The equal width wave (EW) equation is a model partial differential equation for the simulation of one-dimensional wave propagation in nonlinear media with dispersion processes. The EW-Burgers equation models the propagation of nonlinear and dispersive waves with certain dissipative effects. In this work, we derive exact solitary wave solutions for the general form of the EW equation and the gen...

A class of model equations that describe the bi-directional propagation of small amplitude long waves on the surface of shallow water is derived from two-dimensional potential flow equations at various orders of approximation in two small parameters, namely the amplitude parameter a 1⁄4 a=h0 and wavelength parameter b 1⁄4 ðh0=lÞ2, where a and l are the actual amplitude and wavelength of the sur...

A Bose-Einstein condensate (BEC) is a quantum fluid that gives rise to interesting shock wave nonlinear dynamics. Experiments depict a BEC that exhibits behavior similar to that of a shock wave in a compressible gas, eg. traveling fronts with steep gradients. However, the governing Gross-Pitaevskii (GP) equation that describes the mean field of a BEC admits no dissipation hence classical dissip...

We study a dispersive counterpart of the classical gas dynamics problem of the interaction of a shock wave with a counter-propagating simple rarefaction wave often referred to as the shock wave refraction. The dispersive shock wave (DSW) refraction due to its head-on collision with the centred rarefaction wave (RW) is considered in the frameworks of the one-dimensional defocusing nonlinear Schr...

We classify 2+1 dimensional integrable systems with nonlocality of the intermediate long wave type. Links to waterbag system are established. Dimensional reductions constructed in this paper provide dispersive regularisations hydrodynamic equations governing propagation nonlinear waves a shear flow piecewise linear velocity profile (for special values vorticities).

We comment on traveling wave solutions and rational solutions to the 3+1 dimensional Kadomtsev–Petviashvili (KP) equations: (ut + 6uux + uxxx)x ± 3uyy ± 3uzz = 0. We also show that both of the 3+1 dimensional KP equations do not possess the three-soliton solution. This suggests that none of the 3+1 dimensional KP equations should be integrable, and partially explains why they do not pass the Pa...

Dispersive hydrodynamics is the domain of applied mathematics and physics concerned with fluid motion in which internal friction, e.g., viscosity, is negligible relative to wave dispersion. In conservative media such as superfluids, optical materials, and water waves, nonlinearity has the tendency to engender wavebreaking that is mitigated by dispersion. The mathematical framework for such medi...

With the aid of computerized symbolic computation, a new elliptic function rational expansion method is presented by means of a new general ans€atz and is very powerful to uniformly construct more new exact doubly-periodic solutions in terms of rational formal Jacobi elliptic function of nonlinear evolution equations (NLEEs). As an application of the method, we choose a (1+ 1)-dimensional dispe...

We obtain a dispersive long-time decay in weighted energy norms for solutions to the 1D wave equation with generic potential. The decay extends the results obtained by Murata for the 1D Schrödinger equation.

We classify 2+1 dimensional integrable systems with nonlocality of the intermediate long wave type. Links to waterbag system are established. Dimensional reductions constructed in this paper provide dispersive regularisations hydrodynamic equations governing propagation nonlinear waves a shear flow piecewise linear velocity profile (for special values vorticities).