Related papers: KdV-Volterra chain
A one-to-one correspondence between triple Jordan systems and integrable multi-component models of the modified Korteveg--de Vries type is established.
The algebraic geometric approach to $N$-component systems of nonlinear integrable PDE's is used to obtain and analyze explicit solutions of the coupled KdV and Dym equations. Detailed analysis of soliton fission, kink to anti-kink…
We produce a hierarchiy of integrable equations by systematically adding terms to the Lax pair for the lattice modified KdV equation. The equations in the hierarchy are related to one aonother by recursion relations. These recursion…
Multicomponent KdV-systems are defined in terms of a set of structure constants and, as shown by Svinolupov, if these define a Jordan algebra the corresponding equations may be said to be integrable, at least in the sense of having…
The Painlev\'{e} property of coupled, non-autonomous Korteweg-de Vries (KdV) type of systems is studied. The conditions under which the systems pass the Painlev\'{e} test for integrability are obtained. For some of the integrable cases,…
The Volterra and Toda chains equations are considered. A class of special reductions for these equations are derived.
The duality between a class of the Davey-Stewartson type coupled systems and a class of two-dimensional Toda type lattices is discussed. A new coupled system related to the recently found lattice is presented. A method for eliminating…
Even though the KdV and modified KdV equations are nonlinear, we show that suitable linear combinations of known periodic solutions involving Jacobi elliptic functions yield a large class of additional solutions. This procedure works by…
A new approach to double-sub equation method is introduced to construct novel solutions for the nonlinear partial differential equations. It is applied to the Korteweg-de Vries (KdV) equation and yields new complexiton solutions of both the…
Taking the coupled KdV system as a simple example, analytical and nonsingular complexiton solutions are firstly discovered in this letter for integrable systems. Additionally, the analytical and nonsingular positon-negaton interaction…
We establish a relation between the classical non-linear Schr\"odinger equation and the KP hierarchy, and we extend this relation to the quantum case by defining a quantum KP hierarchy. We present evidence that an integrable hierarchy of…
Integrable mixed models have been used as a generalization of traditional integrable models. However, a map from a traditional integrable model to a mixed integrable model is not well understood yet. Here, it is studied the relation between…
Some results about existence, uniqueness, and attractive behaviour of solutions for nonlinear Volterra integral equations with non-convolution kernels are presented in this paper. These results are based on similar ones about nonlinear…
Superpositions of hierarchies of integrable equations are also integrable. The superposed equations, such as the Hirota equations in the AKNS hierarchy, cannot be considered as new integrable equations. Furthermore if one applies the Hirota…
It is well-known that the finite-gap solutions of the KdV equation can be generated by its recursion operator.We generalize the result to a special form of Lax pair, from which a method to constrain the integrable system to a…
We construct a new series of multicomponent integrable PDE systems that contain as particular example (with appropriately chosen parameters) many famous integrable systems including KdV, coupled KdV, Harry Dym, coupled Harry Dym,…
Generalization of the modified KdV equation to a multi-component system, that is expressed by $(\partial u_i)/(\partial t) + 6 (\sum_{j,k=0}^{M-1} C_{jk} u_j u_k) (\partial u_i)/(\partial x) + (\partial^3 u_{i})/(\partial x^3) = 0, i=0, 1,…
Some types of coupled Korteweg de-Vries (KdV) equations are derived from an atmospheric dynamical system. In the derivation procedure, an unreasonable $y$-average trick (which is usually adopted in literature) is removed. The derived models…
We show how to generate coupled KdV hierarchies from Staeckel separable systems of Benenti type. We further show that solutions of these Staeckel systems generate a large class of finite-gap and rational solutions of cKdV hierarchies. Most…
In the example of the Schr\"odinger/KdV equation we treat the theory as equivalence of two concepts of Liouvillian integrability: quadrature integrability of linear differential equations with a parameter (spectral problem) and Liouville's…