Explicit multi-symplectic method for the Zakharov—Kuznetsov equation

We propose an explicit multi-symplectic method to solve the two-dimensional Zakharov-Kuznetsov equation. The method combines the multi-symplectic Fourier pseudospectral method for spatial discretization and the Euler method for temporal discretization. It is verified that the proposed method has corresponding discrete multi-symplectic conservation laws. Numerical simulations indicate that the proposed scheme is characterized by excellent conservation.     

Imaginary eigenvalues of Zakharov–Shabat problems with non-zero background

The focusing Zakharov–Shabat scattering problem on the infinite line with non-zero boundary conditions for the potential is studied, and sufficient conditions on the potential are identified to ensure that the problem admits only purely imaginary discrete eigenvalues. The results, which generalize previous work by Klaus and Shaw, are applicable to the study of solutions of the focusing nonlinear Schrödinger equation with non-zero background.     

A discontinuous Galerkin method for the Vlasov–Poisson system

A discontinuous Galerkin method for approximating the Vlasov–Poisson system of equations describing the time evolution of a collisionless plasma is proposed. The method is mass conservative and, in the case that piecewise constant functions are used as a basis, the method preserves the positivity of the electron distribution function and weakly enforces continuity of the electric field through mesh interfaces and boundary conditions. The performance of the method is investigated by computing several examples and error estimates of the approximation are stated. In particular, computed results are benchmarked against established theoretical results for linear advection and the phenomenon of linear Landau damping for both the Maxwell and Lorentz distributions. Moreover, two nonlinear problems are considered: nonlinear Landau damping and a version of the two-stream instability are computed. For the latter, fine scale details of the resulting long-time BGK-like state are presented. Conservation laws are examined and various comparisons to theory are made. The results obtained demonstrate that the discontinuous Galerkin method is a viable option for integrating the Vlasov–Poisson system.     

Asymptotic-Preserving Particle-In-Cell method for the Vlasov–Poisson system near quasineutrality

This paper deals with the numerical resolution of the Vlasov–Poisson system in a nearly quasineutral regime by Particle-In-Cell (PIC) methods. In this regime, Classical PIC methods are subject to stability constraints on the time and space steps related to the small Debye length and large plasma frequency. Here, we propose an “Asymptotic-Preserving” PIC scheme which is not subjected to these limitations. Additionally, when the plasma period and Debye length are small compared to the time and space steps, this method provides a consistent PIC discretization of the quasineutral Vlasov equation. We perform several one-dimensional numerical experiments which provide a solid validation of the method and its underlying concepts, and compare the method with Classical PIC and Direct-Implicit methods.     

N-soliton solutions for the nonlocal two-wave interaction system via the Riemann–Hilbert method

In this paper, a nonlocal two-wave interaction system from the Manakov hierarchy is investigated via the Riemann–Hilbert approach. Based on the spectral analysis of the Lax pair, a Riemann–Hilbert problem for the nonlocal two-wave interaction system is constructed. By discussing the solutions of this Riemann–Hilbert problem in both the regular and nonregular cases, we explicitly present the N-soliton solution formula of the nonlocal two-wave interaction system. Moreover,the dynamical behaviour of the single-soliton solution is shown graphically.     

Stability criterion for solitons of the Zakharov–Kuznetsov-type equations

Early results concerning the linear stability of the solitons in equation of the KDV-type [1] are generalized to solitons describing by the Zakharov–Kuznetsov-type equation. The linear stability criterion for ground solitons in the Vakhitov–Kolokolov form is derived for such equations with arbitrary nonlinearity. For the power nonlinearity the instability criterion coincides with the condition of the Hamiltonian unboundedness from below. The latter represents the main feature for appearance of collapse in such systems.     

On some new analytical solutions for the nonlinear long–short wave interaction system

This paper deals with exact soliton solutions of the nonlinear long–short wave interaction system, utilizing two analytical methods. The system of coupled long–short wave interaction equations is investigated with the help of two analytical methods, namely, the generalized \(\tan (\phi /2)\)-expansion method and He’s semi-inverse variational method. Moreover, in this paper we generalize two aforementioned methods which give new soliton wave solutions. As a consequence, solutions are including solitons, kink, periodic and rational solutions. Moreover, dark, bright and singular solition solutions of the coupled long–short wave interaction equations have been found. All solutions have been verified back into its corresponding equation with the aid of maple package program. We depicted the physical explanation of the extracted solutions with the free choice of the different parameters by plotting some 3D and 2D illustrations. Finally, we believe that the executed methods are robust and efficient than other methods and the obtained solutions in this paper can help us to understand the soliton waves in the fields of physics and mechanics.     

