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1.
The two-dimensional Toda lattice equation with self-consistent sources is proposed based on its bilinear forms. Casoratian-type solutions and Bäcklund transformation (BT) for the bilinear forms are presented. Starting from the BT, a Lax pair is derived for the 2D Toda lattice with self-consistent sources.  相似文献   

2.
In this paper, we study the generalized coupled integrable dispersionless (GCID) equations and construct two integrable discrete analogues including a semi-discrete system and a full-discrete one. The results are based on the relations among the GCID equations, the sine-Gordon equation and the two-dimensional Toda lattice equation. We also present the N-soliton solutions to the semi-discrete and fully discrete systems in the form of Casorati determinant. In the continuous limit, we show that the fully discrete GCID equations converge to the semi-discrete GCID equations, then further to the continuous GCID equations. By using the integrable semi-discrete system, we design two numerical schemes to the GCID equations and carry out several numerical experiments with solitons and breather solutions.  相似文献   

3.
We construct the coupled generalization of the Nizhnik-Veselov-Novikov (NVN) equation, i.e., the NVN equation with self-consistent sources (ESCS) and solve it using the so-called source generation procedure. We obtain BKP-type and DKP-type Pfaffian solutions of the NVN ESCS. In addition, we give a bilinear Bäcklund transformation and a Lax pair for the NVN ESCS. Furthermore, this Lax pair can be reduced to a Lax pair for the NVN equation.  相似文献   

4.
A new procedure called ‘source generation’ is applied to the pfaffianized KP equation. As a result, the pfaffianized-KP equation with self-consistent sources (ESCS) is obtained. This coupled system cannot only be reduced to the pfaffianized KP equation, but also reduced to the KP equation with self-consistent sources (KPESCS). So the pfaffianized-KP ESCS can be viewed as a pfaffian version of the KPESCS, which indicates the commutativity of the ‘source generation’ procedure and pfaffianization.  相似文献   

5.
In this study, a time semi-discrete Crank-Nicolson (CN) formulation with second-order time accuracy for the non-stationary parabolized Navier-Stokes equations is firstly established. And then, a fully discrete stabilized CN mixed finite element (SCNMFE) formulation based on two local Gauss integrals and parameterfree with the second-order time accuracy is established directly from the time semi-discrete CN formulation. Thus, it could avoid the discussion for semi-discrete SCNMFE formulation with respect to spatial variables and its theoretical analysis becomes very simple. Finaly, the error estimates of SCNMFE solutions are provided.  相似文献   

6.
It is shown that the intrinsic determining equations of a given differential-difference equation (DDE) can be derived by the compatibility between the original equation and the intrinsic invariant surface condition. The (2+1)-dimensional Toda lattice, the special Toda lattice and the DD-KP equation serving as examples are used to illustrate this approach. Then, Bäcklund transformations of the (2+1)-dimensional DDEs including the special Toda lattice, the modified Toda lattice and the DD-KZ equation are presented by using the non-intrinsic direct method. In addition, the Clarkson-Kruskal direct method is developed to find similarity reductions of the DDEs.  相似文献   

7.
In the present paper, we are concerned with the link between the Kadomtsev–Petviashvili–Toda (KP–Toda) hierarchy and the massive Thirring (MT) model. First, we bilinearize the MT model under both the vanishing and nonvanishing boundary conditions. Starting from a set of bilinear equations of two-component KP–Toda hierarchy, we derive multibright solution to the MT model. Then, considering a set of bilinear equations of the single-component KP–Toda hierarchy, multidark soliton and multibreather solutions to the MT model are constructed by imposing constraints on the parameters in two types of tau function, respectively. The dynamics and properties of one- and two-soliton for bright, dark soliton and breather solutions are analyzed in details.  相似文献   

