Two 3 × 3 discrete matrix spectral problems and associated Liouville integrable lattice soliton equations

Two 3 × 3 discrete matrix spectral problems are introduced and the corresponding lattice soliton equations are derived. By means of the discrete trace identity the Hamiltonian structures of the resulting equations are constructed. Liouville integrability of the discrete Hamiltonian systems is proved.     

A non-variational approach to the construction of new ‘higher-order’ conservation laws of the family of nonlinear equations α(ut + 3uux) + β(utxx + 2uxuxx + uuxxx) − γuxxx = 0

In this paper, we study and classify the conservation laws of the combined nonlinear KdV, Camassa–Holm, Hunter–Saxton and the inviscid Burgers equation which arises in, inter alia, shallow water equations. It is shown that these can be obtained by variational methods but the main focus of the paper is the construction of the conservation laws as a consequence of the interplay between symmetry generators and ‘multipliers’, particularly, the higher-order ones.     

The non-traveling wave solutions and novel fractal soliton for the (2 + 1)-dimensional Broer–Kaup equations with variable coefficients

A method is proposed by extending the linear traveling wave transformation into the nonlinear transformation with the (G′/G)-expansion method. The non-traveling wave solutions with variable separation can be constructed for the (2 + 1)-dimensional Broer–Kaup equations with variable coefficients via the method. A novel class of fractal soliton, namely, the cross-like fractal soliton is observed by selecting appropriately the arbitrary functions in the solutions.     

Deautonomisation of differential-difference equations and discrete Painlevé equations

We examine a family of integrable differential-difference equations and obtain their non-autonomous extensions using a discrete/continuous integrability criterion.     

Bifurcations of travelling wave solutions in the (N + 1)-dimensional sine–cosine-Gordon equations

In this paper, the (N + 1)-dimensional sine–cosine-Gordon equations are studied. The existence of solitary wave, kink and anti-kink wave, and periodic wave solutions are proved, by using the method of bifurcation theory of dynamical systems. All possible bounded exact explicit parametric representations of the above travelling solutions are obtained.     

From quantum Euler–Maxwell equations to incompressible Euler equations

The combined quasi-neutral and non-relativistic limit of compressible quantum Euler–Maxwell equations for plasmas is studied in this paper. For well-prepared initial data, it is shown that the smooth solution of compressible quantum Euler–Maxwell equations converges to the smooth solution of incompressible Euler equations by using the modulated energy method. Furthermore, the associated convergence rates are also obtained.     

p-Euler equations and p-Navier–Stokes equations

We propose in this work new systems of equations which we call p-Euler equations and p-Navier–Stokes equations. p-Euler equations are derived as the Euler–Lagrange equations for the action represented by the Benamou–Brenier characterization of Wasserstein-p distances, with incompressibility constraint. p-Euler equations have similar structures with the usual Euler equations but the ‘momentum’ is the signed ($p?1$)-th power of the velocity. In the 2D case, the p-Euler equations have streamfunction-vorticity formulation, where the vorticity is given by the p-Laplacian of the streamfunction. By adding diffusion presented by γ-Laplacian of the velocity, we obtain what we call p-Navier–Stokes equations. If $\gamma =p$, the a priori energy estimates for the velocity and momentum have dual symmetries. Using these energy estimates and a time-shift estimate, we show the global existence of weak solutions for the p-Navier–Stokes equations in ${\mathbb{R}}^d$ for $\gamma =p$ and $p\ge d\ge 2$ through a compactness criterion.     

Boundary layer problem: Navier–Stokes equations and Euler equations

This work is concerned with the boundary layer turbulence, which is an outstanding problem in fluid mechanics. We consider an incompressible viscous fluid in 2D domains with permeable walls. The permeability is described by the Yudovich condition. The goal of the article is to study the fluid behavior at vanishing viscosity (large Reynold’s numbers). We show that the vanishing viscous limit is a solution of the Euler equations with the Yudovich condition on the inflow region of the boundary.     

Nonlinear pseudoparabolic equations as singular limit of reaction–diffusion equations

In this article, a solution of a nonlinear pseudoparabolic equation is constructed as a singular limit of a sequence of solutions of quasilinear hyperbolic equations. If a system with cross diffusion, modelling the reaction and diffusion of two biological, chemical, or physical substances, is reduced then such an hyperbolic equation is obtained. For regular solutions even uniqueness can be shown, although the needed regularity can only be proved in two dimensions.     

Reflectionless Sturm–Liouville equations

We consider compactly supported perturbations of periodic Sturm–Liouville equations. In this context, one can use the Floquet solutions of the periodic background to define scattering coefficients. We prove that if the reflection coefficient is identically zero, then the operators corresponding to the periodic and perturbed equations, respectively, are unitarily equivalent. In some appendices, we also provide the proofs of several basic estimates, e.g., bounds and asymptotics for the relevant m$m$-functions.     

