# Time-dependent defects in integrable soliton equations

## TL;DR

This paper investigates integrable (1+1)-dimensional soliton equations with time-dependent defects, demonstrating that such defects preserve integrability and allowing for peaked soliton solutions.

## Contribution

It introduces a novel approach to modeling time-dependent defects as Bäcklund transformations, showing they do not disrupt integrability and lead to unique peaked soliton solutions.

## Key findings

- Defects modeled as Bäcklund transformations preserve integrability.
- Time-dependent defects allow for peaked soliton solutions.
- The defect condition is evaluated at a moving point c(t).

## Abstract

We study $(1+1)$-dimensional integrable soliton equations with time-dependent defects located at $x=c(t)$, where $c(t)$ is a function of class $C^1$. We define the defect condition as a B\"{a}cklund transformation evaluated at $x=c(t)$ in space rather than over the full line. We show that such a defect condition does not spoil the integrability of the system. We also study soliton solutions that can meet the defect for the system. An interesting discovery is that the defect system admits peaked soliton solutions.