Spectral instability of peakons for a class of cubic quasilinear shallow-water equations

One of the mathematical models describing the behavior of two horizontally infinite adjoining layers of an ideal incompressible liquid under a solid cover moving at different speeds is investigated. At a large difference in the layer velocities, the Kelvin-Helmholtz instability occurs, which leads to a distortion of the interface. At the initial point in time, the interface is not necessarily flat. From a mathematical point of view, the behavior of the liquid layers is described by a system of four quasilinear equations, either hyperbolic or elliptic, in partial derivatives of the first order. Some type shallow water equations are used to construct the model. In the simple version of the model considered in this paper, in the spatially one-dimensional case, the unknowns are the boundary between the liquid layers h(x,t) and the difference in their velocities γ(x,t). The main attention is paid to the case of elliptic equations when |h|<1 and γ>1. An evolutionary Cauchy problem with arbitrary sufficiently smooth initial data is set for the system of equations. The explicit dependence of the Riemann invariants on the initial variables of the problem is indicated. To solve the Cauchy problem formulated in terms of Riemann invariants, a variant of the hodograph method based on a certain conservation law is used. This method allows us to convert a system of two quasilinear partial differential equations of the first order to a single linear partial differential equation of the second order with variable coefficients. For a linear equation, the Riemann-Green function is specified, which is used to construct a two-parameter implicit solution to the original problem. The explicit solution of the problem is constructed on the level lines (isochrons) of the implicit solution by solving a certain Cauchy problem for a system of ordinary differential equations. As a result, the original Cauchy problem in partial derivatives of the first order is transformed to the Cauchy problem for a system of ordinary differential equations, which is solved by numerical methods. Due to the bulkiness of the expression for the Riemann-Green function, some asymptotic approximation of the problem is considered, and the results of calculations, and their analysis are presented.

This paper is concerned with two classes of cubic quasilinear equations, which can be derived as asymptotic models from shallow-water approximation to the 2D incompressible Euler equations. One class of the models has homogeneous cubic nonlinearity and includes the integrable modified Camassa–Holm (mCH) equation and Novikov equation, and the other class encompasses both quadratic and cubic nonlinearities. It is demonstrated here that both these models possess localized peaked solutions. By constructing a Lyapunov function, these peaked waves are shown to be dynamically stable under small perturbations in the natural energy space $H^1$, without restriction on the sign of the momentum density. In particular, for the homogeneous cubic nonlinear model, we are able to further incorporate a higher-order conservation law to conclude orbital stability in $H^1\cap W^{1,4}$. Our analysis is based on a strong use of the conservation laws, the introduction of certain auxiliary functions, and a refined continuity argument.

Saint-Venant equations are a set of quasi-linear partial differential equations that represent the canal hydraulics. It is difficult to adapt to control. In this paper, by using a spatial discretization and linearization method, a simplified state-space model of irrigation canals is derived from the Saint-Venant equations. In order to validate this model, under the same initial and boundary conditions, experimental data of canals was obtained from Simulation and Integration of Control for Canals (SICC) software. Comparisons of data outputs of the simplified model and SICC demonstrate the accuracy of the validation.

The method of non‐standard finite elements was used to develop multilevel difference schemes for linear and quasilinear hyperbolic equations with Dirichlet boundary conditions. A closed form equation of kth‐order accuracy in space and time (O(Δtk, Δxk)) was developed for one‐dimensional systems of linear hyperbolic equations with Dirichlet boundary conditions. This same equation is also applied to quasilinear systems. For the quasilinear systems a simple iteration technique was used to maintain the kth‐order accuracy.Numerical results are presented for the linear and non‐linear inviscid Burger's equation and a system of shallow water equations with Dirichlet boundary conditions.

Mathematical Aspects of Numerical Solution of Hyperbolic Systems. Monographs and Surveys in Pure and Applied Mathematics, Vol. 118

In the control of open-channel, it is difficult to estimate wave propergation time because of the complexity of reflection, superposition and energy attenuation of shallow water waves. Although Saint-Venant equation have reasonable accuracy in describing the motion law of unsteady flow, mathematically it is the first order quasilinear hyperbolic partial differential equations which makes it difficult to be used as control model in the design optimized controller. The Channel Integrator Delay Zero (IDZ) model linearizes the Saint-Venant equation and ensures the accuracy of the response of water level and discharge in high frequency band. However, there is still a considerable difference between the theoretical value and the actual response. In order to avoid this difference caused by theoretical derivation, this paper uses the single channel model of Zhanghe Irrigation District to play online identification of the relevant parameters by using the existing periodic response process of the canal system. The reliability of model identification under step water intake is verified, and the effect of this method on periodic water intakes in long channel is verified. The results show that the identified model can catch most dynamic action of the canal system with water intakes, which ensures its validity to be used for controller design. Meanwhile, it is simple in application.

This note presents a set of systems of two first-order quasi-linear partial differential equations, which can be reduced to the shallow-water equations. This set includes equations describing a two-layered fluid flow.

The quasi-linear partial differential equations known as the shallow-water equations describe the flow of water in open channels or over sloping planes (overland flow). Because no analytic solution exists for these equations, finite-difference methods must be used to obtain solutions. Formulation of finite-difference schemes involves consideration of the convergence of the finite-difference solutions to the true solution of the equation. A theoretical examination of the approximation of several difference operators, both implicit and explicit, is presented and the stability of these schemes is examined empirically for flow over a plane with critical depth downstream boundary condition and a zero inflow upstream boundary condition. A finite-difference scheme based on the method of characteristics was found to be satisfactory in many cases. Explicit methods were found to be not suitable for this problem except in some special cases.

On the Cauchy problem for a class of cubic quasilinear shallow-water equations

We are concerned with the Riemann problem for the system of quasilinear integrodifferential equations governing the propagation of the ro­tational long waves on the surface of an ideal incompressible fluid. The study is based on the generalizations of the notions of characteristics and hyper­bolicity. The theory of simple waves and shocks is developed for the model of vortex shallow water. The Riemann solution is constructed for the special class of initial conditions.

On a two dimensional nonlocal shallow-water model

Wave-breaking phenomena for a new weakly dissipative quasilinear shallow-water waves equation

A nonlocal shallow-water model with the weak Coriolis and equatorial undercurrent effects

We derive a quasilinear shallow water equation directly from the governing equations for gravity water waves within a certain regime for large-amplitude waves which has not been studied so far. Furthermore, we demonstrate local well-posedness of the corresponding Cauchy problem and finally discuss some aspects of the blowup behavior of solutions.

Motivated by the question whether higher-order nonlinear model equations, which go beyond the Camassa-Holm regime of moderate amplitude waves, could point us to new types of waves profiles, we study the traveling wave solutions of a quasilinear evolution equation which models the propagation of shallow water waves of large amplitude. The aim of this paper is a complete classification of its traveling wave solutions. Apart from symmetric smooth, peaked and cusped solitary and periodic traveling waves, whose existence is well-known for moderate amplitude equations like Camassa-Holm, we obtain entirely new types of singular traveling waves: periodic waves which exhibit singularities on both crests and troughs simultaneously, waves with asymmetric peaks, as well as multi-crested smooth and multi-peaked waves with decay. Our approach uses qualitative tools for dynamical systems and methods for integrable planar systems.