# GBDT and explicit solutions for the matrix coupled dispersionless equations (local and nonlocal cases)

## TL;DR

This paper introduces matrix coupled dispersionless equations, constructs explicit solutions, and analyzes their asymptotic behavior, covering local, nonlocal, and scalar cases to advance understanding of these integrable systems.

## Contribution

It provides the first comprehensive explicit solutions and asymptotic analysis for matrix coupled dispersionless equations, including local and nonlocal variants.

## Key findings

- Constructed explicit multipole solutions.
- Derived explicit Darboux and wave matrix functions.
- Analyzed asymptotic behavior in key cases.

## Abstract

We introduce matrix coupled (local and nonlocal) dispersionless equations, construct wide classes of explicit multipole solutions, give explicit expressions for the corresponding Darboux and wave matrix valued functions and consider their asymptotics in some interesting cases. We consider the scalar cases of coupled, complex coupled and nonlocal dispersionless equations as well.