# On a method for constructing the Lax pairs for integrable models via quadratic ansatz

## TL;DR

This paper introduces a quadratic ansatz-based method for constructing Lax pairs of integrable models, demonstrated on KdV and Volterra chain, leading to new Lax pairs and conservation laws.

## Contribution

It presents a novel quadratic form approach to derive nonlinear Lax pairs and their linearization, expanding tools for analyzing integrable systems.

## Key findings

- New Lax pairs for KdV and Volterra chain
- Formal asymptotic expansions of eigenfunctions
- Infinite series of conservation laws obtained

## Abstract

A method for constructing the Lax pairs for nonlinear integrable models is suggested. First we look for a nonlinear invariant manifold to the linearization of the given equation. Examples show that such invariant manifold does exist and can effectively be found. Actually it is defined by a quadratic form. As a result we get a nonlinear Lax pair consisting of the linearized equation and the invariant manifold. Our second step consists of finding an appropriate change of the variables to linearize the found nonlinear Lax pair. The desired change of the variables is again defined by a quadratic form. The method is illustrated by the well-known KdV equation and the modified Volterra chain. New Lax pairs are found. The formal asymptotic expansions for their eigenfunctions are constructed around the singular values of the spectral parameter. By applying the method of the formal diagonalization to these Lax pairs the infinite series of the local conservation laws are obtained for the corresponding nonlinear models.