order are examined in the Caputo sense. The suggested methodology remains unaffected by any assumptions, restrictions, or hypotheses on variables that could potentially pervert the fractional problem. We present numerical findings via visual representations to indicate the usability and performance of fractional order derivatives for depicting water waves in long-wavelength regions. The significance of our proposed scheme is demonstrated by the consistency of analytical results that align with the exact solutions. These derived results demonstrate that SHTM is an effective and powerful scheme for examining the results in the representation of series for time-fractional problems.

On the positron-acoustic Kawahara solitary and cnoidal waves in a non-Maxwellian electron–positron–ion plasma

The dissemination of positron-acoustic (PA) nonlinear structures, including the solitary waves (SWs) and cnoidal waves (CWs), is analyzed in an unmagnetized electron–positron–ion (e–p–i) plasma having inertial cold positrons and inertialess Cairns distributed electrons and Maxwellian positrons as well as immobile positive ions. The reductive perturbation method (RPM) is introduced to reduce the fluid equations to this model to the Korteweg–de Vries (KdV) type equation for studying small amplitude PA waves (PAWs). Moreover, the Kawahara (sometimes called the fifth-order KdV) equation is also obtained to investigate the characteristics of large amplitude PAWs. The effects of related parameters, such as nonthermal parameters, hot positron concentration, electron concentration, and temperature ratios, are numerically examined on the features of SWs and CWs.

Dispersive decay bound of small data solutions to Kawahara equation in a finite time scale

In this article, we prove that small localized data yield solutions to Kawahara-type equations which have linear dispersive decay on a finite time scale depending on the size of the initial data. We use the similar method used by Ifrim and Tataru to derive the dispersive decay bound of the solutions to the KdV equation, with some steps being simpler. This result is expected to be the first result of the small data global bounds of the fifth-order dispersive equations with quadratic nonlinearity.

Utmost travelling wave phenomena to the fractional type nonlinear evolution equation in mathematical physics

In applied physics and engineering, non-linear fractional types of partial differential equations become increasingly prominent as an estimation technique to explain a wide variety of non-linear phenomena. Throughout this study, we choose to use an adaptable extended tanh-function scheme over a conformable derivative in order to construct a comprehensive closed-form traveling wave solution of the time fractional Cahn–Hilliard and modified Kawahara equations. The mentioned equations, have been used as frameworks for various phenomena such as quantum theory, geosciences, water wave mechanics, and solitary waves in shallow water in the porous medium. The different values of the fractional order of the model can be employed successfully to organize the wave velocity. We recognized numerous forms of solutions by using the computational software, namely solitons type, bell types, kink types, periodic types, multiple periodic, single soliton, singular kink, and other types of solutions that are illustrated using 3D, contour, and spherical. The attained results have effectively handled the previously stated phenomena in various fields of engineering and mathematical physics. The results originating in this study were confirmed by the computational software, which was used to rewrite them as non-linear fractional partial differential equations. We ensure that the technique is revised to make it more effective, pragmatic, and credible, and we pursue more comprehensive exact results for traveling waves, and then for solitary waves.

Analytical and numerical studies of the modified Kawahara equation with dual-power law nonlinearities

Analytical and numerical studies of the modified Kawahara equation with dual-power law nonlinearities

A computational strategy for nonlinear time-fractional generalized Kawahara equation using new eighth-kind Chebyshev operational matrices

This paper presents a new method to numerically solve the nonlinear time-fractional generalized Kawahara equations (NTFGKE) with uniform initial boundary conditions (IBCs). A class of modified shifted eighth-kind Chebyshev polynomials (MSEKCPs) is introduced to satisfy the given IBCs. The proposed method is based on using the operational matrices (OMs) for the ordinary derivatives (ODs) and the fractional derivatives (FDs) of MSEKCPs. These OMs are employed together with the spectral collocation method (SCM). Our presented algorithm enables the extraction of efficient and accurate numerical solutions. The convergence of the suggested method and the error analysis have been developed. Three numerical examples are presented to demonstrate the applicability and accuracy of our algorithm. Some comparisons of the presented numerical results with other existing ones are offered to validate the efficiency and superiority of our approach. The presented tables and graphs demonstrate that the proposed approach produces approximate solutions with high accuracy.

