Numerical algorithms for solutions of Korteweg–de Vries equation

The nonlinear Korteweg–de Vries (KdVE) equation is solved numerically using both Lagrange polynomials based differential quadrature and cosine expansion‐based differential quadrature methods. The first test example is travelling single solitary wave solution of KdVE and the second test example is interaction of two solitary waves, whereas the other three examples are wave production from solitary waves. Maximum error norm and root mean square error norm are computed, and numerical comparison with some earlier works is done for the first two examples, the lowest four conserved quantities are computed for all test examples. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

On the uniform decay for the Korteweg–de Vries equation with weak damping

The aim of this work is to consider the Korteweg–de Vries equation in a finite interval with a very weak localized dissipation namely the H^?1‐norm. Our main result says that the total energy decays locally uniform at an exponential rate. Our analysis improves earlier works on the subject (Q. Appl. Math. 2002; LX (1):111–129; ESAIM Control Optim. Calculus Variations 2005; 11 (3):473–486) and gives a satisfactory answer to a problem suggested in (Q. Appl. Math. 2002; LX (1):111–129). Copyright © 2007 John Wiley & Sons, Ltd.     

Discrete artificial boundary conditions for the linearized Korteweg–de Vries equation

We consider the derivation of continuous and fully discrete artificial boundary conditions for the linearized Korteweg–de Vries equation. We show that we can obtain them for any constant velocities and any dispersion. The discrete artificial boundary conditions are provided for two different numerical schemes. In both continuous and discrete case, the boundary conditions are nonlocal with respect to time variable. We propose fast evaluations of discrete convolutions. We present various numerical tests which show the effectiveness of the artificial boundary conditions.© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1455–1484, 2016     

Solving unsteady Korteweg–de Vries equation and its two alternatives

This paper aims to present the generalized Kudryashov method to find the exact traveling wave solutions transmutable to the solitary wave solutions of the ubiquitous unsteady Korteweg–de Vries equation and its two famed alternatives, namely, the regularized long‐wave equation and the time regularized long‐wave equation. The exact analytic solutions of the studied equations are constructed explicitly in three forms, namely, hyperbolic, trigonometric, and rational function. The validity of our solutions is verified with MAPLE by putting them back into the original equation and found correct. Moreover, it has shown that the generalized Kudryashov method is an easy and reliable technique over the existing methods. Copyright © 2016 John Wiley & Sons, Ltd.     

Complexiton solutions of the Korteweg–de Vries equation with self-consistent sources

Complexiton solutions to the Korteweg–de Vires equation with self-consistent sources are presented. The basic technique adopted is the Darboux transformation. The resulting solutions provide evidence that soliton equations with self-consistent sources can have complexiton solutions, in addition to soliton, positon and negaton solutions. This also implies that soliton equations with self-consistent sources possess some kind of analytical characteristics that linear differential equations possess and brings ideas toward classification of exact explicit solutions of nonlinear integrable differential equations.     

Bell polynomial approach to an extended Korteweg–de Vries equation

Under investigation in this paper is an extended Korteweg–de Vries equation. Via Bell polynomial approach and symbolic computation, this equation is transformed into two kinds of bilinear equations by choosing different coefficients, namely KdV–SK‐type equation and KdV–Lax‐type equation. On the one hand, N‐soliton solutions, bilinear Bäcklund transformation, Lax pair, Darboux covariant Lax pair, and infinite conservation laws of the KdV–Lax‐type equation are constructed. On the other hand, on the basis of Hirota bilinear method and Riemann theta function, quasiperiodic wave solution of the KdV–SK‐type equation is also presented, and the exact relation between the one periodic wave solution and the one soliton solution is established. It is rigorously shown that the one periodic wave solution tend to the one soliton solution under a small amplitude limit. Copyright © 2013 John Wiley & Sons, Ltd.     

