Perturbed Korteweg-de Vries Equation Research Articles

Solitary Waves of the Perturbed KdV Equation with Nonlocal Effects

In this paper, the Korteweg–de Vries (KdV) equation is considered, which is a shallow water wave model in fluid mechanic fields. First the existence of solitary wave solutions for the original KdV equation and geometric singular perturbation theory are recalled. Then the existence of solitary wave solutions is established for the equation with two types of delay convolution kernels by using the method of dynamical system, especially the geometric singular perturbation theory, invariant manifold theory and Melnikov method. Finally, the asymptotic behaviors of solitary wave solution are discussed by applying the asymptotic theory. Moreover, an interesting result is found for the equation without backward diffusion effect, there is no solitary wave solution in the case of local delay, but there is a solitary wave solution in the case of nonlocal delay.

Analysis of interaction of lump solutions with kink-soliton solutions of the generalized perturbed KdV equation using Hirota-bilinear approach

Using the Hirota bilinear method (HBM) and Cole-Hopf transformation, we extract the interaction between the lump and kink-soliton solutions for the generalized perturbed Korteweg–de Vries (KdV) equation in this manuscript. In the literature [15], the authors have applied the Cole-Hopf transformation with a value of R to study the particular case of the generalized perturbed KdV equation, which is found to be an inappropriate value. Alqarni et al., computed an appropriate value of R in the literature [29]. We use the correct value of R and get more clearly the outcomes for a lump-single kink soliton, a lump-two kink soliton, and a lump-periodic soliton solution. We display these solutions in 2D and 3D space using Mathematica by selecting the proper values for the parameters.

Solitary-wave dynamics in an [formula omitted]LC nonlinear transmission line with voltage bias

It was recently established that when an LC transmission line with a Schottky varactor is connected to a voltage terminal, a pulse is formed whose amplitude grows exponentially leading eventually to its instability after some finite propagation time. In the present study an R2LC nonlinear transmission line, consisting of a lumped circuit with unit cells composed of a linear inductance and a linear resistance in the primary branch, and a Schottky varactor shunted with a linear resistance in the secondary branch, all driven by a voltage terminal, is examined. Our objective is to investigate the effects of the competition between the two resistances, on the soliton amplification and propagation characteristics in general, along the nonlinear transmission line. The study is carried out in two steps: first, an analytical study is carried out within the framework of the reductive perturbation theory. This first step leads to a perturbed Korteweg–de Vries equation, in which the two resistances as well as the voltage terminal are regarded as perturbations. It is found that the electrical pulse amplitude is either exponentially amplified or exponentially depressed, depending on a specific relationship between the two resistances and the voltage terminal. Next the discrete line equations are directly solved numerically, assuming the analytically obtained pulse as input signal. This second analysis suggests a complex behaviour of the electrical pulse amplitude, in response to the influence of the two resistances. A particularly interesting behaviour observed in numerical simulations is the decay of a single-pulse soliton with propagation, and its burst into two-pulse and eventually multi-pulse soliton patterns after some finite propagation time. The proposed nonlinear transmission-line model can find widespread applications in complex transmission networks requiring high-power signals, as for instance in wideband and microwave digital signals, wireless, radar and sensor array processings.

Approximate Symmetries Analysis and Conservation Laws Corresponding to Perturbed Korteweg–de Vries Equation

The Korteweg–de Vries (KdV) equation is a weakly nonlinear third-order differential equation which models and governs the evolution of fixed wave structures. This paper presents the analysis of the approximate symmetries along with conservation laws corresponding to the perturbed KdV equation for different classes of the perturbed function. Partial Lagrange method is used to obtain the approximate symmetries and their corresponding conservation laws of the KdV equation. The purpose of this study is to find particular perturbation (function) for which the number of approximate symmetries of perturbed KdV equation is greater than the number of symmetries of KdV equation so that explore something hidden in the system.

Solitary Waves and Periodic Waves in a Perturbed KdV Equation

In this paper, we consider a perturbed Korteweg–de Vries (KdV) equation with weak dissipation and Marangoni effects. Main attention is focused on the existence conditions of periodic and solitary wave solutions of the perturbed KdV equation. Based on bifurcation theory of dynamic system and geometric singular perturbation method, the parameter conditions and wave speed conditions for the existence of one periodic solution, one solitary solution and the coexistence of a solitary solution and infinite number of periodic solutions are given. By using Chebyshev criterion to analyze the ratio of Abelian integrals, the monotonicity of wave speed is proved, and the upper and lower bounds of wave speed are obtained.

Multiple Soliton Solutions of the Sawada-Kotera Equation with a Nonvanishing Boundary Condition and the Perturbed Korteweg de Vries Equation by Using the Multiple Exp-Function Scheme

The Sawada-Kotera equation with a nonvanishing boundary condition, which models the evolution of steeper waves of shorter wavelength than those depicted by the Korteweg de Vries equation, is analyzed and also the perturbed Korteweg de Vries (pKdV) equation. For this goal, a capable method known as the multiple exp-function scheme (MEFS) is formally utilized to derive the multiple soliton solutions of the models. The MEFS as a generalization of Hirota’s perturbation method actually suggests a systematic technique to handle nonlinear evolution equations (NLEEs).

