Derive two-dimensional Whitham modulation equations for the 2DTL

Derive Whitham modulation equations for the two-dimensional Toda lattice to provide a quantitative modulation-theoretic description of its dispersive shock waves and their interactions.

Background

The paper analyzes the interaction of oblique dispersive shock waves in the two-dimensional Toda lattice primarily through exact one- and two-soliton solutions. Although the oblique wave profiles reduce locally to one-dimensional Toda traveling waves, the authors state that a genuinely two-dimensional Whitham modulation system is not currently available.

Such a system would place the soliton-based wedge analysis on a firmer theoretical foundation and would enable quantitative treatment of the full dispersive shock waves, rather than only their leading-edge solitons. It would also support stability analyses and the study of regimes not fully resolved by the present approach.

References

No Whitham modulation equations are available for the 2DTL to the best of our knowledge, although their derivation is a particularly intriguing question for future work.

— Wedge problems and dispersive shock waves in the two-dimensional Toda lattice  (2608.27415 - Calabrese et al., 27 Aug 2026) in Remark following Eq. (e:2DTL_obliqueleadingedgeamplitude), Section 4

A more precise analytical study and quantitative numerical investigation of both type~I regimes would require the use of Whitham modulation equations for the 2DTL. Such equations are not available at present, however. Such a study is therefore left as an open problem for future work.

— Wedge problems and dispersive shock waves in the two-dimensional Toda lattice  (2608.27415 - Calabrese et al., 27 Aug 2026) in Section 4.3, subsection “Type I wedges”