The 1 + 1 dimensional Kuramoto–sine-Gordon system consists of a set of N nonlinear coupled equations for N scalar fields θ _i , which constitute the nodes of a complex system. These scalar fields interact by means of Kuramoto nonlinearities over a network of connections determined by N(N − 1)/2 symmetric coupling coefficients a _ij . This system, regarded as a chirally invariant quantum field theory, describes a single decoupled massless field together with N − 1 scalar boson excitations of nonzero mass depending on a _ij , which propagate and interact over the network. For N = 2 the equations decouple into separate sine-Gordon and wave equations. The system allows an extensive array of soliton configurations which interpolate between the various minima of the 2π-periodic potential, including sine-Gordon solitons in both static and time-dependent form, as well as double sine-Gordon solitons which can be imbedded into the system for any N. The precise form of the stable soliton depends critically on the coupling coefficients a _ij . We investigate specific configurations for N = 3 by classifying all possible potentials, and use the symmetries of the system to construct static solitons in both exact and numerical form.

1 + 1 维 Kuramoto-sine-Gordon 系统由一组N 个非线性耦合方程组成，用于N 个标量场θ _i，这些方程构成复杂系统的节点。这些标量场通过 Kuramoto 非线性在由N ( N − 1)/2 个对称耦合系数a _ij确定的连接网络上相互作用。该系统被视为手性不变量子场论，描述了单个解耦的无质量场以及N - 1 个非零质量的标量玻色子激发，取决于a _ij，它们在网络上传播和相互作用。对于N = 2 方程解耦为单独的正弦-戈登方程和波动方程。该系统允许在 2 π周期势的各个最小值之间插入大量孤子配置，包括静态和时间相关形式的正弦-戈登孤子，以及可以嵌入的双正弦-戈登孤子任何N的系统。稳定孤子的精确形式主要取决于耦合系数a _ij。我们通过对所有可能的势进行分类来研究N = 3 的特定配置，并使用系统的对称性以精确和数字形式构建静态孤子。