# Remarks on existence/nonexistence of analytic solutions to higher order KdV equations

## TL;DR

This paper investigates the existence of analytic solutions to higher order KdV equations, revealing that while second-order solutions resemble KdV solutions, multi-soliton solutions do not exist for higher orders.

## Contribution

It demonstrates the nonexistence of multi-soliton solutions in higher order KdV equations beyond second order, extending understanding of their solution structures.

## Key findings

- Extended KdV (KdV2) admits three types of analytic solutions similar to KdV.
- Multi-soliton solutions do not exist for KdV2.
- Higher order KdV equations lack analytic solutions in KdV-like forms.

## Abstract

In this note, we discuss the existence of analytic solutions to the nonlinear wave equations of the higher order than the ubiquitous Korteweg-de Vries (KdV) equation. First, we recall our recent results which show that the extended KdV equation (KdV2), that is, the equation obtained within second-order perturbation approach possesses three kinds of analytic solutions. These solutions have the same functional form as the corresponding KdV solutions. We show, however, that the most intriguing multi-soliton solutions, known for the KdV equation, do not exist for KdV2. Moreover, we show that for the equations obtained in the third order perturbation approach (and then in any higher order) analytic solutions in the forms known from KdV theory do not exist.