8.
A time semi-discrete Crank-Nicolson(CN) formulation with second-order time accuracy for the non-stationary parabolized Navier-Stokes equations is firstly established.And then, a fully discrete stabilized CN mixed finite volume element(SCNMFVE) formulation based on two local Gaussian integrals and parameter-free with the second-order time accuracy is established directly from the time semi-discrete CN formulation so that it could avoid the discussion for semi-discrete SCNMFVE formulation with respect to spatial variables and its theoretical analysis becomes very simple. Finally, the error estimates of SCNMFVE solutions are provided.  相似文献   

9.
The bilinear Bäcklund transformation for the Toda lattice is derived from the Darboux transformation and some novel solutions to the Toda lattice are obtained through a modified bilinear Bäcklund transformation.  相似文献   

10.
New solutions to the ultradiscrete soliton equations, such as the Box–Ball system, the Toda equation, etc. are obtained. One of the new solutions which we call a "negative-soliton" satisfies the ultradiscrete KdV equation (Box–Ball system) but there is not a corresponding traveling wave solution for the discrete KdV equation. The other one which we call a "static-soliton" satisfies the ultradiscrete Toda equation but there is not a corresponding traveling wave solution for the discrete Toda equation. A collision of a soliton with a negative-soliton generates many balls in a box over the capacity of the box in the Box–Ball system, while a collision of a soliton with the static-soliton describes, in the ultradiscrete limit, transmission of a soliton through junctions of a "nonuniform Toda equation." We have obtained exact solutions describing these phenomena.  相似文献   

11.
The explicit solution of the generalized Toda models connected with the root systems of simple Lie algebras is constructed. This solution is obtained by a special projection of geodesic flows on some symmetric spaces on the phase space of the Toda models. In particular, for the usualN-body Toda model the solutions are expressed by minors of the exponent of the Jacobi matrix depending on initial data.This paper has been published early as a preprint ITEP-157 (1978)  相似文献   

12.
In this paper, we construct a new even constrained B(C)-type Toda hierarchy and derive its B(C)-type Block-type additional symmetry. Also we generalize the B(C)-type Toda hierarchy to the N-component B(C)-type Toda hierarchy which is proved to have symmetries of a coupled \(\bigotimes ^NQT_+ \) algebra (N-fold direct product of the positive half of the quantum torus algebra QT).  相似文献   

13.
For time dependent problems, the Schwarz waveform relaxation (SWR) algorithm can be analyzed both at the continuous and semi-discrete level. For consistent space discretizations, one would naturally expect that the semi-discrete algorithm performs as predicted by the continuous analysis. We show in this paper for the reaction diffusion equation that this is not always the case. We consider two space discretization methods—the 2nd-order central finite difference method and the 4th-order compact finite difference method, and for each method we show that the semi-discrete SWR algorithm with Dirichlet transmission condition performs as predicted by the continuous analysis. However, for Robin transmission condition the semi-discrete SWR algorithm performs worse than predicted by the continuous analysis. For each type of transmission conditions, we show that the convergence factors of the semi-discrete SWR algorithm using the two space discretization methods are (almost) equal. Numerical results are presented to validate our conclusions.  相似文献   

14.
This paper studies a Laplace operator on semi-discrete surfaces. A semi-discrete surface is represented by a mapping into three-dimensional Euclidean space possessing one discrete variable and one continuous variable. It can be seen as a limit case of a quadrilateral mesh, or as a semi-discretization of a smooth surface. Laplace operators on both smooth and discrete surfaces have been an object of interest for a long time, also from the viewpoint of applications. There are a wealth of geometric objects available immediately once a Laplacian is defined, e.g., the mean curvature normal. We define our semi-discrete Laplace operator to be the limit of a discrete Laplacian on a quadrilateral mesh, which converges to the semi-discrete surface. The main result of this paper is that this limit exists under very mild regularity assumptions. Moreover, we show that the semi-discrete Laplace operator inherits several important properties from its discrete counterpart, like symmetry, positive semi-definiteness, and linear precision. We also prove consistency of the semi-discrete Laplacian, meaning that it converges pointwise to the Laplace–Beltrami operator, when the semi-discrete surface converges to a smooth one. This result particularly implies consistency of the corresponding discrete scheme.  相似文献   

15.

