Kawahara Equation Research Articles

Lie symmetries of generalized Kawahara equations

We carry out the group classification of a normalized class of generalized Kawahara equations with variable coefficients. Admissible transformations are studied, and the partition of the class into two normalized subclasses is performed. For each of these subclasses, the respective equivalence groupoids are found. As a result, all equations from the class admitting Lie symmetry extensions are revealed.

Solitary wave solutions of nonlinear PDEs using Kudryashov's R function method

Applications of a new function introduced by Kudryashov [Optik. 2020;206:163550] to obtain solitary wave solutions of nonlinear PDEs through their travelling wave reductions are considered. The Kudryashov function, R, satisfying a first-order second degree ODE has several features which significantly assist symbolic calculations, especially for highly dispersive nonlinear equations. A remarkable feature of the Kudryashov function R, is that its even order derivatives are polynomials in R only while its odd order derivatives turn out to be polynomials in R and Rz . The procedure has been illustrated by means of the Schrödinger–Hirota equation, a quartic NLS equation and the fifth-order Kawahara equation as examples. A comparison with the Rayleigh–Ritz variational approach has also been considered for the purposes of illustration. The results obtained here are novel and span the family of solutions for such kind of equations.

Bell-shaped soliton solutions and travelling wave solutions of the fifth-order nonlinear modified Kawahara equation

Abstract The main aim of this work is to investigate numerical solutions of the two different types of the fifth-order modified Kawahara equation namely bell-shaped soliton solutions and travelling wave solutions that occur thereby the different type of the Korteweg–de Vries equation. For this approach, we have used an effective and simple type of finite difference method namely Crank-Nicolson scheme for time integration and third-order modified cubic B-spline-based differential quadrature method for space integration. We preferred the third-order modified cubic B-splines to solve the fifth-order partial differential equation because of by using low energy, less algebraic process and produce better results than earlier works. To display the efficiency and accuracy of the present fresh approach famous test problems namely bell-shaped single soliton that has negative amplitude and travelling wave solutions that have the both of the positive and negative amplitudes are solved and the error norms L 2 and L ∞ are calculated and compared with earlier works. Comparison of the error norms show that present fresh approach obtained superior results than earlier works by using same parameters. At the same time, two lowest invariants of the test problems during the simulations are calculated and reported. Besides those, relative changes of invariants are computed and reported.

Stable soliton solutions to the shallow water waves and ion-acoustic waves in a plasma

The Kawahara equation (KE) and the modified Kawahara equation (MKE) are important modeling equations to interpret shallow water waves with surface tension, magneto-acoustic wave in plasma and gravity waves. In this study, the stable broad-ranging soliton solutions including the well-known bell-shape soliton, anti-bell shape soliton, periodic soliton, compacton, kink, etc. are established through the sine-Gordon expansion (SGE) method. The effect of the physical parameters, namely the dispersion and perturbed dispersive coefficients in the surface elevation are shown through the figure. The surface elevations remain ascertainable for diverse values of the physical and the associated free parameters. The 3D and contour plot of the ascertained solutions clarifies the surface wave properties. The achieved results for each modeling equation provide a significant contribution in analyzing the ion-acoustic waves in plasma, gravity waves, the surface waves in shallow water. This study asserts that the SGE method is reliable, competent, easy and powerful for extracting closed form soliton solutions.

Nonlinear dissipative wave structures in planar and nonplanar geometry with quantum electron exchange-correlation potential

Taking into account the effects of electron exchange-correlation potential, particle collisions, the nonlinear dynamical properties of planar and nonplanar (cylindrical and spherical) electrostatic ion-acoustic waves (IAWs) are investigated theoretically and numerically in quantum plasmas. With the aid of quantum hydrodynamic model and the reductive perturbation technique, the planar and nonplanar dispersive-dissipative equations (modified Kakutani–Kawahara equations) are obtained to elucidate the nonlinear quantum IAW profiles. The influences of the electron exchange-correlation potential, Bohm potential and collisional effects in the wave dynamics are studied. For a typical parametric range, relevant for laboratory and astrophysical environments, it is found that the contribution from exchange-correlation potential to be prominent in comparison to the effect due to the Bohm potential. Also, numerical simulation predicts that the dissipation induced by the particle collisions leads to the evolution of low frequency shock structures in dense quantum plasmas.

Highly efficient approach to numerical solutions of two different forms of the modified Kawahara equation via contribution of two effective methods

The numerical solutions of the two different forms of the modified Kawahara equation namely bell-shaped soliton solutions and travelling wave solutions that occur thereby the different form of the KdV equation have been investigated. To improve the numerical solutions, two efficient methods have been used together. Firstly, Crank–Nicolson discretization algorithm for time integration is used and then fifth-order quintic B-spline based differential quadrature method for space integration is used. To observe the performance of the present algorithm bell-shaped soliton solution and travelling wave solutions are surveyed. The error norms L2 and L∞ are obtained quite less than earlier papers. The invariants and relative changes of invariants are added to sympathize with superior present results.

