Integrable equations of the infinite nonlinear Schrödinger equation hierarchy with time variable coefficients

Area of Science:

• Mathematical Physics
• Nonlinear Dynamics
• Integrable Systems

Background:

• Nonlinear Schrödinger equations are fundamental in describing wave phenomena.
• Integrable systems offer exact solutions and analytical tractability.
• Extending these systems to variable coefficients is a key challenge.

Purpose of the Study:

• To present an infinite hierarchy of integrable nonlinear Schrödinger equations.
• To demonstrate the integrability of this hierarchy using Lax pairs and Darboux transformations.
• To extend the formalism to systems with propagation-variable-dependent coefficients.

Main Methods:

• Development of recurrence relations for the equation hierarchy.
• Construction of Lax pairs for the entire hierarchy.
• Application of Darboux transformations to generate exact solutions.
• Analysis of the applicability of the formalism with variable coefficients.

Main Results:

• An infinite hierarchy of integrable nonlinear Schrödinger equations is established.
• Exact analytical solutions (wavefunctions) are derived using Darboux transformations.
• The Lax pair and Darboux transformation formalisms are shown to be applicable to generalized nonlinear systems with variable coefficients.
• A method for constructing complicated solutions in diversified domains is presented.

Conclusions:

• The presented hierarchy provides a powerful tool for studying nonlinear wave phenomena.
• The integrability and solution methods are robust and extendable to more complex systems.
• This work opens avenues for analyzing generalized nonlinear systems with time-varying properties.