Kortweg-de Vries Research Articles

Mathematical modeling and component generalization of (2+1)-dimensional Schwarz–Kortweg–de Vries model in shallow water waves

This work employs the sub-ODE method to compute new soliton solutions for the component generalization of [Formula: see text]-dimensional Schwarz–Kortweg–de Vries (SKdV) equation in shallow water waves. The rational soliton solutions (RSS), Jacobian elliptic function (JEF), periodic solutions (PS), hyperbolic function solutions (HFS), positive solitons, dark soliton solutions (DSS) and bright soliton solutions (BSS) are constructed with this technique. Solitons-like solutions, HFS and trigonometric-function solutions (TFS) are also found as the JEF’s parameter m approaches values of 1 and 0, respectively. The suggested approach for the nonlinear equations in mathematical physics is concise, simple, effective, and promising.

Nonlinear ion acoustic waves in dense magnetoplasmas: Analyzing interaction solutions of the KdV equation using Wronskian formalism for electron trapping with Landau diamagnetism and thermal excitations

Nonlinear ion acoustic waves in the presence of electron trapping with Landau quantization and thermal excitations induced smearing effects of the Fermi step function are investigated using the two-fluid theory in the vicinity of white dwarfs. In spite of the fact that the Kortweg de Vries (KdV) equation is derived within the confines of small amplitude approximation used in the reductive perturbation theory (RPT), it is found to admit solutions that exhibit outstanding capability of wave enhancement and have applications in the nonlinear wave propagation. It is found that enhancing the quantizing magnetic field and the electron thermal effects modify the spatial scale of the formation of the novel nonlinear structures reported for our model. Interestingly, it is found that, unlike the solitary and periodic structures, the choice of space and time coordinates affects the structure of these solutions. Interaction of these solutions with a stable nonlinear structure are also studied.

Solitons and lumps in the cylindrical Kadomtsev-Petviashvili equation. I. Axisymmetric solitons and their stability.

We revise soliton and lump solutions described by the cylindrical Kadomtsev-Petviashvili (cKP) equation and construct new exact solutions relevant to physical observation. In the first part of this study, we consider basically axisymmetric waves described by the cylindrical Kortweg-de Vries equation and analyze approximate and exact solutions to this equation. Then, we consider the stability of the axisymmetric solitons with respect to the azimuthal perturbations and suggest a criterion of soliton instability. The results of our numerical modeling confirm the suggested criterion and reveal lump emergence in the course of the development of the modulation instability of ring solitons in the unstable case. In the next part of this study, which will follow shortly, we will present exact solutions to the cKP equation describing weakly nonlinear waves in media with positive dispersion subject to the modulation instability of solitons with respect to small azimuthal perturbations.

On the Cauchy problem of defocusing mKdV equation with finite density initial data: Long time asymptotics in soliton-less regions

We investigate the long time asymptotics for the solutions to the Cauchy problem of defocusing modified Kortweg-de Vries (mKdV) equation with finite density initial data. The present paper is the subsequent work of our previous paper [arXiv:2108.03650], which gives the soliton resolution for the defocusing mKdV equation in the central asymptotic sector {(x,t):|ξ|<6} with ξ:=x/t. In the present paper, via the Riemann-Hilbert (RH) problem associated to the Cauchy problem, the long-time asymptotics in the soliton-less regions {(x,t):|ξ|>6,|ξ|=O(1)} for the defocusing mKdV equation are further obtained. It is shown that the leading term of the asymptotics is in compatible with the “background solution” and the error terms are derived via rigorous analysis.

Ion temperature gradient mode linear and nonlinear structures in electron-ion plasma, with stationary charged dust grain

Ion temperature gradient mode nonlinear structures are studied with stationary charged dust. The dynamical role here is played by heavy ions. The small amplitude limit influence of various plasma parameters shows that the phase velocity of the mode modifies the overall plasma dynamics. Further, from the solution of the model equations we obtain Kortweg-de-Vries (KdV) and Korteweg-de-Vries Burger (KdVB) like equations, the solution of which gives solitary and shock wave nonlinear structures. Numerical analysis of these nonlinear structures shows that dust number density and polarity have a substantial impact on these structures. The present observation may be beneficial in space and laboratory plasma investigations.

An approximation of one-dimensional nonlinear Kortweg de Vries equation of order nine.