Matrix Painlevé II equations

Theoretical and Mathematical Physics - We use the Painlevé–Kovalevskaya test to find three matrix versions of the Painlevé II equation. We interpret all these equations as...     

A note on stochastic semilinear equations and their associated Fokker–Planck equations

The main purpose of this paper is to prove existence and uniqueness of (probabilistically weak and strong) solutions to stochastic differential equations (SDE) on Hilbert spaces under a new approximation condition on the drift, recently proposed in [6] to solve Fokker–Planck equations (FPE), extended in this paper to a considerably larger class of drifts. As a consequence we prove existence of martingale solutions to the SDE (whose time marginals then solve the corresponding FPE). Applications include stochastic semilinear partial differential equations with white noise and a non-linear drift part which is the sum of a Burgers-type part and a reaction diffusion part. The main novelty is that the latter is no longer assumed to be of at most linear, but of at most polynomial growth. This case so far had not been covered by the existing literature. We also give a direct and more analytic proof for existence of solutions to the corresponding FPE, extending the technique from [6] to our more general framework, which in turn requires to work on a suitable Gelfand triple rather than just the Hilbert state space.     

Quaternionic Monge-Ampère equations

The main result of this article is the existence and uniqueness of the solution of the Dirichlet problem for quaternionic Monge-Ampère equations in quaternionic strictly pseudoconvex bounded domains in ℍ ⁿ . We continue the study of the theory of plurisubharmonic functions of quaternionic variables started by the author at [2].     

About inclusion of Maxwell’s equations in systems relativistic of the invariant equations

Maxwell’s equations, relativistic invariant equations, foundations of difference schemes.     

Existence and concentration behavior of sign-changing solutions for quasilinear Schr?dinger equations equations

We consider the quasilinear Schrdinger equations of the form-ε~2?u + V(x)u- ε~2?(u2)u = g(u), x ∈ R~N,where ε 0 is a small parameter, the nonlinearity g(u) ∈ C~1(R) is an odd function with subcritical growth and V(x) is a positive Hlder continuous function which is bounded from below, away from zero, and infΛV(x) inf ?ΛV(x) for some open bounded subset Λ of RN. We prove that there is an ε0 0 such that for all ε∈(0, ε0],the above mentioned problem possesses a sign-changing solution uε which exhibits concentration profile around the local minimum point of V(x) as ε→ 0~+.     

Vanishing viscosity limit for Navier–Stokes equations with general viscosity to Euler equations

In this paper, we study vanishing viscosity limit of 1-D isentropic compressible Navier–Stokes equations with general viscosity to isentropic Euler equations. Firstly, we improve estimates of the entropy flux, then we obtain that the weak solution of the isentropic Euler equations is the inviscid limit of the isentropic compressible Navier–Stokes equations with general viscosity using the compensated compactness frame recently established by G.-Q. Chen and M. Perepelitsa.     

Relationship between the Udwadia–Kalaba equations and the generalized Lagrange and Maggi equations

In their paper “A New Perspective on Constrained Motion,” F. E. Udwadia and R. E. Kalaba propose a new form of matrix equations of motion for nonholonomic systems subject to linear nonholonomic second-order constraints. These equations contain all of the generalized coordinates of the mechanical system in question and, at the same time, they do not involve the forces of constraint. The equations under study have been shown to follow naturally from the generalized Lagrange and Maggi equations; they can be also obtained using the contravariant form of the motion equations of a mechanical system subjected to nonholonomic linear constraints of second order. It has been noted that a similar method of eliminating the forces of constraint from differential equations is usually useful for practical purposes in the study of motion of mechanical systems subjected to holonomic or classical nonholonomic constraints of first order. As a result, one obtains motion equations that involve only generalized coordinates of a mechanical system, which corresponds to the equations in the Udwadia–Kalaba form.     

A numerical study of variable depth KdV equations and generalizations of Camassa–Holm-like equations

In this paper we numerically study the KdV-top equation and compare it with the Boussinesq equations over uneven bottoms. We use here a finite-difference scheme that conserves a discrete energy for the fully discrete scheme. We also compare this approach with the discontinuous Galerkin method. For the equations obtained in the case of stronger nonlinearities and related to the Camassa–Holm equation, we find several finite difference schemes that conserve a discrete energy for the fully discrete scheme. Because of its accuracy for the conservation of energy, our numerical scheme is also of interest even in the simple case of flat bottoms. We compare this approach with the discontinuous Galerkin method.