# Quantum Integrable System.

In this article, we will talk about integrable systems, especially quantum integrable systems. An integrable system is a system that can be solved by combining integration, differentiation, transformation of variables, solving differential equations, and the four arithmetic operations, and can be expressed as a fully integrable total differential equation [1]. In such a system, as the number of elements increases, the expression for the conservation law increases. Therefore, in the limit of infinite number of elements, the number of conserved quantities diverges to positive infinity. This is the case for the Toda lattice and the KdV equation in soliton fields. The latter can be solved by Fourier series expansion in terms of trigonometric functions including coordinates and time [2].

In quantum integrable systems, the number of conserved quantities commutative with the Hamiltonian increases with the number of elements. Examples include quantum solitons, one-dimensional quantum gases, and one-dimensional Hubbard models (such as the XXZ system) [3]. One of the most familiar is the nonlinear engineering interaction of carrier waves in optical fibers [4]. Since optical fiber is a medium that cannot be described by the Lorentz model, the dielectric susceptibility becomes a nonlinear tensor and nonlinear optical effects occur. This is the sum of the interaction of the light waves through the medium and the interaction of the light waves with the medium itself. Examples include self-modulation, self-steepening (interaction of light with the medium), Raman scattering, induced Raman scattering (interaction of multiple light waves with the medium through the medium), and four-light wave mixing (interaction between light waves through the medium). The result is a nonlinear Schrodinger equation whose complex amplitude is a quantum integrable system [5].

The examples are simple, but the general theory is difficult, and I have to admit that I do not understand it very well. Originally, this would be defined using a leaf layer structure, but unfortunately I am not familiar with differential geometry, so I cannot guarantee the content. However, there are many examples of integrable systems in the references, and I hope you will refer to them on your own.

[1] http://www.nara-wu.ac.jp/initiative-MPI/images/nwu_only/Inoue-file.pdf

[2] http://gandalf.math.kyushu-u.ac.jp/~kaji/ohp/2007_master.pdf

[3] https://www.math.sci.hokudai.ac.jp/~wakate/mcyr/2018/pdf/00100_sato_jun.pdf

[4] https://optipedia.info/laser/fiberlaser/nonlinear-schrodinger/