论文标题
非线性Schrodinger方程的可整合旋转器/四元素概括
Integrable spinor/quaternion generalizations of the nonlinear Schrodinger equation
论文作者
论文摘要
提出了NLS方程的可集成概括，其中动态复杂变量$ u（t，x）$被一对动态复杂变量$（u_1（t，x），u_2（t，x）$和$ i $替换为Pauli Matrix $ j $。通过在结果的2组分系统中添加非局部项来保留集成性。获得了进一步的集成概括，涉及动态标量变量和附加的非局部项。对于每个系统，宽松对和双汉顿公式均来自基于对称谎言代数的零曲率框架，并使用Hasimoto变量。该系统均显示出等同于双通常流量和Schrodinger地图方程，从而推广了NLS方程在$ r^3 $中的众所周知的对等效性，而Schrodinger Map等式则以$ S^2 $为单位。此外，这两个集成系统都描述了带有$ $（u_1（t，x），u_2（t，x））$的旋转器/四元素NLS型方程，被视为旋转变量或等效的Quaternion变量。
An integrable generalization of the NLS equation is presented, in which the dynamical complex variable $u(t,x)$ is replaced by a pair of dynamical complex variables $(u_1(t,x),u_2(t,x))$, and $i$ is replaced by a Pauli matrix $J$. Integrability is retained by the addition of a nonlocal term in the resulting 2-component system. A further integrable generalization is obtained which involves a dynamical scalar variable and an additional nonlocal term. For each system, a Lax pair and a bi-Hamiltonian formulation are derived from a zero-curvature framework that is based on symmetric Lie algebras and that uses Hasimoto variables. The systems are each shown to be equivalent to a bi-normal flow and a Schrodinger map equation, generalizing the well-known equivalence of the NLS equation to the bi-normal flow in $R^3$ and the Schrodinger map equation in $S^2$. Furthermore, both of the integrable systems describe spinor/quaternion NLS-type equations with the pair $(u_1(t,x),u_2(t,x))$ being viewed as a spinor variable or equivalently a quaternion variable.
