We consider two physically and mathematically distinct regularization mechanisms of scalar
hyperbolic conservation laws. When the flux is convex, the combination of diffusion and …

Extended shallow water wave equations are derived, using the method of asymptotic
expansions, from the Euler (or water wave) equations. These extended models are valid one …

The concept of soliton gas was introduced in 1971 by Zakharov as an infinite collection of
weakly interacting solitons in the framework of Korteweg–de Vries (KdV) equation. In this …

We review the spectral theory of soliton gases in integrable dispersive hydrodynamic
systems. We first present a phenomenological approach based on the consideration of …

A negative effective mass can be realized in quantum systems by engineering the
dispersion relation. A powerful method is provided by spin-orbit coupling, which is currently …

Solitons and breathers are localized solutions of integrable systems that can be viewed as
“particles” of complex statistical objects called soliton and breather gases. In view of the …

We explore the dynamics and interactions of multiple bright droplets and bubbles, as well as
the interactions of kinks with droplets and with antikinks, in the extended one-dimensional …

The Riemann problem of the high‐order Jaulent–Miodek (JM) equation with initial data of
step discontinuity is explored by Whitham modulation theory, which is a modified version of …

A new numerical method for solving the Serre–Green–Naghdi (SGN) equations describing
dispersive waves on shallow water is proposed. From the mathematical point of view, the …

We analyze the large-n behavior of soliton solutions of the integrable focusing nonlinear
Schrödinger equation with associated spectral data consisting of a single pair of conjugate …