Lump, rogue wave, multi-waves and Homoclinic breather solutions for (2+1)-Modified Veronese Web equation

This work addresses the four main inducements: Lump, rogue wave, Homoclinic breather and multi-wave solutions for (2+1)-Modified Veronese Web (MVW) equation via Hirota bilinear approach and the ansatz technique. This model is a linearly degenerate integrable nonlinear partial differential equation (NLPDE) and can also be used to admit a differential covering with nonremoval physical parameters. By assuming the function [Formula: see text] in the Hirota bilinear form of the presented model as the general quadratic function, trigonometric function and exponential function form, also with appropriate set of parameters, we have prevented the lump, rogue wave, breather and multi-wave solutions successfully. A precise compatible wave transformation is utilized to obtain multi-wave solutions of governing model. Also, the motion track of the lump, Rogue wave and multi-waves is also explained both physically and theoretically. These new results contain some special arbitrary constants that can be useful to spell out diversity in qualitative features of wave phenomena.

Hyperplane-Approximation-Based Method for Many-Objective Optimization Problems with Redundant Objectives

For a many-objective optimization problem with redundant objectives, we propose two novel objective reduction algorithms for linearly and, nonlinearly degenerate Pareto fronts. They are called LHA and NLHA respectively. The main idea of the proposed algorithms is to use a hyperplane with non-negative sparse coefficients to roughly approximate the structure of the PF. This approach is quite different from the previous objective reduction algorithms that are based on correlation or dominance structure. Especially in NLHA, in order to reduce the approximation error, we transform a nonlinearly degenerate Pareto front into a nearly linearly degenerate Pareto front via a power transformation. In addition, an objective reduction framework integrating a magnitude adjustment mechanism and a performance metric [Formula: see text] are also proposed here. Finally, to demonstrate the performance of the proposed algorithms, comparative experiments are done with two correlation-based algorithms, LPCA and NLMVUPCA, and with two dominance-structure-based algorithms, PCSEA and greedy [Formula: see text]MOSS, on three benchmark problems: DTLZ5(I,M), MAOP(I,M), and WFG3(I,M). Experimental results show that the proposed algorithms are more effective.

Global Riemann solvers for several 3 × 3 systems of conservation laws with degeneracies

We study several [Formula: see text] systems of conservation laws, arising in the modeling of two-phase flow with rough porous media and traffic flow with rough road condition. These systems share several features. The systems are of mixed type, with various degeneracies. Some families are linearly degenerate, while others are not genuinely nonlinear. Furthermore, along certain curves in the domain, the eigenvalues and eigenvectors of different families coincide. Most interestingly, in some suitable Lagrangian coordinate, the systems are partially decoupled, where some unknowns can be solved independently of the others. Finally, in special cases, the systems reduce to some [Formula: see text] models, which have been studied in the literature. Utilizing the insights gained from these features, we construct global Riemann solvers for all these models. Possible treatments on the Cauchy problems are also discussed.

Local exact boundary controllability of entropy solutions to linearly degenerate quasilinear hyperbolic systems of conservation laws

In this paper, we study the local exact boundary controllability of entropy solutions to linearly degenerate quasilinear hyperbolic systems of conservation laws with characteristics of constant multiplicity. We prove the two-sided boundary controllability, the one-sided boundary controllability and the two-sided boundary controllability with fewer controls, by applying the strategy used in [T. Li and L. Yu, J. Math. Pures et Appl. 107 (2017) 1–40; L. Yu, Chinese Ann. Math., Ser. B (To appear)]. Our constructive method is based on the well-posedness of semi-global solutions constructed by the limit of ε-approximate front tracking solutions to the mixed initial-boundary value problem with general nonlinear boundary conditions, and on some further properties of both ε-approximate front tracking solutions and limit solutions.

Classical solutions to the relativistic Euler equations for a linearly degenerate equation of state

We are concerned with the global existence and blowup of the classical solutions to the Cauchy problem of one-dimensional isentropic relativistic Euler equations (Chaplygin gas, pressureless perfect fluid and stiff matter) with linearly degenerate characteristics. We at first derive the exact representation formula for all the fluids by the property of linearly degenerate. Then for the Chaplygin gas and the pressureless perfect fluid, we give a classification of the initial data that leads to the global existence and the blowup of the classical solution, respectively. We construct, especially, a class of initial data that contributes to the formation of “cusp-type” singularity and study the evolution of the solution after blowup by introducing a weak solution called delta shock wave. At last, for the stiff matter, we show that this system is indeed a linear system and prove the global existence of the classical solution to this fluid.

The piston problem in hyperelasticity with the stored energy in separable form

The piston problem for a hyperelastic hyperbolic conservative model where the stored energy is given in separable form is studied. The eigenfields corresponding to the hyperbolic system are of three types: linearly degenerate fields (corresponding to the contact characteristics), the fields which are genuinely nonlinear in the sense of Lax (corresponding to longitudinal waves), and, finally, nonlinear fields which are not genuinely nonlinear (corresponding to transverse waves). Taking the initial state free of stresses, we presented possible auto-similar solutions to the piston problem. In particular, we have shown that the equations admit transverse shock waves having a remarkable property: the solid density is decreasing through such a shock, it is thus a ‘rarefaction’ shock.