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A search for integrable evolution equations with Lax pairs over the octonions

Stephen C. Anco, Philic Lam, Thomas Wolf

TL;DR

This work extends integrable theory to octonion-valued fields by systematically constructing Lax pairs for octonion evolution equations using a scaling-based undetermined-coefficient approach. Reformulating the Lax pair in an isospectral framework allows nonassociative octonion products to be handled, and a polynomial ansatz with octonion-based basis expansions yields several new integrable equations, including a KdV-type equation $oldsymbol{u}_t=oldsymbol{ extalpha}(oldsymbol{u}oldsymbol{u}_x+oldsymbol{u}_xoldsymbol{u})+oldsymbol{u}_{xxx}$ and multiple mKdV-type families with rich Lax-pair structures. The results reveal two main mKdV-type families and a KdV-type family, each accompanied by several Lax pairs, with real reductions connecting to scalar KdV/mKdV forms and higher-weight constructions yielding additional, novel variants. The findings broaden the scope of inverse scattering techniques to non-associative algebras and suggest future work on broader scaling weights, complexifications, and REDUCE-based automation. $${ extstyle oldsymbol{u}_t=oldsymbol{ extalpha}(oldsymbol{u}oldsymbol{u}_x+oldsymbol{u}_xoldsymbol{u})+oldsymbol{u}_{xxx}}$$ and related Lax structures underpin these contributions.

Abstract

Four new integrable evolutions equations with operator Lax pairs are found for an octonion variable. The method uses a scaling ansatz to set up a general polynomial form for the evolution equation and the Lax pair, using KdV and mKdV scaling weights. A condition for linear differential operators to be a Lax pair over octonions is formulated and solved for the unknown coefficients in the polynomials.

A search for integrable evolution equations with Lax pairs over the octonions

TL;DR

This work extends integrable theory to octonion-valued fields by systematically constructing Lax pairs for octonion evolution equations using a scaling-based undetermined-coefficient approach. Reformulating the Lax pair in an isospectral framework allows nonassociative octonion products to be handled, and a polynomial ansatz with octonion-based basis expansions yields several new integrable equations, including a KdV-type equation and multiple mKdV-type families with rich Lax-pair structures. The results reveal two main mKdV-type families and a KdV-type family, each accompanied by several Lax pairs, with real reductions connecting to scalar KdV/mKdV forms and higher-weight constructions yielding additional, novel variants. The findings broaden the scope of inverse scattering techniques to non-associative algebras and suggest future work on broader scaling weights, complexifications, and REDUCE-based automation. and related Lax structures underpin these contributions.

Abstract

Four new integrable evolutions equations with operator Lax pairs are found for an octonion variable. The method uses a scaling ansatz to set up a general polynomial form for the evolution equation and the Lax pair, using KdV and mKdV scaling weights. A condition for linear differential operators to be a Lax pair over octonions is formulated and solved for the unknown coefficients in the polynomials.

Paper Structure

This paper contains 10 sections, 1 theorem, 39 equations.

Table of Contents

  1. Introduction
  2. Outline of Method
  3. KdV scaling-weight equations
  4. Results
  5. mKdV scaling-weight equations
  6. Results I
  7. Results II
  8. Higher weight $w_\mathbf{L}$
  9. Concluding remarks
  10. Acknowledgements

Key Result

Proposition 2.1

A Lax pair laxpair is equivalent to the linear system This system augmented by $L\psi=\lambda\psi$ is isospectral, namely $\lambda_t =0$.

Theorems & Definitions (1)

  • Proposition 2.1