Logarithmic nonlinear Schrödinger equation and irrotational, compressible flows: an exact solution

Area of Science:

• Fluid Dynamics
• Theoretical Physics
• Nonlinear Dynamics

Background:

• Compressible fluid flow analysis often involves complex nonlinear equations.
• Exact solutions are valuable for validating numerical methods and understanding fundamental flow behaviors.
• The Madelung transformation offers a unique approach to nonlinear fluid dynamics problems.

Purpose of the Study:

• To theoretically investigate a class of irrotational, isentropic, and compressible flows.
• To derive exact solutions for fluid properties and velocities using a novel mathematical framework.
• To demonstrate the applicability of the method across different dimensionalities.

Main Methods:

• Formulation of density and velocity potential using the Madelung transformation.
• Solving the resulting nonlinear Schrödinger equation via similarity variables.
• Derivation of an exact analytical solution applicable for any specific heat ratio.

Main Results:

• Explicit expressions for fluid properties and velocities in terms of time and spatial coordinates.
• Density is analytically described as a Gaussian function of the similarity variable.
• Temperature is found to be solely a function of time.

Conclusions:

• The Madelung transformation provides a powerful tool for analyzing compressible flows.
• The derived exact solutions are general and applicable in 1D, 2D, and 3D geometries.
• The method facilitates exact formulation of complex scenarios, such as a 1D gas column with mass injection.