相关论文: A Lax Pair for 2D Euler Equation

We analysis the symmetries of the reflection equation for open $XYZ$ model and find their solutions $K^{\pm}$ case by case. In the general open boundary conditions, the Lax pair for open one-dimensional $XYZ$ spin-chain is given.

We study the C_2 Ruijsenaars-Schneider(RS) model with interaction potential of trigonometric type. The Lax pairs for the model with and without spectral parameter are constructed. Also given are the involutive Hamiltonians for the system.…

A semi-discrete Lax pair formed from the differential system and recurrence relation for semi-classical orthogonal polynomials, leads to a discrete integrable equation for a specific semi-classical orthogonal polynomial weight. The main…

A three-step method due to Nijhoff and Bobenko & Suris to derive a Lax pair for scalar partial difference equations (P\Delta Es) is reviewed. The method assumes that the P\Delta Es are defined on a quadrilateral, and consistent around the…

Lax representation in terms of $2\times 2$ matrices is constructed for a separable multiply--periodic system splitting on two tori. Hyperelliptic Kleinian functions and their reduction to elliptic functions are used.

The equations that define the Lax pairs for generalized principal chiral models can be solved for any nondegenerate bilinear form on $su(2)$. The solution is dependent on one free variable that can serve as the spectral parameter.

For the 2D and 3D Euler equations, their existing exact solutions are often in linear form with respect to variables x,y,z. In this paper, the Clarkson-Kruskal reduction method is applied to reduce the 2D incompressible Euler equations to a…

A method for constructing the Lax pairs for nonlinear integrable models is suggested. First we look for a nonlinear invariant manifold to the linearization of the given equation. Examples show that such invariant manifold does exist and can…

We construct Lax pairs for a wide class of St\"ackel systems by applying the multi-parameter St\"ackel transform to Lax pairs of a suitably chosen systems from the seed class. For a given St\"ackel system, the obtained set of non-equivalent…

In this note we contribute two results to the theory of the $2D$ Euler equations in vorticity form on the full plane. First, we establish a generalized Lagrangian representation of weak (in general measure-valued) solutions, which includes…

We derive an efficient algorithm to find solutions to Euler's concordant form problem and rational points on elliptic curves associated with this problem.

Even in two dimensions, the spectrum of the linearized Euler operator is notoriously hard to compute. In this paper we give a new geometric calculation of the essential spectrum for 2D flows. This generalizes existing results---which are…

Matrix differential-difference Lax pairs play an essential role in the theory of integrable nonlinear differential-difference equations. We present sufficient conditions which allow one to simplify such a Lax pair by matrix gauge…

We prove two results of strong continuity with respect to the initial datum for bounded solutions to the Euler equations in vorticity form. The first result provides sequential continuity and holds for a general bounded solution. The second…

We expand the completeness study instigated in [J. Math. Phys. 50 (2009), 103516, 29 pages] which found all $2\times2$ Lax pairs with non-zero, separable terms in each entry of each Lax matrix, along with the most general nonlinear systems…

We present a solution for Nash problem for stable toric varieties. We also introduce Nash problem for pairs and prove it for toric pairs.

We rigorously construct the first steady traveling wave solutions of the 2D incompressible Euler equation that take the form of a contiguous vortex-patch dipole, which can be viewed as the vortex-patch counterpart of the well-known…

We establish an explicit formula for the general solution of the Benjamin-Ono equation on the torus and on the line. Contents 1. Introduction 1 1.1. The Benjamin-Ono equation 1 1.2. The Lax pair 2 1.3. The explicit formula on the torus 3…

In this paper, we derived the parametric solution of Euler and Elkies, xyz(x+y+z) = a, in an elementary manner. In addition we proved there are infinitely many parametric solutions of Euler's and Elkies's family of solutions.

An integrable two-component nonlinear Schr\"odinger equation in $2+1$ dimensions is presented. The singular manifold method is applied in order to obtain a three-component Lax pair. The Lie point symmetries of this Lax pair are calculated…