This research presents the approximate solution of nonlinear Korteweg-de Vries equation of order nine by a hybrid staggered one-dimensional Haar wavelet collocation method. In literature, the underlying equation is derived by generalizing the bilinear form of the standard nonlinear KdV equation. The highest order derivative is approximated by Haar series, whereas the lower order derivatives are attained by integration formula introduced by Chen and Hsiao in 1997. The findings are shown in the form of tables and a figure, demonstrating the proposed technique’s convergence, robustness, and ease of application in a small number of collocation points.

Abundant solitary wave solutions of Gardner’s equation using new [formula omitted]-model expansion method

The Gardner’s equation is the generalization of Kortweg-de Vries equation and modified Kortweg-de Vries equation having real life applications in different fields of science, such as, hydrodynamics, quantum field theory, etc. Moreover, it has applications in fluid mechanics where it explains surface and internal waves. This manuscript addresses application of the new ϕ6-model expansion method to Gardner’s equation for getting its solitary wave solutions. Such new results aid in getting noteworthy advances in various disciplines of science and technology. Periodic, kink, anti-kink, bright, dark, w-shaped and singular soliton solutions of the governing equation have been constructed. The proposed method takes into account a variety of closed form solutions using the Jacobi elliptic functions. The graphical illustration of some of the obtained solutions has been presented for a suitable choice of free parameters to depict the dynamic nature of the obtained wave solutions.

Characteristics of supernonlinear and coexistence features for electron-acoustic waves in an adiabatic quantum plasma

Bifurcation of quantum electron-acoustic waves (QEAWs) is studied in an adiabatic quantum plasma with nonextensively distributed hot electrons. Using reductive perturbation technique, modified Kortweg de Vries (KdV) equation is derived with dual power nonlinearity for highly nonlinear QEAWs. Using Galilean transformation the modified KdV equation is reduced to a planar dynamical system with three equilibrium points. Applying phase plane analysis periodic wave solutions and supernonlinear periodic wave solutions for QEAWs are perceived. It is found that the amplitude of the nonlinear structures is indirectly proportional to the ratio of number densities ( $$\mu $$ ), the cold to hot electron temperature ratio ( $$\sigma $$ ) and the nonextensivity parameter (q), we have explained physically that confronts our results. In addition, coexistence of superperiodic and quasiperiodic phenomena, quasiperiodic and chaotic phenomena and chaotic, superperiodic and quasiperiodic phenomena for QEAWs are observed for appropriate initial values.

On Monotonic Pattern in Periodic Boundary Solutions of Cylindrical and Spherical Kortweg–De Vries–Burgers Equations

We studied, for the Kortweg–de Vries–Burgers equations on cylindrical and spherical waves, the development of a regular profile starting from an equilibrium under a periodic perturbation at the boundary. The regular profile at the vicinity of perturbation looks like a periodical chain of shock fronts with decreasing amplitudes. Further on, shock fronts become decaying smooth quasi-periodic oscillations. After the oscillations cease, the wave develops as a monotonic convex wave, terminated by a head shock of a constant height and equal velocity. This velocity depends on integral characteristics of a boundary condition and on spatial dimensions. In this paper the explicit asymptotic formulas for the monotonic part, the head shock and a median of the oscillating part are found.

On the interaction of nonlinear ion acoustic solitary waves in non-ideal plasma incorporated with Cairns-Gurevich distributed electrons

A rigorous theoretical study of the head-on collision (HoC) of two ion acoustic solitons in unmagnetized non-ideal plasmas containing warm fluid ions and Cairns-Gurevich distributed electrons is presented. The extended Poincare-Lighthill-Kuo (PLK) method is employed and coupled modified Kortweg-de Vries (mKdV) equations describing the system are derived. The nonlinear evolution equations for the colliding solitons, the corresponding phase shifts and trajectories of the two solitary waves are calculated and analyzed numerically. It is found that the process of solitons collision strongly depends on the hybrid nonthermal trapped electron parameter. Shortly, it is detected that the collision phase shifts increase with the increment of the nonthermal parameter in the case of ideal plasma. Though, for ideal plasma case, the phase shifts are constant and larger than that of non-ideal case. The gained results could be useful in clarifying the requisite features of localized electrostatic disturbance in the Earth’s magnetosphere where such distribution exists.

