无耗散哈密顿系统中的算术选择规则
Arithmetic selection rules in dispersionless Hamiltonian systems
AI总结：
该研究推导了单项式电荷密度的刘维尔可积性条件对应的选择规则，研究了组合多项式序列与1+1维刘维尔可积哈密顿场论的对应关系，得到了Motzkin系统等相关结论。
AI中文摘要：
在本研究中，我们推导了由任意次幂单项式电荷密度的刘维尔可积性条件所施加的选择规则。对于某一单项式哈密顿系统，该选择规则可简化为负佩尔方程，其解生成了一组相互对合的无穷多运动积分。此外，我们研究了1+1维组合多项式序列与刘维尔可积哈密顿场论之间的对应关系，证明了Motzkin系统与Levi系统的无耗散极限一致，而二项式系统等价于无耗散导数非线性薛定谔方程。我们还表明，二项式哈密顿模型可约化为无粘性Burgers方程，其高阶电荷生成了广义Burgers型守恒律。
英文摘要：
In this work, we derive selection rules imposed by Liouville integrability conditions for monomial charge densities with arbitrary powers. For a certain monomial Hamiltonian system, the selection rules reduce to a negative Pell equation, and its solutions generate an infinite set of integrals of motion that are mutually in involution. Furthermore, we study the correspondence between combinatorial polynomial sequences and Liouville integrable Hamiltonian field theories in 1+1 dimensions. We show that the Motzkin system coincides with the dispersionless limit of the Levi system, while the binomial system is equivalent to the dispersionless derivative nonlinear Schrödinger equation. Additionally, we show that the binomial Hamiltonian model admits a reduction to the inviscid Burgers equation and its higher-order charges generate generalized Burgers-type conservation laws.
