Perturbed KdV Equation Research Articles

Approximate conservation laws of perturbed partial differential equations

This paper presents a general result on approximate conservation laws of perturbed partial differential equations. A method of constructing approximate conservation laws to systems of perturbed partial differential equations is given, which is based on approximate Noether symmetries of approximate and standard adjoint systems of the original system. The relationship between the Noether symmetry operators of approximate and standard adjoint system is established. As a result, the approach is applied to the perturbed wave equation and the perturbed KdV equation.

Multiple-soliton solutions of the perturbed KdV equation

The perturbed KdV equation through O ( ε ) is investigated. The Hirota’s bilinear method is mainly used in this work. The study highlights the multiple-soliton solutions and the multiple singular soliton solutions of the perturbed KdV equation.

Renormalization Group Method for Soliton Evolution in a Perturbed KdV Equation

We show that the conventional renormalization group method can be used to give an analytic description of the evolution of a soliton in a perturbed KdV equation. The renormalization group equations describe the deformation of the soliton as the effect of perturbations. The method is concise and easy to understand and the results obtained agree with the results of other approaches.

Two-component description of dynamical systems that can be approximated by solitons: The case of the ion acoustic wave equations of plasma physics

A new approach to the perturbative analysis of dynamical systems, which can be described approximately by soliton solutions of integrable non-linear wave equations, is employed in the case of small-amplitude solutions of the ion acoustic wave equations of plasma physics. Instead of pursuing the traditional derivation of a perturbed KdV equation, the ion velocity is written as a sum of two components: elastic and inelastic. In the single-soliton case, the elastic component is the full solution. In the multiple-soliton case, it is complemented by the inelastic component. The original system is transformed into two evolution equations: An asymptotically integrable Normal Form for ordinary KdV solitons, and an equation for the inelastic component. The zero-order term of the elastic component is a single-soliton or multiple-soliton solution of the Normal Form. The inelastic component asymptotes into a linear combination of single-soliton solutions of the Normal Form, with amplitudes determined by soliton interactions, plus a second-order decaying dispersive wave. Satisfaction of a conservation law by the inelastic component and of mass conservation by the disturbance to the ion density is determined solely by the initial data and/or boundary conditions imposed on the inelastic component. The electrostatic potential is a first-order quantity. It is affected by the inelastic component only in second order. The charge density displays a triple-layer structure. The analysis is carried out through the third order.

Waves in an elastic tube filled with a heterogeneous fluid of variable viscosity

By treating the artery as a prestressed thin elastic tube and the blood as an incompressible heterogeneous fluid with variable viscosity, we studied the propagation of weakly non-linear waves in such a composite medium through the use of reductive perturbation method. By assuming a variable density and a variable viscosity for blood in the radial direction we obtained the perturbed Korteweg–deVries equation as the evolution equation when the viscosity is of order of ε 3 / 2 . We observed that the perturbed character is the combined result of the viscosity and the heterogeneity of the blood. A progressive wave type of solution is presented for the evolution equation and the result is discussed. The numerical results indicate that for a certain value of the density parameter sigma , the wave equation loses its dispersive character and the evolution equation degenerates. It is further shown that, for the perturbed KdV equation both the amplitude and the wave speed decay in the time parameter τ .

From single- to multiple-soliton solutions of the perturbed KdV equation

The solution of the perturbed KdV equation (PKDVE), when the zero-order approximation is a multiple-soliton wave, is constructed as a sum of two components: elastic and inelastic. The elastic component preserves the elastic nature of soliton collisions. Its perturbation series is identical in structure to the series-solution of the PKDVE when the zero-order approximation is a single soliton. The inelastic component exists only in the multiple-soliton case, and emerges from the first order and onwards. Depending on initial data or boundary conditions, it may contain, in every order, a plethora of inelastic processes. Examples are given of sign-exchange soliton–anti-soliton scattering, soliton–anti-soliton creation or annihilation, soliton decay or merging, and inelastic soliton deflection. The analysis has been carried out through third order in the expansion parameter, exploiting the freedom in the expansion to its fullest extent. Both elastic and inelastic components do not modify soliton parameters beyond their values in the zero-order approximation. When the PKDVE is not asymptotically integrable, the new expansion scheme transforms it into a system of two equations: The Normal Form for ordinary KdV solitons, and an auxiliary equation describing the contribution of obstacles to asymptotic integrability to the inelastic component. Through the orders studied, the solution of the latter is a conserved quantity, which contains the dispersive wave that has been observed in previous works.