A survey on fractal fractional nonlinear Kawahara equation theoretical and computational analysis

With the use of the Caputo, Caputo-Fabrizio (CF), and Atangana-Baleanu-Caputo (ABC) fractal fractional differential operators, this study offers a theoretical and computational approach to solving the Kawahara problem by merging Laplace transform and Adomian decomposition approaches. We show the solution’s existence and uniqueness through generalized and advanced version of fixed point theorem. We present a precise and efficient method for solving nonlinear partial differential equations (PDEs), in particular the Kawahara problem. Through careful error analysis and comparison with precise solutions, the suggested method is validated, demonstrating its applicability in solving the nonlinear PDEs. Moreover, the comparative analysis is studied for the considered equation under the aforementioned operators.

On the radius of analyticity for a Korteweg–de Vries–Kawahara equation with a weak damping term

We consider the Cauchy problem for an equation of Korteweg–de Vries–Kawahara type with initial data in the analytic Gevrey spaces. By using linear, bilinear and trilinear estimates in analytic Bourgain spaces, we establish the local well-posedness of this problem. By using an approximate conservation law, we extend this to a global result in such a way that the radius of analyticity of solutions is uniformly bounded below by a fixed positive number for all time.

Numerical Investigation of Fractional Kawahara Equation via Haar Scale Wavelet Method

The Kawahara equation is a fifth-order dispersive equation that plays a significant role in explaining the creation of non-linear water waves in the long-wavelength region. In this research, the Kawahara equation is solved numerically using the novel Haar scale-3 wavelet method in conjunction with the collocation method. The quasilinearisation approach and the Caputo derivative are used to characterise the non-linearity and fractional behaviour of the equation, respectively. To verify that the findings obtained are legitimate, residual and error estimates are generated. A thorough comparison is made between the present solutions and the numerical findings that have already been published in the literature, which demonstrates the advantages and effectiveness of the suggested technique. The Haar wavelet method reveals a dynamic system of alternative solutions for a wide variety of physical parameters.

Unconditional well-posedness for the periodic Boussinesq and Kawahara equations

<abstract><p>In this article, we obtain new results on the unconditional well-posedness for a pair of periodic nonlinear dispersive equations using an abstract framework introduced by Kishimoto. This framework is based on a normal form reductions argument coupled with a number of crucial multilinear estimates.</p></abstract>

SPECTRAL THEORY AND DIFFERENTIAL EQUATIONS ON THE KAWAHARA TYPE SINGULAR DIFFERENTIAL EQUATION

Abstract. We study the fifth-order degenerate differential equation of Kawahara type given in the space 𝐿2(−∞,∞). We assume that the coefficients of the equation are unbounded functions. Sufficient conditions for the correctness of the mentioned equation, as well as the maximum regularity estimate of its solution were shown in the work. According to the obtained results, if some intermediate coefficients of the equation grow rapidly to infinity then the requirement of constant sign of its smaller coefficient can be removed. Similarly, the minor coefficient can be an unbounded function. Although the coefficients of the equation are assumed to be smooth functions, no additional requirements are imposed on their first and higher order derivatives. Keywords: Kawahara equation, variable coefficient, singularity, differential operator, generalized solution, norm,

Stabilization results for delayed fifth-order KdV-type equation in a bounded domain

Studied here is the Kawahara equation, a fifth-order Korteweg-de Vries type equation, with time-delayed internal feedback. Under suitable assumptions on the time delay coefficients, we prove that the solutions of this system are exponentially stable. First, considering a damping and delayed system, with some restriction of the spatial length of the domain, we prove that the energy of the Kawahara system goes to $ 0 $ exponentially as $ t \rightarrow \infty $. After that, by introducing a more general delayed system, and by introducing suitable energies, we show using the Lyapunov approach, that the energy of the Kawahara equation goes to zero exponentially, considering the initial data small and a restriction in the spatial length of the domain. To remove these hypotheses, we use the compactness-uniqueness argument which reduces our problem to prove an observability inequality, showing a semi-global stabilization result.

Numerical analysis of the fractional nonlinear waves of fifth-order KdV and Kawahara equations under Caputo operator

<p>This study delved into the analytical investigation of two significant nonlinear partial differential equations, namely the fractional Kawahara equation and fifth-order Korteweg-De Vries (KdV) equations, utilizing advanced analytical techniques: the Aboodh residual power series method and the Aboodh transform iterative method. Both equations were paramount in various fields of applied mathematics and physics due to their ability to describe diverse nonlinear wave phenomena. Here, we explored using the Aboodh methods to efficiently solve these equations under the framework of the Caputo operator. Through rigorous analysis and computational simulations, we demonstrated the efficacy of the proposed methods in providing accurate and insightful solutions to the time fractional Kawahara equation and fifth-order KdV equations. Our study advanced the understanding of nonlinear wave dynamics governed by fractional calculus, offering valuable insights and analytical tools for tackling complex mathematical models in diverse scientific and engineering applications.</p>