New exact solutions for the space-time fractional Kawahara equation

The Kawahara equation is a model of capillary-gravity water wave and plasma waves. In this paper, a direct method based on the Jacobi elliptic functions is presented to get analytical solutions of the space-time fractional Kawahara equation. Moreover, new exact solutions are obtained for a nonlinear ordinary differential equation (dF/dξ)2 = PF4(ξ) + QF2(ξ) + R, which appears as an auxiliary equation for solving many partial differential equations. Some notes, theorems, and corollaries are also given for this equation. Solving this auxiliary equation, the solutions of the Duffing equation can also be found. Besides, three test examples are presented to illustrate the application of the method and some solutions are demonstrated by the graphics.

The initial-boundary value problem for the Kawahara equation on the half-line

This paper concerns the initial-boundary value problem of the Kawahara equation posed on the right and left half-lines. We prove the local well-posedness in the low regularity Sobolev space. We introduce the Duhamel boundary forcing operator, which is introduced by Colliander and Kenig (Commun Partial Differ Equ 27:2187–2266, 2002) in the context of Airy group operators, to construct solutions on the whole line. We also give the bilinear estimate in $$X^{s,b}$$ space for $$b < \frac{1}{2}$$ , which is almost sharp compared to IVP of Kawahara equation (Chen et al. in J Anal Math 107:221–238, 2009; Jia and Huo in J Differ Equ 246:2448–2467, 2009).

Nonclassical symmetries and analytic solutions to Kawahara equation

The focus of this paper is on Kawahara equation (KE) which models diverse variety of waves enforced in many branches of applied sciences. Symmetries to the considered equation are successfully obtained by practicing nonclassical method. According to nonclassical symmetries, KE has been reduced to numerous ODEs. Further, these ODEs are treated such that new analytic solutions are determined by applying power series, ansätz and [Formula: see text]-expansion method.

Discontinuous shock solutions of the Whitham modulation equations as zero dispersion limits of traveling waves

Whitham modulation theory describes the zero dispersion limit of nonlinear disperesive partial differential equations (PDEs) by a system of conservation laws for the parameters of modulated periodic traveling waves (TWs). In this work, admissible, discontinuous, weak solutions of the Whitham modulation equations—termed Whitham shocks—are identified with zero dispersion limits of TW solutions to higher order dispersive PDEs. The far-field behavior of the TW solutions satisfies the Rankine–Hugoniot jump conditions for the Whitham modulation equations. Generally, the numerically computed traveling waves represent heteroclinic connections between two periodic orbits of an ordinary differential equation. The focus here is on the fifth order Korteweg–de Vries equation where three admissible one-parameter families of Whitham shocks are identified as solution components to the generalized Riemann problem for the Whitham modulation equations. Admissible KdV5–Whitham shocks are generally undercompressive, i.e., all characteristic families pass through the shock. The heteroclinic TWs that limit to admissible Whitham shocks are found to be ubiquitous in numerical simulations of smoothed step initial conditions for other higher order dispersive equations including the Kawahara equation (with third and fifth order dispersion) and a nonlocal model of weakly nonlinear gravity-capillary waves with full dispersion. Whitham shocks are linked to recent studies of nonlinear higher order dispersive waves in optics and ultracold atomic gases. The approach presented here provides a novel method for constructing new TW solutions to dispersive nonlinear wave equations and a framework to identify physically relevant, admissible shock solutions of the Whitham modulation equations.

Performance of compact and non-compact structure preserving algorithms to traveling wave solutions modeled by the Kawahara equation

The main contribution of this article is to introduce new compact fourth-order, standard fourth-order, and standard second-order finite difference schemes for solving the Kawahara equation, the fifth-order partial derivative equation. The conservation of mass only of the numerical solution obtained by the compact fourth-order finite difference scheme is proven. However, the standard fourth-order and standard second-order finite difference schemes can preserve both mass and energy. The stability is also proven by von Neumann analysis. According to analysis for numerical experiments, the order of accuracy for each scheme and the computational efficiency of the compact scheme are presented. To validate the potential of the presented methods, we also consider long-time behavior. Finally, results obtained from the compact scheme are superior than those from the non-compact schemes.

New exact solutions of space and time fractional modified Kawahara equation

One of the important nonlinear evolution equations in mathematical physics is the modified Kawahara equation. In this work, by utilizing an analytic method based on the Jacobi elliptic functions, a group of exact solutions of the fractional modified Kawahara equation had been found. The method can be applied to all of the time, space and space–time fractional equations. Four problems had also been given to demonstrate the application of the method and some solutions had been illustrated by the graphics.