Analysis of an extended two-lane lattice hydrodynamic model considering mixed traffic flow and self-stabilization effect

PurposeThe purpose of this paper is to explore the impact of the mixed traffic flow, self-stabilization effect and the lane changing behavior on traffic flow stability.Design/methodology/approachAn extended two-lane lattice hydrodynamic model considering mixed traffic flow and self-stabilization effect is proposed in this paper. Through linear analysis, the stability conditions of the extended model are derived. Then, the nonlinear analysis of the model is carried out by using the perturbation theory, the modified Kortweg–de Vries equation of the density of the blocking area is derived and the kink–antikink solution about the density is obtained. Furthermore, the results of theoretical analysis are verified by numerical simulation.FindingsThe results of numerical simulation show that the increase of the proportion of vehicles with larger maximum speed or larger safe headway in the mix flow are not conducive to the stability of traffic flow, while the self-stabilization effect and lane changing behavior is positive to the alleviation of traffic congestion.Research limitations/implicationsThis paper does not take into account the factors such as curve and slope in the actual road environment, which will have more or less influence on the stability of traffic flow, so there is still a certain gap with the real traffic environment.Originality/valueThe existing two-lane lattice hydrodynamic models are rarely discussed in the case of mixed traffic flow. The improved model proposed in this paper can better reflect the actual traffic, which can also provide a theoretical reference for the actual traffic governance.

Numerical Simulation of Modified Kortweg-de Vries Equation by Linearized Implicit Schemes

In this paper, we are going to derive four numerical methods for solving the Modified Kortweg-de Vries (MKdV) equation using fourth Pade approximation for space direction and Crank Nicolson in the time direction. Two nonlinear schemes and two linearized schemes are presented. All resulting schemes will be analyzed for accuracy and stability. The exact solution and the conserved quantities are used to highlight the efficiency and the robustness of the proposed schemes. Interaction of two and three solitons will be also conducted. The numerical results show that the interaction behavior is elastic and the conserved quantities are conserved exactly, and this is a good indication of the reliability of the schemes which we derived. A comparison with some existing is presented as well.

Interactions between exotic multi-valued solitons of the (2+1)-dimensional Korteweg-de Vries equation describing shallow water wave

Korteweg-de Vries equation governs the weakly nonlinear long wave whose phase speed reaches a simple maximum of wave with the infinite length in shallow water wave. The exponential-form variable separation solution of (2+1)-dimensional Kortweg-de Vries equation is found via the two-function method, and this solution covers many special combined solutions including sinh-cosh,sin-cos,sech-tanh,csch-coth,sec-tan and csc-cot solutions. From the exponential-form solution with choosing suitable functions, inelastic interactions between special multi-valued solitons with two loops such as anti-bell-shaped, anti-peak-shaped semifoldons and anti-foldon are graphically and analytically studied. By the asymptotic analysis, phase shift and its difference during interactions between multi-valued solitons are analytically given.

On instability of solitons in the 2d cubic Zakharov–Kuznetsov equation

We consider the critical generalized Zakharov–Kuznetsov (ZK) equation, $$u_t + \partial _{x_1}(\Delta u + u^3) = 0, (x_1,x_2) \in {\mathbb {R}}^2$$ . In Farah et al. (Instability of solitons in the 2d cubic Zakharov–Kuznetsov equation, arXiv:1711.05907 , 2017), we proved that solitons are unstable for this equation following the strategy by Martel and Merle (GAFA Geom Funct Anal 11:74–123, 2001) in their study of the critical generalized Kortweg–de Vries equation. The main ingredient used in Farah et al. (Instability of solitons in the 2d cubic Zakharov–Kuznetsov equation, arXiv:1711.05907 , 2017) was the new pointwise decay estimates in two dimensions together with monotonicity properties of solutions. In this paper, we show that using only monotonicity properties and not relying on pointwise estimates, thus, greatly simplifying the approach, we can prove an instability of solitons, though a slightly weaker version.

An Analytical Technique to Solve the System of Nonlinear Fractional Partial Differential Equations

The Kortweg–de Vries equations play an important role to model different physical phenomena in nature. In this research article, we have investigated the analytical solution to system of nonlinear fractional Kortweg–de Vries, partial differential equations. The Caputo operator is used to define fractional derivatives. Some illustrative examples are considered to check the validity and accuracy of the proposed method. The obtained results have shown the best agreement with the exact solution for the problems. The solution graphs are in full support to confirm the authenticity of the present method.

Solitary kinetic Alfvén waves in dense plasmas with relativistic degenerate electrons and positrons

Propagation of solitary kinetic Alfvén waves (KAWs) is investigated in small but finite (particle-to-magnetic pressure ratio) collisionless dense plasma whose constituents are nondegenerate warm ions, and relativistic degenerate electrons and positrons. Through the use of reductive perturbation technique, Kortweg–de Vries equation is derived to obtain small amplitude localized wave solution of KAWs. The effects of plasma positron concentration, electron relativistic degeneracy parameter, ion thermal temperature and obliqueness parameter on solitary KAWs are studied. The results of this theoretical investigation are aimed at elucidating characteristics of kinetic Alfvén solitary waves in relativistic degenerate e–p–i plasmas found in dense astrophysical objects specifically neutron stars and white dwarfs.

