Noether symmetries, solutions and conserved quantities of a new (3+1)-dimensional Painlevé integrable fifth-order equation with third-order temporal dispersion: Multi-analytical approaches applicable in optics, oceanography and astronomy

This article comprehensively reveals the analytical investigations carried out on a new (3+1)-dimensional Painlevé integrable fifth-order equation with third-order temporal dispersion. The well-celebrated Noether's theorem is engaged to comprehensively construct conserved vectors of the underlying equation. The use of the popular theorem makes it possible to calculate interesting nonlocal conservation laws, thus producing diverse conserved quantities of notes which are associated with the integrable model. In addition, the equation is reduced to its corresponding nonlinear ordinary differential equation using a plane wave transformation. Direct integration of the achieved differential equation occasions the emergence of elliptic integral solution and Weierstrass function solution, where the former constitutes the most general exact solution of the underlying equation. Moreover, two standard analytical techniques are deployed to obtain closed-form solutions of the understudy model. In consequence, one achieves both complex and non-complex exponential function solutions. Additionally, the physical meaning of the results is put on the front burner by revealing the wave dynamics of these solutions.