Integrable nonlinear Schrödinger equation on simple networks: connection formula at vertices

Area of Science:

• Nonlinear dynamics
• Mathematical physics
• Complex systems

Background:

• The nonlinear Schrödinger equation (NLSE) is a fundamental model in various scientific fields.
• Integrability of NLSE on complex network structures with varying parameters remains an active research area.

Purpose of the Study:

• To investigate the conditions under which NLSE on networks with bond-dependent nonlinearity becomes completely integrable.
• To analyze the properties of soliton solutions and transmission probabilities in such network systems.

Main Methods:

• Derivation of connection formulas at vertices based on norm and energy conservation.
• Analysis of soliton solutions for the completely integrable NLSE on a 1D chain.
• Generalization of network structures beyond branched chains to include star, tree, and loop graphs.
• Calculation of transmission probabilities using reflectionless soliton propagation.

Main Results:

• NLSE on networks with specific nonlinearity sum rules at vertices exhibits complete integrability and infinite constants of motion.
• Soliton solutions on individual bonds are scaled versions of universal 1D solutions, modified by bond nonlinearity.
• Transmission probabilities on outgoing bonds are inversely proportional to the bond's nonlinearity strength.

Conclusions:

• The study establishes conditions for complete integrability of NLSE on diverse network topologies.
• The findings provide insights into soliton behavior and propagation in complex nonlinear systems.
• Numerical evidence supports the theoretical predictions regarding soliton transmission probabilities.