We classify integrable Hamiltonian equations of the form $$u_t={\mathrm{\partial }}_x\left(\frac{\delta H}{\delta u}\right),H=\int h(u,w)\hspace{0.17em}\text{d}x\text{d}y,$$ where the Hamiltonian density h(u, w) is a function of two variables: dependent variable u and the non-locality $w={\mathrm{\partial }}_x^{-1}{\mathrm{\partial }}_{y}u$. Based on the method of hydrodynamic reductions, the integrability conditions are derived (in the form of an involutive PDE system for the Hamiltonian density h). We show that the generic integrable density is expressed in terms of the Weierstrass σ-function: h(u, w) = σ(u) e^w. Dispersionless Lax pairs, commuting flows and dispersive deformations of the resulting equations are also discussed.

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关于一类2+1维可积哈密顿方程

我们对形式的可积哈密顿方程进行分类 $${你}_{吨}={\mathrm{\partial }}_X\left(\frac{\delta H}{\delta 你}\right),H=\int H(你,w)\hspace{0.17em}\text{d}X\text{d}\mathrm{是的},$$ 其中哈密顿密度h ( u , w ) 是两个变量的函数：因变量u和非局部性$w={\mathrm{\partial }}_X^{-1}{\mathrm{\partial }}_{\mathrm{是的}}你$. 基于流体动力学简化的方法，推导出可积性条件（以哈密顿密度h的对合偏微分方程的形式）。我们证明了通用可积密度用 Weierstrass σ函数表示：h ( u , w ) =  σ ( u ) e ^w。还讨论了所得方程的无色散 Lax 对、通勤流和色散变形。