378 research outputs found
Computer classification of integrable coupled KdV-like systems
The foundations of the symmetry approach to the classification problem of integrable non-linear evolution systems are briefly described. Within the framework of the symmetry approach the ten-parametric family of the third order non-linear evolution coupled KdV-like systems is investigated. The necessary integrability conditions lead to an over-determined non-linear algebraic system. To solve that system an effective method based on its structure has been used. This allows us to obtain the complete list of integrable systems of a given type. All computation has been completed on the basis of computer algebra systems FORMAC and REDUCE
Classification of polynomial integrable systems of mixed scalar and vector evolution equations. I
We perform a classification of integrable systems of mixed scalar and vector
evolution equations with respect to higher symmetries. We consider polynomial
systems that are homogeneous under a suitable weighting of variables. This
paper deals with the KdV weighting, the Burgers (or potential KdV or modified
KdV) weighting, the Ibragimov-Shabat weighting and two unfamiliar weightings.
The case of other weightings will be studied in a subsequent paper. Making an
ansatz for undetermined coefficients and using a computer package for solving
bilinear algebraic systems, we give the complete lists of 2nd order systems
with a 3rd order or a 4th order symmetry and 3rd order systems with a 5th order
symmetry. For all but a few systems in the lists, we show that the system (or,
at least a subsystem of it) admits either a Lax representation or a linearizing
transformation. A thorough comparison with recent work of Foursov and Olver is
made.Comment: 60 pages, 6 tables; added one remark in section 4.2.17 (p.33) plus
several minor changes, to appear in J.Phys.
Numerical integration of coupled Korteweg-de Vries System
We introduce a numerical method for general coupled Korteweg-de Vries
systems. The scheme is valid for solving Cauchy problems for arbitrary number
of equations with arbitrary constant coefficients. The numerical scheme takes
its legality by proving its stability and convergence which gives the
conditions and the appropriate choice of the grid sizes. The method is applied
to Hirota-Satsuma (HS) system and compared with its known explicit solution
investigating the influence of initial conditions and grid sizes on accuracy.
We also illustrate the method to show the effects of constants with a
transition to non-integrable cases.Comment: 11 pages, 13 figure
The Integrability of New Two-Component KdV Equation
We consider the bi-Hamiltonian representation of the two-component coupled
KdV equations discovered by Drinfel'd and Sokolov and rediscovered by Sakovich
and Foursov. Connection of this equation with the supersymmetric
Kadomtsev-Petviashvilli-Radul-Manin hierarchy is presented. For this new
supersymmetric equation the Lax representation and odd Hamiltonian structure is
given
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