Nonplanar dust-acoustic waves and chaotic motions in Thomas Fermi dusty plasmas

The properties of linear and nonlinear nonplanar dust acoustic (DA) solitary waves and chaotic behavior are investigated in an unmagnetized Thomas Fermi dusty plasma, whose components are degenerate electrons, ions, and negatively charged inertial cold dust grains. A linear dispersion relation is obtained and solved numerically. It has been observed that linear excitation characteristics are influenced by radial distance r, geometric term ν, and ion-to-electron Fermi temperature ratio σi. We have also noted that the addition of a geometrical term in dispersion relation gives damping along the radial axis. A modified Korteweg-de Vries (KdV) equation is derived by employing the reductive perturbation technique, and its numerical solutions are obtained. The modified KdV equation is discussed for cold dust grains in planar and nonplanar frameworks. Upon the introduction of external periodic perturbation, the perturbed modified KdV equation is studied in planar geometry via some qualitative and quantitative approaches. The perturbed KdV equation can give rise to the periodic, quasiperiodic, and chaotic motions for DA waves. The strength of the external perturbation and dust concentration h play the major role of the switching parameter in the transition of dynamic motion. The developed chaos can be weakened with the variation of dust concentration h. It has been observed that the dust concentration affects the dynamics of DA waves in planar geometry which is an important observation in this study.

On shoaling of solitary waves

One of the classic analytical predictions of shoaling-wave amplification is Green’s law – the wave amplitude grows proportional to $h^{-1/4}$, where $h$ is the local water depth. Green’s law is valid for linear shallow-water waves unidirectionally propagating in a gradually varying water depth. On the other hand, conservation of mechanical energy shows that the shoaling-wave amplitude of a solitary wave grows like $a\propto h^{-1}$, if the waveform maintains its solitary-wave identity. Nonetheless, some recent laboratory and field measurements indicate that growth of long waves during shoaling is slower than what is predicted by Green’s law. Obvious missing factors in Green’s law are the nonlinearity and frequency-dispersion effects as well as wave reflection from the beach, whereas the adiabatic shoaling process does not recognize the transformation of the waveform on a beach of finite slope and length. Here we first examine this problem analytically based on the variable-coefficient perturbed Korteweg–de Vries equation. Three analytical solutions for different limits are obtained: (1) Green’s law for the linear and non-dispersive limit, (2) the slower amplitude growth rate for the linear and dispersive limit, as well as (3) nonlinear and non-dispersive limit. Then, in order to characterize the shoaling behaviours for a variety of incident wave and beach conditions, we implement a fifth-order pseudo-spectral numerical model for the full water-wave Euler theory. We found that Green’s law is not the norm but is limited to small incident-wave amplitudes when the wavelength is still small in comparison to the beach length scale. In general, the wave amplification rate during shoaling does not follow a power law. When the incident wave is finite, the shoaling amplification becomes faster than that of Green’s law when the ratio of the wavelength to the beach length is small, but becomes slower when the length ratio increases. We also found that the incident wave starts to amplify prior to its crest arriving at the beach toe due to the wave reflection. Other prominent characteristics and behaviours of solitary-wave shoaling are discussed.

The third-order perturbed Korteweg-de Vries equation for shallow water waves with a non-flat bottom

The goal of this work is to investigate, analytically and numerically, the dynamics of gravity water waves with the effects of the small surface tension and the bottom topography taken into account. Using a third-order perturbative approach of the Boussinesq equation, we obtain a new third-order perturbed Korteweg-de Vries (KdV) equation which includes nonlinear, dispersive, nonlocal and mixed nonlinear-dispersive terms, describing shallow water waves with a non-flat bottom and the surface tension. We show by numerical simulations, for various bottom shapes, that this new third-order perturbed KdV equation can support the propagation of solitary waves, whose profiles strongly depend on the surface tension. In particular, we show that the instability observed in the numerical simulation can be suppressed by the inclusion of small surface tension.

Multiple and exact soliton solutions of the perturbed Korteweg-de Vries equation of long surface waves in a convective fluid via Painlevé analysis, factorization, and simplest equation methods.

In this research, the surface waves of a horizontal fluid layer open to air under gravity field and vertical temperature gradient effects are studied. The governing equations of this model are reformulated and converted to a nonlinear evolution equation, the perturbed Korteweg-de Vries (pKdV) equation. We investigate the latter equation, which includes dispersion, diffusion, and instability effects, in order to examine the evolution of long surface waves in a convective fluid. Dispersion relation of the pKdV equation and its properties are discussed. The Painlevé analysis is applied not only to check the integrability of the pKdV equation but also to establish the Bäcklund transformation form. In addition, traveling wave solutions and a general form of the multiple-soliton solutions of the pKdV equation are obtained via Bäcklund transformation, the simplest equation method using Bernoulli, Riccati, and Burgers' equations as simplest equations, and the factorization method.