On a reduction of the generalized Darboux–Halphen system

The equations for the general Darboux–Halphen system obtained as a reduction of the self-dual Yang–Mills can be transformed to a third-order system which resembles the classical Darboux–Halphen system with a common additive terms. It is shown that the transformed system can be further reduced to a constrained non-autonomous, non-homogeneous dynamical system. This dynamical system becomes homogeneous for the classical Darboux–Halphen case, and was studied in the context of self-dual Einstein's equations for Bianchi IX metrics. A Lax pair and Hamiltonian for this reduced system is derived and the solutions for the system are prescribed in terms of hypergeometric functions.     

On the electrical resistivity of the quasicrystalline decagonal Al–Mn system

A simple effective-medium approach is used to study anisotropy properties of the electrical resistivity in the quasicrystalline decagonal Al–Mn system.     

A finite element method for the numerical solution of the coupled Cahn–Hilliard and Navier–Stokes system for moving contact line problems

In this paper, a semi-implicit finite element method is presented for the coupled Cahn–Hilliard and Navier–Stokes equations with the generalized Navier boundary condition for the moving contact line problems. In our method, the system is solved in a decoupled way. For the Cahn–Hilliard equations, a convex splitting scheme is used along with a P1-P1 finite element discretization. The scheme is unconditionally stable. A linearized semi-implicit P2-P0 mixed finite element method is employed to solve the Navier–Stokes equations. With our method, the generalized Navier boundary condition is extended to handle the moving contact line problems with complex boundary in a very natural way. The efficiency and capacity of the present method are well demonstrated with several numerical examples.     

Development of rotating magnetic field system for the β-NMR method

A rotating magnetic field system has been developed to determine the sign of magnetic (μ) and quadrupole (Q) moments by means of the β-ray-detected nuclear magnetic resonance (β-NMR) method. The rotating magnetic field is obtained using two crossed Helmholtz-like coils at right angles. The phase angles of the alternating currents for the two coils are shifted by 90°. Thus the resulting magnetic field is circularly polarized. By the adjustment of the phase-angle difference, a right- or left-circularly polarized magnetic field is produced. Productions of right- or left-rotating magnetic fields including a linear component were confirmed using pickup-coils.     

On the dispersionless Davey–Stewartson system: Hamiltonian vector field Lax pair and relevant nonlinear Riemann–Hilbert problem for dDS-II system

Letters in Mathematical Physics - In this paper, the semiclassical limit of Davey–Stewartson system is studied. It shows that the dispersionless limited integrable system of hydrodynamic...     

On the phase diagram of a two-dimensional electron–hole system

The phase diagram for a system of spatially separated electrons and holes in coupled quantum wells or graphene double layers is studied in the framework of a BCS-like mean-field approach and a Landau expansion in terms of the pairing order parameter. We find a second order transition between an electron–hole plasma and a BCS phase, as well as a first-order transition between the BCS phase and a bosonic Mott phase of tightly bound electron–hole pairs without phase coherence. The electron–hole plasma exists at low and at high densities for weak interaction, the BCS phase at moderate density and the Mott phase at high density and strong interaction.     

A least-squares/finite element method for the numerical solution of the Navier–Stokes-Cahn–Hilliard system modeling the motion of the contact line

In this article we discuss the numerical solution of the Navier–Stokes-Cahn–Hilliard system modeling the motion of the contact line separating two immiscible incompressible viscous fluids near a solid wall. The method we employ combines a finite element space approximation with a time discretization by operator-splitting. To solve the Cahn–Hilliard part of the problem, we use a least-squares/conjugate gradient method. We also show that the scheme has the total energy decaying in time property under certain conditions. Our numerical experiments indicate that the method discussed here is accurate, stable and efficient.     

On eigenvalue estimates for the Dirac–Witten-type operators

We give optimal lower bounds for the eigenvalues of the Dirac–Witten-type operators associated with the _e0$e_0$-Killing connection and imaginary Killing connection, in terms of the mean curvature and the scalar curvature. The limiting cases are then studied and lead to interesting geometric situations.     

On nonlocal symmetries for the Krichever–Novikov equation

We construct new infinite hierarchies of nonlocal symmetries and cosymmetries for the Krichever–Novikov equation using the inverse of the fourth-order recursion operator of the latter.     

On equations solvable by the inverse scattering method for the Schrödinger operator

The new approach to deriving nonlinear evolution equations solvable by the inverse scattering method is proposed. In particular, this approach allows one to describe all equations solvable by the inverse scattering method for the Schrödinger operator.     

On the Darboux integrability of a three–dimensional forced–damped differential system

In 2011 Pehlivan proposed a three–dimensional forced–damped autonomous differential system which can display simultaneously unbounded and chaotic solutions. This untypical phenomenon has been studied recently by several authors. In this paper we study the opposite to its chaotic motion, i.e. its integrability, mainly the existence of polynomial, rational and Darboux first integrals through the analysis of its invariant algebraic surfaces and its exponential factors.