Numerical solution for the Kawahara equation using local RBF-FD meshless method

When the number of nodes increases more than thousands, the arising system of global radial basis functions (RBFs) method becomes dense and ill-conditioned. To solve this difficulty, local RBFs generated finite difference method (RBF-FD) were introduced. RBF-FD method is based on local stencil nodes and so it has a sparsity system. The main goal in this work is to develop the RBF-FD method in order to obtain numerical solution for the Kawahara equation as a time dependent partial differential equation that appears in the shallow water and acoustic waves in plasma. For this purpose, we have discretized the temporal and spatial derivatives with a finite difference, θ-weighted, and RBF-FD methods. Then by applying the collocation technique at the grid nodes, a system of linear equations is obtained which gives the numerical solution of the Kawahara equation. The stability analysis is given. The efficiency and accuracy of the proposed approach are tested by four examples. In addition, a comparison between proposed method and the methods, RBFs, differential quadrature (DQM), cosine expansion based differential quadrature (CDQ) and modified cubic B-Spline differential quadrature (MCBC-DQM) is shown.

Construction of soliton solutions for modified Kawahara equation arising in shallow water waves using novel techniques

The modified Kawahara equation also called Korteweg-de Vries (KdV) equation of fifth-order arises in shallow water wave and capillary gravity water waves. This study is based on the generalized Riccati equation mapping and modified the F-expansion methods. Several types of solitons such as Bright soliton, Dark-lump soliton, combined bright dark solitary waves, have been derived for the modified Kawahara equation. The obtained solutions have significant applications in applied physics and engineering. Moreover, stability of the problem is presented after being examined through linear stability analysis that justify that all solutions are stable. We also present some solution graphically in 3D and 2D that gives easy understanding about physical explanation of the modified Kawahara equation. The calculated work and achieved outcomes depict the power of the present methods. Furthermore, we can solve various other nonlinear problems with the help of simple and effective techniques.

A conditional local existence result for the generalized nonlinear Kawahara equation

In this paper, a generalized nonlinear Kawahara equation with time and space‐dependent coefficient is considered. We show that the construction of solution with a standard fixed point method can be accomplished so that the well‐posedness in for some is proved under a specific nonpositive differential condition. We introduce an adequate weight function to define the energy and a nondegenerate condition is assumed on the fifth dispersive coefficient.

The dynamics of new dual-mode Kawahara equation: interaction of dual-waves solutions and graphical analysis

In this paper, we introduce a new dual-mode Kawahara equation with new dissipative, nonlinearity and interaction phase velocity parameters. Also, we study the solutions of this model by using the tanh-scheme, Kudryashov-scheme and the sine–cosine function methods. Dynamics and shapes of the obtained solutions are illustrated by using graphical 2D and 3D plots. Finally, the interaction of the obtained dual-waves has been linked with the change of the phase-velocity parameter.

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.

Lipschitz stability of an inverse problem for the Kawahara equation with damping

The aim of this paper is to establish a stability result regarding the inverse problem of retrieving the damping coefficient in Kawahara equation. We first establish an internal Carleman estimate for the linearized problem with the help of Dirichlet-Neumann type boundary conditions. Using the obtained Carleman estimate and the regularity of solutions for the Kawahara equation, we prove the Lipschitz type stability and uniqueness of the considered inverse problems.

On the normalized ground states for the Kawahara equation and a fourth order NLS

We consider the Kawahara model and two fourth order semi-linear Schrodinger equations in any spatial dimension. We construct the corresponding normalized ground states, which we rigorously show to be spectrally stable. For the Kawahara model, our results provide a significant extension in parameter space of the current rigorous results. In fact, our results establish (modulo an additional technical assumption, which should be satisfied at least generically), spectral stability for all normalized waves constructed therein - in all dimensions, for all acceptable values of the parameters. This, combined with the results of [ 5 ], provides orbital stability, for all normalized waves enjoying the non-degeneracy property. The validity of the non-degeneracy property for generic waves remains an intriguing open question. At the same time, we verify and clarify recent numerical simulations of the spectral stability of these solitons. For the fourth order NLS models, we improve upon recent results on spectral stability of very special, explicit solutions in the one dimensional case. Our multidimensional results for fourth order anisotropic NLS seem to be the first of its kind. Of particular interest is a new paradigm that we discover herein. Namely, all else being equal, the form of the second order derivatives (mixed second derivatives vs. pure Laplacian) has implications on the range of existence and stability of the normalized waves.

Some exact solutions of (1+1)-dimensional Kaup-system and seventh-order Kawahara equation

Some exact solutions of (1+1)-dimensional Kaup-system and seventh-order Kawahara equation