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Quasi-continuum models for large-jump DSWs when p̃ > 1

Develop a quasi-continuum partial differential equation approximation for the discrete lattice du_n/dt + u_n^{\widetilde{p}}(u_{n+1}−u_{n-1}) = 0 that accurately captures the dispersive shock wave dynamics arising from the Riemann problem when \widetilde{p} > 1 and the jump \Delta = u^- − u^+ is large, potentially by employing different slow-variable scalings than X = \epsilon n, T = \epsilon t and/or an amplitude scaling of the field u.

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Background

The paper studies the non-integrable discrete lattice du_n/dt + u_n{\widetilde{p}}(u_{n+1} − u_{n-1}) = 0 that discretizes the conservation law u_t + [2/(\widetilde{p}+1) u{\widetilde{p}+1}]_x = 0 and observes both dispersive shock waves (DSWs) and rarefaction waves (RWs) in Riemann problems.

Two quasi-continuum PDE models are derived: a non-regularized model u_T + u{\widetilde{p}}(2u_X + (\epsilon2/3)u_{XXX}) = 0 and a regularized BBM-type model (u{1−\widetilde{p}})_T − (\epsilon2/6)(u{1−\widetilde{p}})_{XXT} = −2(1−\widetilde{p})u_X. Numerical comparisons show these approximations work well for small \widetilde{p} and small jumps, but degrade for \widetilde{p} > 1 with large \Delta.

The authors identify the need for alternative continuum scalings or amplitude rescalings to improve the approximation of lattice DSWs in the challenging regime \widetilde{p} > 1 and large jump \Delta.

References

There are still a variety of interesting open questions remaining, and we only list a few of them. Firstly, we have mentioned in the comparison section \ref{sec: numerical vali} that, for the case when $\widetilde{p} > 1$, both quasi-continuum models failed to yield a good approximation to the DSW of the lattice eq: extension 2 if the jump $\Delta$ is large. This particular failure of the two quasi-continuum models will naturally lead one to think of other potential dispersive quasi-continuum models which can be possibly derived by performing a different set of change of spatial and temporal variables rather than Eq.~eq: Slow variables and also a possible scaling of the amplitude of the wave field of $u$.

eq: extension 2:

dundt+(un)p~(un+1−un−1)=0.\frac{d u_{n}}{dt} + \left(u_n\right)^{\widetilde{p}}\left(u_{n+1} - u_{n-1}\right) = 0.

eq: Slow variables:

X=ϵn,T=ϵt,X = \epsilon n, \quad T = \epsilon t,

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