Of the Lax Representation of the Derivative Nonlinear Schrödinger Equation Using the Kaup–newell Spectral Problem

In modern mathematical physics and the theory of nonlinear differential equations, the study of Schrödinger-type equations plays a crucial role. Specifically, derivative Schrödinger equations are widely used to model quantum mechanics, plasma physics, nonlinear optics, and various wave phenomena. Investigating the solutions of such equations in infinite domains, particularly within the class of rapidly decreasing functions (Schwartz space), is of significant theoretical and practical interest. This paper is devoted to the integration of the derivative Schrödinger equation within this specific function class, thoroughly analyzing the smoothness, behavior, and estimates of the solutions.