Perturbation of a two-soliton solution of the Korteweg-de vries equation in the case of close amplitudes

We investigate the interaction process for two solitons with close amplitudes under a small perturbation. The leading term of the formal asymptotic solution is found as the sum of two solitons with slowly varying parameters. The equations of slow variations are derived for the soliton phase shifts. The effects related to the interaction between the perturbed solitons can compensate the velocity difference in some conditions, which can result in the formation of the so-called quasi-stationary soliton pair. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 118, No. 3, pp. 434–440, March, 1999.     

Perturbation of a singular solution to the Liouville equation

We construct an asymptotic (with respect to a small parameter) solution of the Cauchy problem for the perturbed Liouville equation in the case where the unperturbed solution has singularities on timelike lines. We propose a modification of the Krylov-Bogoliubov method that, in particular, allows us to find the asymptotic corrections to the singularity lines. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 118, No. 3, pp. 390–397, March, 1999.     

Numerical solution to a linearized KdV equation on unbounded domain

Exact absorbing boundary conditions for a linearized KdV equation are derived in this paper. Applying these boundary conditions at artificial boundary points yields an initial‐boundary value problem defined only on a finite interval. A dual‐Petrov‐Galerkin scheme is proposed for numerical approximation. Fast evaluation method is developed to deal with convolutions involved in the exact absorbing boundary conditions. In the end, some numerical tests are presented to demonstrate the effectiveness and efficiency of the proposed method.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

Topological exact soliton solution of the power law KdV equation

This paper obtains the exact 1-soliton solution of the perturbed Korteweg-de Vries equation with power law nonlinearity. The topological soliton solutions are obtained. The solitary wave ansatz is used to carry out this integration. The domain restrictions are identified in the process and the parameter constraints are also obtained. It has been proved that topological solitons exist only when the KdV equation with power law nonlinearity reduces to simply KdV equation.     

Numerical solution for the KdV equation based on similarity reductions

The present paper is devoted to the development of a new scheme to solve the KdV equation locally on sub-domains using similarity reductions for partial differential equations. Each sub-domain is divided into three-grid points. The ordinary differential equation deduced from the similarity reduction can be linearized, integrated analytically and then used to approximate the flux vector in the KdV equation. The arbitrary constants in the analytical solution of the similarity equation can be determined in terms of the dependent variables at the grid points in each sub-domain. This approach eliminates the difficulties associated with boundary conditions for the similarity reductions over the whole solution domain. Numerical results are obtained for two test problems to show the behavior of the solution of the problems. The computed results are compared with other numerical results.     

Asymptotics of the first correction in the perturbation of theN-soliton solution to the KdV equation

We consider a triple Fourier-type integral that represents a solution to the KdV equation linearized on anN-soliton potential. Assuming that the parameters of the potential depend on the slow timet, we construct an asymptotics of this integral as 0 uniform with respect tox, t up to large timet ^–1.Translated fromMatematicheskie Zametki, Vol. 58, No. 2, pp. 204–217, August, 1995.The work was financially supported in part by the Russian Foundation for Basic Research under grant No. 94-01-00193a.     

1-Soliton solution of the generalized KdV equation with generalized evolution

This paper obtains the 1-soliton solution of three variants of the generalized KdV equation with generalized evolution. The solitary wave ansatz is used to carry out the integration of such equation. The parameter domain is also identified in the process. A couple of conserved quantities are also calculated for each of these variants. The numerical simulations are also carried out.     

A direct method for deriving a multisoliton solution to the fifth order KdV equation

A new representation of N-soliton solution of the fifth order KdV equation is obtained by using Bäcklund transformation method. We show a direct method for constructing the novel N-soliton solution by performing an appropriate limiting procedure on the known soliton solutions.     

Low regularity solution for a nonlocal perturbation of KdV equation

The Cauchy problem for a nonlocal perturbation of KdV equation is considered by the Fourier restriction norm method. Local well-posedness for initial data in $H^s({\mathbb{R}})(s > -\frac{3}{4})$H^s({\mathbb{R}})(s > -\frac{3}{4}) and global results for data in L²(\mathbbR)L^2({\mathbb{R}}) are obtained.     

Low regularity solution for a nonlocal perturbation of KdV equation

The Cauchy problem for a nonlocal perturbation of KdV equation is considered by the Fourier restriction norm method. Local well-posedness for initial data in and global results for data in are obtained. The second author was supported by the National Natural Science Foundation of China Grant No.10526011.     

A simple method to construct soliton-like solution of the general KdV equation with external force

A simple and direct method is described to construct the soliton-like solution for the general KdV equation with external force. Crucial to the method is the assumption that the solution chosen is a special truncated expansion.     

Topological and non-topological exact soliton solution of the power law KdV equation

This paper obtains the exact 1-soliton solution of the perturbed Korteweg–de Vries equation with power law nonlinearity. Both topological as well as non-topological soliton solutions are obtained. The solitary wave ansatz is used to carry out this integration. The domain restrictions are identified in the process and the parameter constraints are also obtained. Finally, the numerical simulations are implemented in the paper.     

Topological 1-soliton solution of the generalized KdV equation with generalized evolution

In this paper, the topological 1-soliton solution to the generalized Korteweg-de Vries equation with generalized evolution will be obtained. The solitary wave ansatz methods will be applied to obtain the solutions. In the process, the constraints relations between the soliton parameters will also be determined.     

Cosine expansion-based differential quadrature method for numerical solution of the KdV equation

Numerical solution of the Korteweg–de Vries equation is obtained using space-splitting technique and the differential quadrature method based on cosine expansion (CDQM). The details of the CDQM and its implementation to the KdV equation are given. Three test problems are studied to demonstrate the accuracy and efficiency of the proposed method. Accuracy and efficiency are discussed by computing the numerical conserved laws and L₂, L_∞ error norms.     

带强迫项变系数组合KdV方程的有理展开式精确解

利用符号计算软件Maple,在一个新的广义的Riccati方程有理展开法的帮助下,求出了带强迫项变系数组合KdV方程的有理展开式的精确解,该方法还可被应用到其他变系数非线性发展方程中去.     

A new integrable equation that combines the KdV equation with the negative‐order KdV equation

In this work, we develop a new integrable equation by combining the KdV equation and the negative‐order KdV equation. We use concurrently the KdV recursion operator and the inverse KdV recursion operator to construct this new integrable equation. We show that this equation nicely passes the Painlevé test. As a result, multiple soliton solutions and other soliton and periodic solutions are guaranteed and formally derived.     

Computing solutions to a forced KdV equation

In this paper a special forced Korteweg–de Vries (KdV) equation is considered. This equation is established by recent studies as a simple mathematical model of describing the physics of a shallow layer of fluid subject to external forcing. It serves as an analytical model of tsunami generation by submarine landslides. The bilinear form for this equation is obtained with the aid of Hirota’s method. Some of its one-, two- and three-soliton as well as breather-type soliton solutions and other interesting solutions are derived.     

Solitary wave solution for the generalized KdV equation with time-dependent damping and dispersion

The solitary wave solution of the generalized KdV equation is obtained in this paper in presence of time-dependent damping and dispersion. The approach is from a solitary wave ansatze that leads to the exact solution. A particular example is also considered to complete the analysis.