Generating a chain of maps which preserve the same integral as a given map

We generalise the concept of duality to systems of ordinary difference equations (or maps). We propose a procedure to construct a chain of systems of equations which are dual, with respect to an integral $H$, to the given system, by exploiting the integral relation, defined by the upshifted version and the original version of $H$. When the numerator of the integral relation is biquadratic or multi-linear, we point out conditions where a dual fails to exists. The procedure is applied to several two-component systems obtained as periodic reductions of 2D lattice equations, including the nonlinear Schr"{o}dinger system, the two-component potential Korteweg-De Vries equation, the scalar modified Korteweg-De Vries equation, and a modified Boussinesq system.

💡 Research Summary

The paper extends the notion of duality, previously defined for scalar ordinary difference equations, to systems of ordinary difference equations (maps) with multiple components. Starting from a 2d‑dimensional state space M=S^{2d} with variables (u,v), the authors consider a map F that shifts each component forward and replaces the last components u_d and v_d by rational functions f₁(u,v) and g₁(u,v). Assuming the original system possesses n integrals H₁,…,H_n, they form a linear combination H=∑α_k H_k, which is also an integral. The key observation is that the difference ΔH=H(u′,v′)−H(u,v) vanishes on the orbit, and its numerator N, viewed as a polynomial in the new variables u_{d+1}=f₁(u,v) and v_{d+1}=g₁(u,v), determines whether a dual system can be constructed.

If N is bi‑quadratic in (u_{d+1},v_{d+1}) and does not satisfy certain coefficient relations (equations (2.3)–(2.4) in the paper), the authors show that the equation H(u_{d+1},v_{d+1})=0 can be solved for the “other” root, yielding alternative functions f₂(u,v) and g₂(u,v). This produces two new maps, (f₁,g₂) and (f₂,g₁), each preserving the same integral H. By iterating the process—substituting f_k or g_k back into the integral relation and solving for the complementary component—one generates a chain of dual maps (f_k,g_k). The chain terminates when a pair repeats, i.e., f_{k}=f_{k+1} and g_{k}=g_{k+1}.

The authors also discuss cases where N is merely bilinear; then the integral relation determines a unique partner, and no dual can be formed via the proposed procedure. This observation is formalised in Proposition 2.1, which gives necessary and sufficient conditions for the non‑existence of duals when N is bi‑quadratic.

To assess the integrability of the generated maps, the paper adopts a complexity measure based on degree growth. For a rational map, the degree d_n of the n‑th iterate is recorded; linear growth signals linearizability, polynomial growth indicates integrability (in the Liouville‑Arnold sense), and exponential growth points to non‑integrability. The authors compute d_n for the first 20–30 iterates using affine initial data, which is sufficient to infer the asymptotic behaviour.

The methodology is applied to several concrete examples derived from periodic reductions of two‑dimensional lattice equations:

1. Nonlinear Schrödinger (NLS) system – The integrals are multilinear, making N linear rather than bi‑quadratic; consequently, the procedure yields no dual maps.

2. Two‑component potential KdV (pKdV) system – After a (2,1) reduction, the map is four‑dimensional. The integral H₁ is bilinear, so it does not generate duals, but H₂ is bi‑quadratic. Using H₂, a closed chain of four dual maps is produced. The first dual is linearizable, the second exhibits polynomial degree growth (hence integrable), and the third shows exponential growth (non‑integrable).

3. (2,4) reduction of the scalar modified KdV equation – The structure mirrors the pKdV case, leading to a similar chain of dual maps with analogous integrability properties.

4. Modified Boussinesq system – By employing a generic (d+1,−1) reduction, the authors obtain a d‑dimensional map. For a particular integral, they construct a closed chain of six dual systems. One dual coincides with the original map, another is a periodic map, and the remaining four are interrelated, forming a d‑dimensional generalisation of the alternating QR‑T map. Degree growth analysis shows that two of these are integrable (polynomial growth) while the others are non‑integrable (exponential growth).

Throughout the examples, the authors demonstrate that the existence of a dual chain depends critically on the algebraic form of the integral’s numerator. When the numerator is bi‑quadratic and the coefficient conditions of Proposition 2.1 are violated, a rich hierarchy of dual maps emerges, some of which inherit integrability from the original system, while others become chaotic.

In conclusion, the paper provides a systematic algorithm for generating families of maps that preserve a given integral, extends duality to multi‑component systems, and supplies practical criteria (based on the bi‑quadratic structure and coefficient constraints) for the existence of such duals. The degree‑growth complexity test serves as an efficient diagnostic for integrability of the resulting maps. This framework opens new avenues for discovering integrable discrete dynamical systems and for constructing non‑integrable examples with prescribed conserved quantities.