Initial conditions for the cylindrical Korteweg‐de Vries equation

In this paper, we find suitable initial conditions for the cylindrical Korteweg‐de Vries equation by first solving exactly the initial‐value problem for localized solutions of the underlying axisymmetric linear long‐wave equation. The far‐field limit of the solution of this linear problem then provides, through matching, an initial condition for the cylindrical Korteweg‐de Vries equation. This initial condition is associated only with the leading wave front of the far‐field limit of the linear solution. The main motivation is to resolve the discrepancy between the exact mass conservation law, and the “mass” conservation law for the cylindrical Korteweg‐de Vries equation. The outcome is that in the linear initial‐value problem all the mass is carried behind the wave front, and then the “mass” in the initial condition for the cylindrical Korteweg‐de Vries equation is zero. Hence, the evolving solution in the cylindrical Korteweg‐de Vries equation has zero “mass.” This situation arises because, unlike the well‐known unidirectional Korteweg‐de Vries equation, the solution of the initial‐value problem for the axisymmetric linear long‐wave problem contains both outgoing and ingoing waves, but in the cylindrical geometry, the latter are reflected at the origin into outgoing waves, and eventually the total outgoing solution is a combination of these and those initially generated.     

Exponential finite-difference method applied to Korteweg–de Vries equation for small times

The Korteweg–de Vries equation is numerically solved by using the exponential finite-difference technique. The accuracy of computed solutions is examined by comparison with other numerical and analytical solutions using two examples. The close results agreement between the current results and the exact solutions confirms that the proposed finite-difference procedure is an effective technique for the solution of the Korteweg–de Vries equation at the small times.     

New conservative finite volume element schemes for the modified Korteweg–de Vries equation

In this paper, three conservative finite volume element schemes are proposed and compared for the modif ied Korteweg–de Vries equation, especially with regard to their accuracy and conservative properties. The schemes are constructed basing on the discrete variational derivative method and the finite volume element method to inherit the properties of the original equation. The theoretical analysis show that three schemes are conservative under suitable boundary conditions as well as unconditionally linear stability. Numerical experiments are given to confirm the theoretical results and the capacity of the proposed methods for capturing the solitary wave phenomena. Copyright © 2016 John Wiley & Sons, Ltd.     

On the exact solutions,lie symmetry analysis,and conservation laws of Schamel–Korteweg–de Vries equation

In this work, we study the integrability aspects of the Schamel–Korteweg–de Vries equation that play an important role in studying the effect of electron trapping on the nonlinear interaction of ion‐acoustic waves by including a quasi‐potential. Lie symmetry analysis together with the simplest equation method and Kudryashov method is used to obtain exact traveling wave solutions for this equation. In addition, conservation laws are constructed using two different techniques, namely, the multiplier method and the new conservation theorem. Using the conservation laws and symmetries of the underlying equation, double reduction and exact solution were also constructed. Copyright © 2017 John Wiley & Sons, Ltd.     

Derivation of Korteweg–de Vries flow equations from nonlinear Schrödinger equation

We perform a multiple scales analysis on the nonlinear Schrödinger (NLS) equation in the Hamiltonian form together with the Hamiltonian function. We derive, as amplitude equations, Korteweg–de Vries (KdV) flow equations in the bi-Hamiltonian form with the corresponding Hamiltonian functions.     

Derivation of Korteweg–de Vries flow equations from the short‐wave model equation

We perform a multiple‐time scales analysis and compatibility condition to the short‐wave model equation. We derive Korteweg–de Vries flow equation in the bi‐Hamiltonian form as an amplitude equation. Copyright © 2013 John Wiley & Sons, Ltd.     

Multiphase averaging and the inverse spectral solution of the Korteweg—de Vries equation

Inverse spectral theory is used to prescribe and study equations for the slow modulations of N-phase wave trains for the Korteweg-de Vries (KdV) equation. An invariant representation of the modulational equations is deduced. This representation depends upon certain differentials on a Riemann surface. When evaluated near ∞ on the surface, the invariant representation reduces to averaged conservations laws; when evaluated near the branch points, the representation shows that the simple eigenvalues provide Riemann invariants for the modulational equations. Integrals of the invariant representation over certain cycles on the Riemann surface yield “conservation of waves.” Explicit formulas for the characteristic speeds of the modulational equations are derived. These results generalize known results for a single-phase traveling wave, and indicate that complete integrability can induce enough structure into the modulational equations to diagonalize (in the sense of Riemann invariants) their first-order terms.     

