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Wave structures for non-power potentials in discrete conservation laws

Investigate the discrete lattice du_n/dt + (1/2)\,\Phi'(u_n)(u_{n+1}−u_{n-1}) derived from the continuum conservation law u_t + [\Phi(u)]_x = 0 with non-power potentials \Phi(u) (e.g., exponential), and determine whether Riemann problems generate wave structures beyond dispersive shock waves and rarefaction waves; characterize any such structures.

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Background

The paper primarily considers power-law potentials leading to a lattice of the form du_n/dt + u_n{\widetilde{p}}(u_{n+1} − u_{n-1}) = 0, where DSWs and RWs were observed and analyzed with quasi-continuum models and DSW fitting.

The authors propose extending the paper to non-power potentials \Phi(u) in the general discrete conservation law du_n/dt + (1/2)\,\Phi'(u_n)(u_{n+1} − u_{n-1}) = 0 to explore whether different dispersive structures emerge, potentially enriching the phenomenology beyond DSWs and RWs.

References

There are still a variety of interesting open questions remaining, and we only list a few of them. In addition, we notice that one can also investigate the lattice in Eq.~eq: general discrete conservation law where the potential $\Phi$ is not a power function. For instance, an interesting alternative potential can be the exponential function. The reason to change the potential $\Phi$ is as follows. We recall that we have only discovered the DSW and RW in the lattice eq: general discrete conservation law when a power-type potential is applied. Hence, a different type of potential may excite wave structures other than the DSW and RW so that the lattice becomes more versatile as a media to model various nonlinear dispersive waves.

eq: general discrete conservation law:

ut+[Φ(u)]x=0,u_t + \left[\Phi(u)\right]_x = 0,

— Quasi-continuum approximations for nonlinear dispersive waves in general discrete conservation laws (2509.04630 - Yang, 4 Sep 2025) in Conclusions and future directions