Spontaneously generated waves in perturbed evolution equations

The Burgers and KdV equations provide approximations to more complex dynamical systems. Both equations can be viewed as conservation laws and are integrable. When the small terms, left out in the derivation of the two equations from the original systems, are re-instated, the resulting perturbed equations may lose the conservation-law structure. In addition, integrability may be lost even in the asymptotic sense (i.e. in the order-by-order perturbation analysis). These losses are independent occurrences. However, they can both be traced to a specific contribution, which may exist in the original perturbation and/or may surface in the higher-order analysis. Identification of that contribution is based on the observation that, in the case of a single-wave (front, soliton) zero-order approximation to the solution, the conservation-law structure of the perturbed equation is preserved, and obstacles to asymptotic integrability do not emerge. We first identify the component in the perturbation which is responsible for the fact that the perturbed equation does not have a conservation-law structure. A driving term of an identical structure may emerge in the equations which determine higher-order contributions to the solution and lead to the loss of asymptotic integrability. That both properties are not lost in the case of a single-wave solution is a consequence of the fact that this component in the perturbation vanishes identically. This component can be viewed as a coupling between the solution one is seeking and a new wave that is generated spontaneously by the perturbation. In the case of the Burgers equation, the spontaneously generated wave is a solution of the equation but different from the solution one is seeking. In the case of the perturbed KdV equation, the new wave is constructed from non-KdV solitons.

Nonlinear Analysis in the Aw–Rascle Anticipation Model of Traffic Flow

In this paper, the Aw–Rascle anticipation (ARA) model is discussed from the perspective of the capability to reproduce nonlinear traffic flow behaviors observed in real traffic. For this purpose, a nonlinear traffic flow stability criterion is derived by using a wavefront expansion technique. The result of the nonlinear stability analysis can be used not only to judge the stability evolution of an initial traffic state but also to determine the pressure term in the ARA model. The KdV equation is derived from the ARA model added by the viscous term with the use of the reduction perturbation method. The soliton solution can be analytically obtained from the perturbed KdV equation only near the neutral stability line. Weighted essentially nonoscillatory schemes are employed to simulate the KdV soliton. The numerical results confirm the analytical KdV soliton solution.

Transformations from the unperturbed equation to the perturbed equation with nth order perturbation

In this paper, a new algebra method is developed to find the approximate relations between the perturbed nonlinear PDE with nth order perturbation and the unperturbed equation. Through this relations we can transform the unperturbed equation into the perturbed equation with nth order perturbation. This relations include the partial derivatives of the dependent variables with respect to the independent variables. The perturbed KdV equation with second order perturbation serves as example illustrating this fact.

Freedom in the expansion and obstacles to integrability in multiple-soliton solutions of the perturbed KdV equation

The construction of solutions of the perturbed KdV equation encounters obstacles to asymptotic integrability beyond the first order, when the zero-order approximation is not a single-soliton wave. In the standard analysis, the obstacles lead to the loss of integrability of the Normal Form, resulting in a zero-order term, which does not have the simple structure of the solution of the unperturbed equation. Exploiting the freedom in the perturbative expansion, an algorithm is proposed that shifts the effect of the obstacles from the Normal Form to the higher-order terms. The expansion has been carried out through third order. Through this order, the Normal Form remains integrable, and the zero-order approximation retains the structure of the unperturbed solution. For multiple-soliton solutions, the resulting obstacles decay exponentially away from the soliton-interaction region, which is a finite domain around the origin in the x – t plane. The effect is demonstrated in detail through second order for the two-soliton case, where the obstacles generate a second-order radiative tail that emanates from the origin, and decays exponentially away from the origin. These results suggest a new meaning to “asymptotic” integrability: The zero-order term is determined by an integrable Normal Form, and the higher-order terms in the expansion of the full solution tend to those of the integrable case (when no obstacles exist) asymptotically away from the origin.

New approximate solutions of the perturbed KdV equation

In this Letter, we apply the Lie–Bäcklund group of transformations to solve the perturbed nonlinear evaluation equation. This method is a generalization of Lie point transformation technique. We consider the perturbed KdV equation as an illustration of the approach. New approximate solutions of the perturbed KdV equation stemming from the exact solutions of the unperturbed equation are obtained using the Lie–Bäcklund group method. These approximate solutions of the perturbed KdV equation are the soliton-like solutions.