KdV cnoidal waves in a traffic flow model with periodic boundaries

An optimal-velocity (OV) model describes car motion on a single lane road. In particular, near to the boundary signifying the onset of traffic jams, this model reduces to a perturbed Korteweg–de Vries (KdV) equation using asymptotic analysis. Previously, the KdV soliton solution has then been found and compared to numerical results (see Muramatsu and Nagatani [1]). Here, we instead apply modulation theory to this perturbed KdV equation to obtain at leading order, the modulated cnoidal wave solution. At the next order, the Whitham equations are derived, which have been modified due to the equation perturbation terms. Next, from this modulation system, a family of spatially periodic cnoidal waves are identified that characterise vehicle headway distance. Then, for this set of solutions, we establish the relationship between the wave speed, the modulation term and the driver sensitivity. This analysis is confirmed with comparisons to numerical solutions of the OV model. As well, the long-time behaviour of these solutions is investigated.

Whitham theory for perturbed Korteweg–de Vries equation

Original Whitham’s method of derivation of modulation equations is applied to systems whose dynamics is described by a perturbed Korteweg–de Vries equation. Two situations are distinguished: (i) the perturbation leads to appearance of right-hand sides in the modulation equations so that they become non-uniform; (ii) the perturbation leads to modification of the matrix of Whitham velocities. General form of Whitham modulation equations is obtained in both cases. The essential difference between them is illustrated by an example of so-called ‘generalized Korteweg–de Vries equation’. Method of finding steady-state solutions of perturbed Whitham equations in the case of dissipative perturbations is considered.

Asymptotic Multiphase Solitonlike Solutions of the Cauchy Problem for a Singularly Perturbed Korteweg–de-Vries Equation with Variable Coefficients

UDC 517.9 We describe the set of initial conditions under which the Cauchy problem for a singularly perturbed Korteweg–de-Vries equation with variable coefficients has an asymptotic multiphase solitonlike solution. The notion of manifold of initial values for which the above-mentioned solution exists is proposed for the analyzed Cauchy problem. The statements on the estimation of the difference between the exact and constructed asymptotic solutions are proved for the Cauchy problem.

Modulation theory for the steady forced KdV–Burgers equation and the construction of periodic solutions

We present a multiple-scale perturbation technique for deriving asymptotic solutions to the steady Korteweg–de Vries (KdV) equation, perturbed by external sinusoidal forcing and Burger’s damping term, which models the near resonant forcing of shallow water in a container. The first order solution in the perturbation hierarchy is the modulated cnoidal wave equation. Using the second order equation in the hierarchy, a system of differential equations is found describing the slowly varying properties of the cnoidal wave. We analyse the fixed point solutions of this system, which correspond to periodic solutions to the perturbed KdV equation. These solutions are then compared to the experimental results of Chester and Bones (1968).

A note on the progressive wave solution of the perturbed Korteweg–deVries equation with variable dissipation

In this note, utilizing the method presented in Engelbrecht (1991) and Demiray (2002) we have studied the propagation of progressive waves in perturbed KdV equation with variable dissipation coefficient. The cases of constant and variable dissipation coefficient are studied separately and the results are compared with each other. It is seen that for both cases the amplitude and wave speed decrease with increasing time.

Analytical and numerical studying of the perturbed Korteweg-de Vries equation

The perturbed Korteweg-de Vries equation is considered. This equation is used for the description of one-dimensional viscous gas dynamics, nonlinear waves in a liquid with gas bubbles and nonlinear acoustic waves. The integrability of this equation is investigated using the Painlevé approach. The condition on parameters for the integrability of the perturbed Korteweg-de Vries equation equation is established. New classical and nonclassical symmetries admitted by this equation are found. All corresponding symmetry reductions are obtained. New exact solutions of these reductions are constructed. They are expressed via trigonometric and Airy functions. Stability of the exact solutions of the perturbed Korteweg-de Vries equation is investigated numerically.

Asymptotic multiphase Σ-solutions to the singularly perturbed Korteweg–de Vries equation with variable coefficients

The paper deals with constructing the asymptotic solution to the singularly perturbed Korteweg–de Vries equation with variable coefficients. The notion of an asymptotic multiphase Σ-solution is proposed, and an algorithm of its construction in a neighborhood of the point t = 0 is given. Some theorems concerning the exactness, with which such local asymptotic solution satisfies the equation under study are proved.

Two-Phase Solitonlike Solutions of the Cauchy Problem for a Singularly Perturbed Korteweg-De-Vries Equation with Variable Coefficients

We describe a set of initial conditions for which the Cauchy problem for a singularly perturbed Korteweg–de-Vries equation with variable coefficients has an asymptotic two-phase solitonlike solution. The notion of the manifold of initial data of the Cauchy problem for which this solution exists is proposed.

Asymptotic m-phase soliton-type solutions of a singularly perturbed Korteweg–de Vries equation with variable coefficients. II

We propose an algorithm for the construction of asymptotic two-phase solitonlike solutions of the Korteweg-de Vries equation with a small parameter at the higher derivative.

Asymptotic m-phase soliton-type solutions of a singularly perturbed Korteweg–de Vries equation with variable coefficients

Asymptotic m-phase soliton-type solutions of a singularly perturbed Korteweg–de Vries equation with variable coefficients