Elliptic solutions of isentropic ideal compressible fluid flow in (3+1) dimensions

TL;DR

This paper develops a method to find new elliptic solutions for isentropic ideal compressible fluid flow in four-dimensional space, focusing on bounded solutions expressed via Weierstrass P-functions, even under gradient catastrophe conditions.

Contribution

It introduces a modified conditional symmetry method to construct bounded elliptic solutions involving Weierstrass P-functions for (3+1)-dimensional fluid flow equations.

Findings

Derived new classes of elliptic solutions expressed with Weierstrass P-functions.

Showed solutions remain bounded despite gradient catastrophe.

Provided examples including bumps, kinks, and multi-wave solutions.

Abstract

A modified version of the conditional symmetry method, together with the classical method, is used to obtain new classes of elliptic solutions of the isentropic ideal compressible fluid flow in (3+1) dimensions. We focus on those types of solutions which are expressed in terms of the Weierstrass P-functions of Riemann invariants. These solutions are of special interest since we show that they remain bounded even when these invariants admit the gradient catastrophe. We describe in detail a procedure for constructing such classes of solutions. Finally, we present several examples of an application of our approach which includes bumps, kinks and multi-wave solutions.