Logarithmic Schrödinger-like equation as a model for magma transport

Abstract

We show that, under a suitable assumption on the pressure tensor, the mass and momentum balance equations of hydrodynamical theory, introduced in the early 1980s by many authors to describe matrix separation, yield the equations of the Madelung fluid that are equivalent to the Schrödinger-like equation with logarithmic nonlinearity. This equation has solitary-waves solutions as required by many experimental volcanic models.

Schrödinger Equations with Logarithmic Self-Interactions: From Antilinear PT-Symmetry to the Nonlinear Coupling of Channels

“…One of the best known examples is provided by [16,17] in which the toy models are considered in the form of the nonlinear logarithmic Schrödinger Equation (5) with the wave-function solutions ψ ∈ L 2 ( d ) studied in an interval of time t ∈ (t 0 , t 1 ). This equation, along with its relativistic analogue, finds multiple applications in the physics of quantum fields and particles [49][50][51][52][53][54][55], extensions of quantum mechanics [16,56], optics and transport or diffusion phenomena [57][58][59][60], nuclear physics [61,62], the theory of dissipative systems and quantum information [63][64][65][66][67][68], the theory of superfluidity [69][70][71][72] and the effective models of the physical vacuum and classical and quantum gravity [73][74][75][76], where one can utilize the well-known fluid/gravity analogy between inviscid fluids and pseudo-Riemannian manifolds [77][78][79][80][81]. The relativistic analogue of Equation (5) is obtained by replacing the derivative part with the d'Alembert operator and is not considered here.…”

On the linearity of the Schrödinger equation

“…The logarithmic Schrödinger equation outlived these defeats but not as a fundamental theory. Owing to its unique properties it has been used as an exactly soluble model of nonlinear phenomena in nonlinear optics [16,17], in nuclear physics [14], in the study of dissipative systems [15], in geophysics [18], and even in computer science [19]. The logarithmic nonlinearity is also theoretically appealing due to its connection with stochastic dynamics [22][23][24].…”

Application of Initial Boundary Value Problem on Logarithmic Wave Equation in Dynamics

“…This type of equation arising from many applications in many branches of physics such as nuclear physics, optics and geophysics [7,9,10]. where is a finite interval [a,b], the parameter measures the force of the nonlinear interaction and the nonlinear effects in quantum mechanics are very small.…”