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.     

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.     

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.     

A nonhomogeneous boundary value problem for the Kuramoto–Sivashinsky equation in a quarter plane

We study the initial boundary value problem for the one‐dimensional Kuramoto–Sivashinsky equation posed in a half line with nonhomogeneous boundary conditions. Through the analysis of the boundary integral operator, and applying the known results of the Cauchy problem of the Kuramoto–Sivashinsky equation posed on the whole line , the initial boundary value problem of the Kuramoto–Sivashinsky equation is shown to be globally well‐posed in Sobolev space for any s >?2. Copyright © 2017 John Wiley & Sons, Ltd.     

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     

The general analytical and numerical solution for the modified KdV equation with convergence analysis

We investigate the analytical and numerical solutions of the modified Kortweg de Vries equation by applying the idea of commutative hypercomplex mathematics, He's homotopy perturbation method as a simple particular procedure, and the Runge–Kutta discontinuous Galerkin methods. Moreover, we discuss at great length the convergence conditions for this equation by using the Banach fixed point theory, which could provide a good iteration algorithm. Finally, we compare the homotopy perturbation method with some standard ideas same as the Runge–Kutta discontinuous Galerkin method by some numerical illustrations. Copyright © 2012 John Wiley & Sons, Ltd.     

A numerical method for solving KdV equation with multilevel B-spline quasi-interpolation

In this article, we use a multilevel quartic spline quasi-interpolation scheme to solve the one-dimensional nonlinear Korteweg–de Vries (KdV) equation which exhibits a large number of physical phenomena. The presented scheme is obtained by using the second-order central divided difference of the spatial derivative to approximate the third-order spatial derivative, and the forward divided difference to approximate the temporal derivative, where the spatial derivative is approximated by the proposed quasi-interpolation operator. Compared to other numerical methods, the main advantages of our scheme are the higher accuracy and lower computational complexity. Meanwhile, the algorithm is very simple and easy to implement. Numerical experiments in this article also show that our scheme is feasible and valid.     

On the optimal focusing of solitons and breathers in long‐wave models

Conditions of optimal (synchronized) collisions of any number of solitons and breathers are studied within the framework of the Gardner equation (GE) with positive cubic nonlinearity, which in the limits of small and large amplitudes tends to other long‐wave models, the classic and the modified Korteweg–de Vries equations. The local solution for an isolated soliton or breather within the GE is obtained. The wave amplitude in the focal point is calculated exactly. It exhibits a linear superposition of partial amplitudes of the solitons and breathers. The crucial role of the choice of proper soliton polarities and breather phases on the cumulative wave amplitude in the focal point is demonstrated. Solitons are most synchronized when they have alternating polarities. The straightforward link to the problem of synchronization of envelope solitons and breathers in the focusing nonlinear Schrödinger equation is discussed (then breathers correspond to envelope solitons propagating above a condensate).     

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.     

On the controllability of the fifth-order Korteweg–de Vries equation

In this paper, we consider the fifth-order Korteweg–de Vries equation in a bounded interval. We prove that this equation is locally well-posed when endowed with suitable boundary conditions, and establish a result of local controllability to the trajectories.     

Analyticity for Kuramoto–Sivashinsky‐type equations in two spatial dimensions

I. Stratis In this work, we investigate the analyticity properties of solutions of Kuramoto–Sivashinsky‐type equations in two spatial dimensions, with periodic initial data. In order to do this, we explore the applicability in three‐dimensional models of a spectral method, which was developed by the authors for the one‐dimensional Kuramoto–Sivashinsky equation. We introduce a criterion, which provides a sufficient condition for analyticity of a periodic function u∈C^∞, involving the rate of growth of ?ⁿu, in suitable norms, as n tends to infinity. This criterion allows us to establish spatial analyticity for the solutions of a variety of systems, including Topper–Kawahara, Frenkel–Indireshkumar, and Coward–Hall equations and their dispersively modified versions, once we assume that these systems possess global attractors. Copyright © 2015 John Wiley & Sons, Ltd.     

Explicit Complex-Valued Solutions of the Korteweg–de Vries Equation on the Half-Line and on the Whole-Line

The paper deals with the initial-value problems for the Korteweg–de Vries (KdV) equations on the half-line and on the whole-line for complex-valued measurable and exponentially decreasing potentials. The time evolution equation for the reflection coefficient is derived and then a one-to-one correspondence between the scattering data and the solution of the KdV equation is shown. Families of exact solutions of the KdV equation are represented for the class of reflection-free potentials, in which the inverse scattering problem associated with the KdV equation can be solved exactly. Some helpful examples of soliton solutions of the KdV equation are provided.     

A robust semi-explicit difference scheme for the Kuramoto–Tsuzuki equation

In this paper, we propose a robust semi-explicit difference scheme for solving the Kuramoto–Tsuzuki equation with homogeneous boundary conditions. Because the prior estimate in L_∞-norm of the numerical solutions is very hard to obtain directly, the proofs of convergence and stability are difficult for the difference scheme. In this paper, we first prove the second-order convergence in L₂-norm of the difference scheme by an induction argument, then obtain the estimate in L_∞-norm of the numerical solutions. Furthermore, based on the estimate in L_∞-norm, we prove that the scheme is also convergent with second order in L_∞-norm. Numerical examples verify the correction of the theoretical analysis.     

Attractor for nonlinear Schrödinger equation coupling with stochastic weakly damped, forced KdV equation

The nonlinear Schrödinger equation coupling with stochastic weakly damped, forced KdV equation with additive noise can be solved pathwise, and the unique solution generates a random dynamical system. Then we prove that the system possesses a global weak random attractor.     

Interaction of solitons and the effect of radiation for the generalized KdV equation

We consider the generalized KdV equation with nonlinearity $u^4$ and small dispersion $\epsilon$. Our main result consists in the conclusion that, in the leading term with respect to $\epsilon$, the solitary waves interact in this model like KdV solitons but both small corrections of the soliton parameters and small radiation term appear as a result of the waves collision. Our main tool is the numerical simulation.     

New exact solutions for a generalization of the Korteweg-de Vries equation (KdV6)

The generalized tanh-coth method is used to construct periodic and soliton solutions for a new integrable system, which has been derived from an integrable sixth-order nonlinear wave equation (KdV6). The system is formed by two equations. One of the equations may be considered as a Korteweg-de Vries equation with a source and the second equation is a third-order linear differential equation.     

Determination of a source term in a partial differential equation arising in finance

In this article, we discuss the Lipschitz stability for an inverse problem of determining the source term in option pricing. The main tool for establishing the result is the Carleman estimate.     

Exact solitary wave solutions to a combined KdV and mKdV equation

An exact travelling wave kink soliton to a combination KdV and mKdV equations is given by using an effective homogeneous balance method, and a two‐dimensional generalization is also discussed. Copyright © 2000 John Wiley & Sons, Ltd.