Higher Painlevé transcendents as special solutions of some nonlinear integrable hierarchies

It is well known that the self-similar solutions of the Korteweg-de Vries equation and the modified Korteweg-de Vries equation are expressed via the solutions of the first and second Painlevé equations. In this paper we solve this problem for all equations from the Korteveg-de Vries, modified Korteweg-de Vries, Kaup-Kupershmidt, Caudrey-Dodd-Gibbon and Fordy-Gibbons hierarchies. We show that the self-similar solutions of equations corresponding to hierarchies mentioned above can be found by means of the general solutions of higher-order Painlevé hierarchies introduced more than ten years ago.     

Symmetry reductions for a dissipation-modified KdV equation

In this paper we give a group classification for a dissipation-modified Korteweg-de Vries equation by means of the Lie method of the infinitesimals. We prove that, by using the nonclassical method, we get several new solutions which are unobtainable by Lie classical symmetries. We obtain nonclassical symmetries that reduce the dissipation-modified Korteweg-de Vries equation to ordinary equations with the Painlevé property. These solutions have not been derived elsewhere by the singular manifold method.     

Three types of exact solutions to Wick-type generalized stochastic Korteweg-de Vries equation

A Wick-type generalized stochastic Korteweg-de Vries equation is researched. By means of Hermite transformation, white noise theory and Riccati equation mapping method, three types of exact solutions to the generalized stochastic Korteweg-de Vries equation, which include the functional solutions of hyperbolic-exponential type, trigonometric-exponential type and exponential type, are derived.     

非线性KdV-mKdV方程的新的复合函数解

利用孤立子方程KdV-mKdV的朗斯基解的形式和结构,我们提出了朗斯基形式展开法,运用这一方法获得了KdV-mKdV方程的丰富的新的复合函数解,并且朗斯基行列式中的元素不满足任何线性偏微分方程组.所得到的复合函数解是使用其它的方法得不到的.     

A modified tanh–coth method for solving the KdV and the KdV–Burgers’ equations

In this work we use a modified tanh–coth method to solve the Korteweg-de Vries and Korteweg-de Vries–Burgers’ equations. The main idea is to take full advantage of the Riccati equation that the tanh-function satisfies. New multiple travelling wave solutions are obtained for the Korteweg-de Vries and Korteweg-de Vries–Burgers’ equations.     

On asymptotic behavior of solutions to Korteweg-de Vries type equations related to vortex filament with axial flow

We study the global existence and asymptotic behavior in time of solutions to the Korteweg-de Vries type equation called as “Hirota” equation. This equation is a mixture of cubic nonlinear Schrödinger equation and modified Korteweg-de Vries equation. We show the unique existence of the solution for this equation which tends to the given “modified” free profile by using the two asymptotic formulae for some oscillatory integrals.     

On a KdV equation with higher-order nonlinearity: Traveling wave solutions

In this work, the improved tanh-coth method is used to obtain wave solutions to a Korteweg-de Vries (KdV) equation with higher-order nonlinearity, from which the standard KdV and the modified Korteweg-de Vries (mKdV) equations with variable coefficients can be derived as particular cases. However, the model studied here include other important equations with applications in several fields of physical and nonlinear sciences. Periodic and soliton solutions are formally derived.     

A finite difference method for the Korteweg-de Vries and the Kadomtsev-Petviashvili equations

A linearized implicit finite difference method for the Korteweg-de Vries equation is proposed and straightforwardly extended to the Kadomtsev-Petviashvili equation. We investigate the order of accuracy of the method and prove the method to be unconditionally linearly stable. The numerical experiments for the Korteweg-de Vries and the Kadomtsev-Petviashvili equations are carried out with various conditions. Numerical results for the collision of two lump type solitary wave solutions to the Kadomtsev-Petviashvili equation are also reported.     

New solutions of the Schwarzian Korteweg-de Vries equation in 2+1 dimensions based on weak symmetries

We consider the (2+1)-dimensional integrable Schwarzian Korteweg-de Vries equation. Using weak symmetries, we obtain a system of partial differential equations in 1+1 dimensions. Further reductions lead to second-order ordinary differential equations that provide new solutions expressible in terms of known functions. These solutions depend on two arbitrary functions and one arbitrary solution of the Riemann wave equation and cannot be obtained by classical or nonclassical symmetries. Some of the obtained solutions of the Schwarzian Korteweg-de Vries equation exhibit a wide variety of qualitative behaviors; traveling waves and soliton solutions are among the most interesting. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 3, pp. 380–390, June, 2007.     

