Summation of multiplicative-inverse series for the non-integrable coupled mKdV system

Determine whether the ratio equalities for the multiplicative inverse coefficient sequences derived from the series U(ζ)/ζ, V(ζ)/ζ, and W(ζ)/ζ—specifically, (tilde U_{n+1}(ζ))/(tilde U_n(ζ)) = (tilde U_{n+2}(ζ))/(tilde U_{n+1}(ζ)), (tilde V_{n+1}(ζ))/(tilde V_n(ζ)) = (tilde V_{n+2}(ζ))/(tilde V_{n+1}(ζ)), and (tilde W_{n+1}(ζ))/(tilde W_n(ζ)) = (tilde W_{n+2}(ζ))/(tilde W_{n+1}(ζ)) for n ≥ 1—admit any nontrivial solutions that yield a convergent geometric series, and, if so, construct the resulting closed-form generating functions for U(ξ), V(ξ), and W(ξ) corresponding to the non-integrable coupled modified KdV system given by equations (ex2eq1)–(ex2eq3) after the traveling-wave reduction and substitution e^{σ1 ξ} = ζ.

Background

The paper introduces a modified method of variation of parameters (MMVP) to obtain exact solutions for systems of nonlinear ODEs, and applies it to a non-integrable coupled mKdV-type system. MMVP yields a series solution for the traveling-wave reduction of this system, but, due to non-integrability, summing the series to a closed form is challenging.

To address summation, the authors use the multiplicative inverse of the power series and propose two techniques: truncating the series and summing it by imposing a common-ratio condition so that the series becomes geometric. In Example 1, they attempt to enforce common-ratio equalities on the multiplicative inverse coefficient sequences (tilde U_n, tilde V_n, tilde W_n) but are unable to find any suitable solution for these equations for any n, preventing construction of a generating function via this route.

This unresolved step limits the use of the multiplicative inverse summation technique for the non-integrable case without parameter restrictions, leaving open whether a direct geometric-series summation of the inverse series is possible and what the corresponding closed forms would be if it is.