When a measure \(\varPsi(x)\) on the real line is subjected to the modification \(d\varPsi^{(t)}(x) = e^{-tx} d \varPsi(x)\), then the coefficients of the recurrence relation of the orthogonal polynomials in \(x\) with respect to the measure \(\varPsi^{(t)}(x)\) are known to satisfy the so-called Toda lattice formulas as functions of \(t\). In this paper we consider a modification of the form \(e^{-t(\mathfrak{p}x+ \mathfrak{q}/x)}\) of measures or, more generally, of moment functionals, associated with orthogonal L-polynomials and show that the coefficients of the recurrence relation of these L-orthogonal polynomials satisfy what we call an extended relativistic Toda lattice. Most importantly, we also establish the so called Lax pair representation associated with this extended relativistic Toda lattice. These results also cover the (ordinary) relativistic Toda lattice formulations considered in the literature by assuming either \(\mathfrak{p}=0\) or \(\mathfrak{q}=0\). However, as far as Lax pair representation is concern, no complete Lax pair representations were established before for the respective relativistic Toda lattice formulations. Some explicit examples of extended relativistic Toda lattice and Langmuir lattice are also presented. As further results, the lattice formulas that follow from the three term recurrence relations associated with kernel polynomials on the unit circle are also established.

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16.
罗振东 《计算数学》2014,36(4):355-362
首先给出二维非饱和土壤水流问题基于Crank-Nicolson(CN)方法的具有时间二阶精度的半离散化格式,然后直接从CN时间半离散化格式出发,建立具有时间二阶精度的全离散化CN有限元格式,并给出误差估计,最后用数值例子说明全离散化CN有限元格式的优越性.这种方法可以绕开关于空间变量的半离散化格式的讨论,提高时间离散的精度,极大地减少时间方向的迭代步,从而减少实际计算中截断误差的积累,提高计算精度和计算效率.  相似文献   

17.
We give an integral representation of the wave functions of the quantum N-particle Toda chain with boundary interaction. In the case of the Toda chain with a one-boundary interaction, we obtain the wave function by an integral transformation from the wave functions of the open Toda chain. The kernel of this transformation is given explicitly in terms of -functions. The wave function of the Toda chain with a two-boundary interaction is obtained from the previous wave functions by an integral transformation. In this case, the difference equation for the kernel of the integral transformation admits a separation of variables. The separated difference equations coincide with the Baxter equation.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 142, No. 2, pp. 346–364, February, 2005.  相似文献   

18.
首先给出Sobolev方程关于时间二阶精度的Crank-Nicolson(CN)时间半离散格式,然后直接从时间二阶精度的CN时间半离散格式出发,构造CN全离散化的有限元格式,并给出这种时间二阶精度的CN全离散化有限元解的误差估计.本文研究方法使得理论证明变得更简便, 也是处理Sobolev方程的一种新的尝试.  相似文献   

19.
The relativistic Toda molecule equation (RTM) describes a one-parameter deformation of coefficients of the recurrence relation of a class of biorthogonal polynomials including the Szegö polynomials. In this paper, we present (i) explicit solutions of the discrete relativistic Toda molecule equation (d-RTM), (ii) a new Padé approximation algorithm for a given power series.  相似文献   

20.
NEW SYMPLECTIC MAPS: INTEGRABILITY AND LAX REPRESENTATION   总被引:2,自引:0,他引:2  
NEWSYMPLECTICMAPS:INTEGRABILITYANDLAXREPRESENTATION***ZENGYUNBO*LIYISHEN**ManuscriptreceivedJune26,1995.*DepartmentofAppliedM...  相似文献   

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