An extended method and its application to Whitham–Broer–Kaup equation and two-dimensional perturbed KdV equation

Abstract An extended method is proposed for constructing new forms of exact travelling wave solutions to nonlinear evolution equations by making a more general transformation. For illustration, we apply the extended method to Whitham–Broer–Kaup equation and two-dimensional perturbed KdV equation and successfully cover the previously known travelling wave solutions found by Chen’s method [Chaos, Solitons and Fractals 22 (2004) 675; Int. J. Mod. Phys. C 4 (2004) 595]. Moreover, some new forms of solutions are obtained.

KdV dynamics in the plasma-sheath transition

A mathematical model is formulated to catch the dynamics hidden in the plasma-sheath transition layer and the inner sheath layer for planar motion of a plasma. It is shown that the rescaled potential in the plasma-sheath transition layer and inner layer is governed by a perturbed KdV equation, through which some of the complex interactions and couplings among physical mechanisms acting in the plasma-sheath formation process are elucidated. This model is analyzed and compared with the model used by Fokas in his study of nonlinear dispersive initial boundary value problems [1].

Adomian Decomposition Method and Exact Solutions of the Perturbed KdV Equation**The project supported by Outstanding Youth Foundation under Grant No.19 925 522, Research Fund for Doctoral of Higher Education of China under Grant No. 2 000 024 832 and National Natural Science Foundation of Zhejiang Province of China

The Adomian decomposition method is used to solve the Cauchy problem of the perturbed KdV equation. Three types of exact solitary wave solutions are reobtained via the Adomian's approach by selecting the initial conditions appropriately.

TEMPORAL EVOLUTIONS AND STATIONARY WAVES FOR PERTURBED KDV EQUATION WITH NONLOCAL TERM

Initial value problems as well as stationary solitary and periodic waves are investigated for a perturbed KdV equation including the Hilbert transform; ut + uux + βuxxx + η(ℋux - uxx) = 0 (η > 0). Multi-hump stationary solitary and periodic wave solutions are numerically identified. Furthermore, the close relation between the structure of the stationary waves and the behavior of the temporal evolutions is discussed in comparison with other perturbed KdV equations with different instability and dissipation terms. The results support some general features common to this type of nonlinear evolution equations.

A new algebraic method for finding the line soliton solutions and doubly periodic wave solution to a two-dimensional perturbed KdV equation

A new algebraic method is devised to obtain a series of exact solutions for general nonlinear equations. Compared with the most existing tanh methods, the proposed method gives new and more general solutions. More importantly, the method provides a guideline to classify the various types of the solution according to some parameters. For illustration, we apply the method to solve a new two-dimensional perturbed KdV equation and successfully construct the various kind of exact solutions including line soliton solutions, rational solutions, triangular periodic solutions, Jacobi, and Weierstrass doubly periodic solutions.

Periodic travelling waves and their modulation

A family of periodic travelling wave solutions parameterized by the wavenumber is shown to bifurcate from the trivial solution in a perturbed KdV equation. Studying linearized eigenvalue problem about each periodic travelling wave solution, all of them are shown to be unstable immediately after the bifurcation in contrast to the Eckhaus stability/instability. Analysis from a dynamical viewpoint suggests that “modulated periodic waves” are obtained by a secondary bifurcation from periodic travelling waves as a super critical Hopf bifurcation.

A solitary-wave solution to a perturbed KdV equation

We derive the approximate form and speed of a solitary-wave solution to a perturbed KdV equation. Using a conventional perturbation expansion, one can derive a first-order correction to the solitary-wave speed, but at the next order, algebraically secular terms appear, which produce divergences that render the solution unphysical. These terms must be treated by a regrouping procedure developed by us previously. In this way, higher-order corrections to the speed are obtained, along with a form of solution that is bounded in space. For this particular perturbed KdV equation, it is found that there is only one possible solitary wave that has a form similar to the unperturbed soliton solution.

Localized finite-amplitude disturbances and selection of solitary waves

It turns out that evolution of localized finite-amplitude disturbances in perturbed KdV equation is qualitatively different compared with conventional small-amplitude initial conditions. Namely, relatively fast solitary waves, with one and the same amplitude and velocity, are formed ahead of conventional chaotic-like irregular structures. The amplitude and velocity of the waves, obtained from the asymptotic theory, are in excellent agreement with numerics.

On Integrability of a (2+1)-Dimensional Perturbed KdV Equation

A (2+1)-dimensional perturbed KdV equation, recently introduced by W.X. Ma and B. Fuchssteiner, is proven to pass the Painlevé; test for integrability well, and its 4×4 Lax pair with two spectral parameters is found. The results show that the Painlevé; classification of coupled KdV equations by A. Karasu should be revised.