相关论文: Lax Pairs and Darboux Transformations for Euler Eq…
We obtain the well-known discrete modified Boussinesq equation in two-component form as well as its Lax pair in $3\times3$ matrix form through a 3-periodic reduction technique on the Hirota-Miwa equation and its Lax pair. We describe how…
A new ansatz is presented for a Lax pair describing systems of particles on the line interacting via (possibly nonsymmetric) pairwise forces. Particular cases of this yield the known Lax pairs for the Calogero-Moser and Toda systems, as…
The ``truncation procedure'' initiated by Weiss et al. is best understood as a Darboux transformation. If it leads to the Lax pair of the PDE under study, the B\"acklund transformation follows by an elimination, thus proving the…
This article is a survey concerning the state-of-the-art mathematical theory of the Euler equations of incompressible homogenous ideal fluid. Emphasis is put on the different types of emerging instability, and how they may be related to the…
This note presents Godunov variables and 4-potentials for the relativistic Euler equations of barotropic fluids. The associated additional conservation/ production law has different interpretations for different fluids. In particular it…
We construct linear and quadratic Darboux matrices compatible with the reduction group of the Lax operator for each of the seven known non-Abelian derivative nonlinear Schr\"odinger equations that admit Lax representations. The…
The constraints for evolution equations with some special form of Lax pair are first investigated. We show by examples how the method is rooted in the classical literatures and how the ignored constraints provide nontrivial solutions. Then…
We apply the Darboux transformation to construct new exactly-solvable cases of the two-dimensional massless Dirac equation for potential classes of Lambert-W and inverse exponential type. Both of these classes originate from the Heun…
This article handles in a short manner a few Laplace transform pairs and some extensions to the basic equations are developed. They can be applied to a wide variety of functions in order to find the Laplace transform or its inverse when…
The Darboux transformation on matrix solutions to the generalized coupled dispersionless integrable system based on some non-abelian Lie group, is studied and the solutions are shown to be expressed in terms of quasideterminants. As an…
We propose a method for construction of Darboux transformations, which is a new development of the dressing method for Lax operators invariant under a reduction group. We apply the method to the vector sine-Gordon equation and derive its…
We establish the local-in-time existence of solutions to the relativistic Euler equations representing dynamical liquid bodies in vacuum.
Generalized Euler-Arnold-von Neumann density matrix equations can be solved by a binary Darboux transformation given here in a new form: $\rho[1]=e^{P\ln(\mu/\nu)}\rho e^{-P\ln(\mu/\nu)}$ where $P=P^2$ is explicitly constructed in terms of…
We derive a permutability theorem for the Christoffel, Goursat and Darboux transformations of isothermic surfaces. As a consequence we obtain a simple proof of a relation between Darboux pairs of minimal surfaces in Euclidean space, curved…
A Lax system in three variables is presented, two equations of which form the Lax pair of the stationary Davey-Stewartson II equation. With certain nonlinear constraints, the full integrability condition of this Lax system contains the…
In the recent comment quoted in the title (arXiv:1407.7852v1), a comment is presented on our recent work which derive a generalized nonlinear wave solution formula for mixed coupled nonlinear Sch\"{o}dinger equations by performing the…
A noncommutative version of the semi-discrete Toda equation is considered. A Lax pair and its Darboux transformations and binary Darboux transformations are found and they are used to construct two families of quasideterminant solutions.
Darboux transformations in one independent variable have found numerous applications in various field of mathematics and physics. In this paper we show that the extension of these transformations to two dimensions can be used to decouple…
We consider a classical problem of Computer Algebra: symbolic solution of PDEs. We transform the famous Darboux theorems on differential transformations of hyperbolic operator into the space of invariants. We introduce a new idea -- $X$-…
Algebraic Bargmann and Darboux transformations for equations of a more general form than the Schr\"odinger ones with an additional functional dependence h(r) in the right-hand side of equations are constructed. The suggested generalized…