On the stability of KdV multi-solitons

We consider the stability of multi- or n-soliton solutions to the Korteweg-de Vries equation (KdV) posed on the real line. It is shown that in the standard variational characterization of KdV multi-solitons as critical points, the n-solitons actually realize non-isolated constrained minimizers. (The case n = 1 was already known to Benjamin; see [6].) From this fact a precise dynamic stability result for multi-solitons follows, namely, that initial data close to a given n-soliton evolves in time so as to remain close (in the Hⁿ(??) Sobolev norm) to the n-dimensional manifold of all n-solitons with appropriate wave speeds, i.e., to the set of constrained minimizers. Our techniques are also applicable to other Hamiltonian systems with several conserved quantities. In particular the inverse scattering formalism of KdV is not explicitly exploited. © 1993 John Wiley & Sons, Inc.     

A characteristic description of orthonormal wavelet on subspace of

In this paper, by means of variational iteration method numerical and explicit solutions are computed for some fifth-order Korteweg-de Vries equations, without any linearization or weak nonlinearity assumptions. These equations are the Kawahara equation, Lax’s fifth-order KdV equation and Sawada–Kotera equation. Comparison with Adomian decomposition method reveals that the variational iteration method is easier to be implemented. We conclude that the method is a promising method to various kinds of fifth-order Korteweg-de Vries equations.     

Primitive solutions of the Korteweg–de Vries equation

Theoretical and Mathematical Physics - We survey recent results connected with constructing a new family of solutions of the Korteweg-de Vries equation, which we call primitive solutions. These...     

Self-similar planar curves related to modified Korteweg-de Vries equation

We exhibit a time reversible geometric flow of planar curves which can develop singularities in finite time within the uniform topology. The example is based on the construction of selfsimilar solutions of modified Korteweg-de Vries equation of a given (small) mean.     

一类新的矩阵型修正Korteweg-de Vries方程的Riemann-Hilbert方法与精确解北大核心CSCD

在本文中,一类新的矩阵型修正Korteweg-de Vries(简记为mmKdV)方程被首次通过RiemannHilbert方法研究,而且,这一方程可通过选取特殊的势矩阵来降阶为我们熟知的耦合型修正Kortewegde Vries方程.从方程对应的Lax对的谱分析入手,作者成功地建立了方程对应的Riemann-Hilbert问题.在无反射势的特殊条件下,mmKdV方程的精确解可由Riemann-Hilbert问题的解给出.而且,基于特殊势矩阵所对应的特殊对称性,作者可以对原有的孤子解进行分类,从而得到一些有趣的解的现象,比如呼吸孤子、钟形孤子等.     

Reduction of the dressing chain of the Schrödinger operator

We reduce the problem of constructing real finite-gap solutions of the focusing modified Korteweg-de Vries equation, to the dressing chain of the Schrödinger operator. We show that the Schrödinger operator spectral curve corresponding to such a solution is real. We give some restrictions on the initial data for the chain that lead to such solutions. We also consider a soliton, reduction. We obtain compact representations for the multisoliton and breather solutions of the modified Korteweg-de Vries equations; these representations can be useful in developing the perturbation theory for various applied problems.     

Comments on “New exact periodic solitary-wave solution of MKdV equation”

By means of a so-called extended homoclinic test method, Guo et al. (2009) [1] obtained exact periodic solitary-wave solutions of the modified Korteweg-de Vries (MKdV) equation. We point out that the bilinear form of MKdV equation given in above paper is not true, so that the obtained solutions in the above paper are not actual solutions of the equation. In this paper, we apply an extended ansatz to obtain some new exact periodic solitary-wave solutions to MKdV equation.     

Analytic investigation of the (2 + 1)-dimensional Schwarzian Korteweg-de Vries equation for traveling wave solutions

By means of the two distinct methods, the Exp-function method and the extended (G′/G)-expansion method, we successfully performed an analytic study on the (2 + 1)-dimensional Schwarzian Korteweg-de Vries equation. We exhibited its further closed form traveling wave solutions which reduce to solitary and periodic waves. New rational solutions are also revealed.     

Limit symmetry of the Korteweg-de Vries equation and its applications

We discuss a symmetry of the Korteweg-de Vries (KdV) equation. This symmetry can be related to the squared eigenfunction symmetry by a limit procedure. As applications, we consider the similarity reduction of the KdV equation and a KdV equation with new self-consistent sources. We derive some solutions via a bilinear approach.     

Stability of periodic traveling waves for complex modified Korteweg-de Vries equation

We study the existence and stability of periodic traveling-wave solutions for complex modified Korteweg-de Vries equation. We also discuss the problem of uniform continuity of the data-solution mapping.