Singular solutions of the KdV equation and the inverse scattering method

The paper is devoted to the construction of singular solutions of the KdV equation. The presentation is based on a variant of the inverse scattering method for singular solutions of nonlinear equations developed in previous works of the authors.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 133, pp. 17–37, 1984.In conclusion we wish to express our gratitude to F. Calogero, who drew our attention to works [4–6], an to mention that the generalization of these works proposed here is based on the early paper [13] of L. D. Faddeev.     

The multi-soliton solutions of the CH-γ equation

This paper solves the integrable CH-γ equation for analytical multiple soliton solutions with the Darboux transformation method. Some properties of the soliton solutions are different from the CH equation. This work was partially supported by the National Natural Science Foundation of China (Grant No. 10401022) and the Research Grants Council of Hong Kong     

修正的广义Vakhnenko方程的周期解

利用Hirota双线性方法以及Rienmann theta函数,构造了含两个任意常系数的修正的广义Vakhnenko方程的周期解.特别是在极限情况下,可以由方程的周期解得到其孤子解.     

Solution of the inverse scattering problem for the Landau-Lifshits equation by the Gel'fand-Levitan-Marchenko method

Moscow Institute of Steel and Alloys. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 23, No. 3, pp. 73–75, July–September, 1989.     

A series of abundant exact travelling wave solutions for a modified generalized Vakhnenko equation using auxiliary equation method

In this paper, a series of abundant exact travelling wave solutions is established for a modified generalized Vakhnenko equation by using auxiliary equation method. These solutions can be expressed by Jacobi elliptic function. When Jacobi elliptic functions modulus m→1 or 0, the travelling wave solutions degenerate to four types of solutions, namely, the soliton solutions, the hyperbolic function solutions, the trigonometric function solutions, constant solutions.     

The singular solutions of a nonlinear evolution equation taking continuous part of the spectral data into account in inverse scattering method

A procedure for finding the solutions of the Vakhnenko–Parkes equation by means of the inverse scattering method is described. Both the bound state spectrum and the continuous spectrum are considered in the associated eigenvalue problem. The suggested special form of the singularity function gives rise to periodic solutions. The interaction of a soliton with a one-mode periodic wave is studied.     

Solving the one-loop soliton solution of the Vakhnenko equation by means of the Homotopy analysis method

A powerful, easy-to-use analytic technique for nonlinear problems, namely the Homotopy analysis method, is applied to solve the Vakhnenko equation, a nonlinear equation with loop soliton solutions governing the propagation of high-frequency waves in a relaxing medium. By means of the transformation of independent variables, an analysis one-loop soliton solution expressed by a series of exponential functions is obtained, which agrees well with the exact solution. This indicates the validity and great potential of the Homotopy analysis method in solving complicated solitary wave problems.     

The interaction processes of the N-soliton solutions for an extended generalization of Vakhnenko equation

The N-soliton solutions for an extended generalization of Vakhnenko equation are calculated by applying the improved Hirota method and the variable transformations. The N-soliton solutions can be expressed explicitly. Furthermore, the interaction processes are discussed for the N-soliton solutions.     

The method of fundamental solutions for the calculation of the eigenvalues of the Helmholtz equation

We investigate the application of the method of fundamental solutions (MFS) for the calculation of the eigenvalues of the Helmholtz equation in the plane subject to homogeneous Dirichlet boundary conditions. We present results for circular and rectangular geometries.     

Application of the inverse scattering method to the equation σ xt =eσ

Theoretical and Mathematical Physics -     

The inverse scattering problem for a perturbed difference Hill equation

We consider the inverse scattering problem for the difference analog of a perturbed Hill equation. The perturbation coefficients are recovered from the periodic coefficients and from the scattering data.     

Group analysis of the modified generalized Vakhnenko equation

In this paper the Lie symmetry group, the corresponding symmetry reductions and invariant solutions of the modified generalized Vakhnenko equation are determined. Moreover a numerical algorithm that is based on a Lie symmetry group preserving scheme is applied to the ordinary differential equations obtained by symmetry reduction.     

Numerical realization of the inverse scattering method

Institute of Metal Physics, Ural Scientific Center of the USSR Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 83, No. 3, pp. 323–333, June, 1990.     

The Kirsch-Kress method for inverse scattering by infinite locally rough interfaces

This paper is concerned with the inverse problem of reconstructing an infinite, locally rough interface from the scattered field measured on line segments above and below the interface in two dimensions. We extend the Kirsch-Kress method originally developed for inverse obstacle scattering problems to the above inverse transmission problem with unbounded interfaces. To this end, we reformulate our inverse problem as a nonlinear optimization problem with a Tikhonov regularization term. We prove the convergence of the optimization problem when the regularization parameter tends to zero. Finally, numerical experiments are carried out to show the validity of the inversion algorithm.