He's homotopy perturbation method for solving Korteweg‐de Vries Burgers equation with initial condition

In this article, we present the Homotopy Perturbation Method (Shortly HPM) for obtaining the numerical solutions of the Korteweg‐de Vries Burgers (KdVB) equation. The series solutions are developed and the reccurance relations are given explicity. The initial approximation can be freely chosen with possibly unknown constants which can be determined by imposing the boundary and initial conditions. The results reveal that HPM is very simple and effective. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Generalized solitary waves in a finite‐difference Korteweg‐de Vries equation

Generalized solitary waves with exponentially small nondecaying far field oscillations have been studied in a range of singularly perturbed differential equations, including higher order Korteweg‐de Vries (KdV) equations. Many of these studies used exponential asymptotics to compute the behavior of the oscillations, revealing that they appear in the solution as special curves known as Stokes lines are crossed. Recent studies have identified similar behavior in solutions to difference equations. Motivated by these studies, the seventh‐order KdV and a hierarchy of higher order KdV equations are investigated, identifying conditions which produce generalized solitary wave solutions. These results form a foundation for the study of infinite‐order differential equations, which are used as a model for studying lattice equations. Finally, a lattice KdV equation is generated using finite‐difference discretization, in which a lattice generalized solitary wave solution is found.     

Rigorous Justification of the Whitham Modulation Equations for the Generalized Korteweg–de Vries Equation

In this paper, we consider the spectral stability of spatially periodic traveling wave solutions of the generalized Korteweg–de Vries equation to long‐wavelength perturbations. Specifically, we extend the work of Bronski and Johnson by demonstrating that the homogenized system describing the mean behavior of a slow modulation (WKB) approximation of the solution correctly describes the linearized dispersion relation near zero frequency of the linearized equations about the background periodic wave. The latter has been shown by rigorous Evans function techniques to control the spectral stability near the origin, that is, stability to slow modulations of the underlying solution. In particular, through our derivation of the WKB approximation we generalize the modulation expansion of Whitham for the KdV to a more general class of equations which admit periodic waves with nonzero mean. As a consequence, we will show that, assuming a particular nondegeneracy condition, spectral stability near the origin is equivalent with the local well‐posedness of the Whitham system.     

Error analysis for solving the Korteweg‐de Vries equation by a Legendre pseudo‐spectral method

A Legendre pseudo‐spectral method is proposed for the Korteweg‐de Vries equation with nonperiodic boundary conditions. Appropriate base functions are chosen to get an efficient algorithm. Error analysis is given for both semi‐discrete and fully discrete schemes. The numerical results confirm to the theoretical analysis. © (2000) John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 513–534, (2000)     

Introduction to nonclassical shock solutions of the modified Burgers‐Korteweg‐de Vries equation

A general property of nonlinear hyperbolic equations is the eventual formation of discontinuities in the propagating signal. These discontinuities are not uniquely defined by the initial data for the problem and a central issue is the identification of acceptable weak solutions. Particular difficulties arise when the hyperbolic system ceases to be genuinely nonlinear in some of its characteristic fields. This equates in the case of a scalar law to the lack of convexity in the flux function. Here a representative example is provided by the modified Korteweg‐de Vries‐Burgers equation which exhibits a quadratic as well as a cubic nonlinear term and arises in a variety of engineering applications including weakly nonlinear waves in fluidized beds and two‐layer fluid flows. Its solutions have the distinguishing feature to generate undercompressive or nonclassical shocks in the hyperbolic limit with dispersion and dissipation balanced. The resulting rich variety of wave phenomena: shocks which emanate rather than absorb characteristics, compound shocks and shock fan combinations, which have no counterpart in classical shock theories is discussed.     

Global well-posedness of Korteweg–de Vries equation in

We prove that the Korteweg–de Vries initial-value problem is globally well-posed in and the modified Korteweg–de Vries initial-value problem is globally well-posed in . The new ingredient is that we use directly the contraction principle to prove local well-posedness for KdV equation in H^−3/4 by constructing some special resolution spaces in order to avoid some ‘logarithmic divergence’ from the high–high interactions. Our local solution has almost the same properties as those for H^s (s>−3/4) solution which enable us to apply the I-method to extend it to a global solution.     

Computability of Solutions of the Korteweg‐de Vries Equation

In this paper we study computability of the solutions of the Korteweg‐de Vries (KdV) equation u_t + uu_x + u_xxx = 0. This is one of the open problems posted by Pour‐El and Richards [25]. Based on Bourgain's new approach to the initial value problem for the KdV equation in the periodic case, we show that the periodic solution u (x, t) of the KdV equation is computable if the initial data is computable.