Particle acceleration by ion-acoustic solitons in plasma

We propose a new acceleration mechanism for charged particles by using cylindrical or spherical non-linear acoustic waves propagating in ion-electron plasma. The acoustic wave, which is described by the cylindrical or spherical Kortweg-de Vries equation, grows in its wave height as the wave shrinks to the center. Charged particles confined by the electric potential accompanied with the shrinking wave get energy by repetition of reflections. We obtain power law spectrums of energy for accelerated particles. As an application, we discuss briefly that high energy particles coming from the Sun are produced by the present mechanism.

Green’s functions for higher order nonlinear equations

The well-known Green’s function method has been recently generalized to nonlinear second-order differential equations. In this paper, we study possibilities of exact Green’s function solutions of nonlinear differential equations of higher order. We show that, if the nonlinear term satisfies a generalized homogeneity property, then the nonlinear Green’s function can be represented in terms of the homogeneous solution. Specific examples and a numerical analysis support the advantage of the method. We show how, for the Bousinesq and Kortweg–de Vries equations, we are forced to introduce higher order Green’s functions to obtain the solution to the inhomogeneous equation. The method proves to work also in this case supporting our generalization that yields a closed form solution to a large class of nonlinear differential equations, providing also a formula easily amenable to numerical evaluation.

Numerical solution of the generalized, dissipative KdV–RLW–Rosenau equation with a compact method

The nonlinear dynamics of the one–dimensional, generalized Korteweg–de Vries–regularized–long wave–Rosenau (KdV–RLW–Rosenau) equation with second– and fourth–order dissipative terms subject to initial Gaussian conditions is analyzed numerically by means of three–point, fourth–order accurate, compact finite differences for the discretization of the spatial derivatives and a trapezoidal method for time integration. By means of a Fourier analysis and global integration techniques, it is shown that the signs of both the fourth–order dissipative and the mixed fifth–order derivative terms must be negative. It is also shown that an increase of either the linear drift or the nonlinear convection coefficients results in an increase of the steepness, amplitude and speed of the right–propagating wave, whereas the speed and amplitude of the wave decrease as the power of the nonlinearity is increased, if the amplitude of the initial Gaussian condition is equal to or less than one. It is also shown that the wave amplitude and speed decrease and the curvature of the wave’s trajectory increases as the coefficients of the second– and fourth–order dissipative terms are increased, while an increase of the RLW coefficient was found to decrease both the damping and the phase velocity, and generate oscillations behind the wave. For some values of the coefficients of both the fourth–order dissipative and the Rosenau terms, it has been found that localized dispersion shock waves may form in the leading part of the right–propagating wave, and that the formation of a train of solitary waves that result from the breakup of the initial Gaussian conditions only occurs in the absence of both Rosenau’s, Kortweg–de Vries’s and second– and fourth–order dissipative terms, and for some values of the amplitude and width of the initial condition and the RLW coefficient. It is also shown that negative values of the KdV term result in steeper, larger amplitude and faster waves and a train of oscillations behind the wave, whereas positive values of that coefficient may result in negative phase and group velocities, no wave breakup and oscillations ahead of the right–propagating wave.

Wave Propagation Analysis in a One – Dimensional Lattice with Long Range Interactions and Its Continuum Model

Fermi-Pasta-Ulam (FPU) lattice known as a nonlinear lattice includes nonlinear terms explicitly. Therefore, we can evaluate nonlinear effects of FPU lattice straightforwardly. It includes, however, only the nearest neighbor interactions. In some cases, long range interaction plays an important role in dynamical properties in real crystals. We take the second nearest neighbor interactions into account in FPU lattice. We derive the nonlinear wave equation (modified Kortweg-de Vries – type equation) from the nonlocal FPU lattice as a continuum limit. Linear dispersion relationship of the wave equation is discussed. We also find as exact solution of soliton in the nonlinear wave equation analytically. We reveal that the velocity of the soliton is related to the second nearest neighbor interactions. We analytically evaluate the temporal evolution of the soliton and estimate the spectrum of the solitary wave using two dimensional FFT in case of different interactions. The result shows that the second nearest neighbor interactions affects the appearance of the lean of peak.