Self-adjointness and conservation laws for Kadomtsev–Petviashvili–Burgers equation

In this work we study the Kadomtsev–Petviashvili–Burgers equation, which is a natural model for the propagation of the two-dimensional damped waves. We show that the equation is nonlinear self-adjoint and it will become strict self-adjoint or weak self-adjoint in some equivalent form. By using Ibragimov’s theorem on conservation laws we find some conservation laws for this equation.     

Lie symmetries and conservation laws of the Hirota–Ramani equation

In this paper, Lie symmetry method is performed for the Hirota–Ramani (H–R) equation. We will find the symmetry group and optimal systems of Lie subalgebras. Furthermore, preliminary classification of its group invariant solutions, symmetry reduction and nonclassical symmetries are investigated. Finally conservation laws of the H–R equation are presented.     

Existence and conservation laws for the Boltzmann–Fermi–Dirac equation in a general domain

We prove an existence theorem for the Boltzmann–Fermi–Dirac equation for integrable collision kernels in possibly bounded domains with specular reflection at the boundaries, using the characteristic lines of the free transport. We then obtain that the solution satisfies the local conservations of mass, momentum and kinetic energy thanks to a dispersion technique.     

Symmetry reduction,exact group-invariant solutions and conservation laws of the Benjamin–Bona–Mahoney equation

We show that the Benjamin–Bona–Mahoney (BBM) equation with power law nonlinearity can be transformed by a point transformation to the combined KdV–mKdV equation, that is also known as the Gardner equation. We then study the combined KdV–mKdV equation from the Lie group-theoretic point of view. The Lie point symmetry generators of the combined KdV–mKdV equation are derived. We obtain symmetry reduction and a number of exact group-invariant solutions for the underlying equation using the Lie point symmetries of the equation. The conserved densities are also calculated for the BBM equation with dual nonlinearity by using the multiplier approach. Finally, the conserved quantities are computed using the one-soliton solution.     

Complete classification of local conservation laws for generalized Cahn–Hilliard–Kuramoto–Sivashinsky equation

In this paper, we consider nonlinear multidimensional Cahn–Hilliard and Kuramoto–Sivashinsky equations that have many important applications in physics and chemistry, and a certain natural generalization of these two equations to which we refer to as the generalized Cahn–Hilliard–Kuramoto–Sivashinsky equation. For an arbitrary number of spatial independent variables, we present a complete list of cases when the latter equation admits nontrivial local conservation laws of any order, and for each of those cases, we give an explicit form of all the local conservation laws of all orders modulo trivial ones admitted by the equation under study. In particular, we show that the original Kuramoto–Sivashinsky equation admits no nontrivial local conservation laws, and find all nontrivial local conservation laws for the Cahn–Hilliard equation.     

Almost conservation laws and global rough solutions of the defocusing nonlinear wave equation on ℝ2

In this article, we investigate the initial value problem(IVP) associated with the defocusing nonlinear wave equation on ?² as follows:

$\{\begin{array}{l}{?}_{tt}u-\Delta u=-u^3,\\ u\left(0,x\right)=u_0\left(x\right),{?}_{t}u\left(0,x\right)=u_1\left(x\right),\end{array}$

where the initial data (u₀, u₁) ? H^s(?²) × H^s?1(?²). It is shown that the IVP is global well-posedness in H^s(?²) × H^s?1(?²) for any 1 > s > 2/5. The proof relies upon the almost conserved quantity in using multilinear correction term. The main difficulty is to control the growth of the variation of the almost conserved quantity. Finally, we utilize linear-nonlinear decomposition benefited from the ideas of Roy [1].     

Bifurcation and asymptotics for the Lane–Emden–Fowler equation

We are concerned with the Lane–Emden–Fowler equation ?Δu=λf(u)+a(x)g(u) in $\text{Ω}$, subject to the Dirichlet boundary condition u=0 on $\text{?Ω,}$ where $\text{Ω?}\text{R}{}^{\mathrm{N}}$ is a smooth bounded domain, λ is a positive parameter, $\text{a}\text{:}\text{Ω}\text{→[0,∞)}$ is a Hölder function, and f is a positive nondecreasing continuous function such that f(s)/s is nonincreasing in (0,∞). The singular character of the problem is given by the nonlinearity g which is assumed to be unbounded around the origin. In this Note we discuss the existence and the uniqueness of a positive solution of this problem and we also describe the precise decay rate of this solution near the boundary. The proofs rely essentially on the maximum principle and on elliptic estimates. To cite this article: M. Ghergu, V.D. R?dulescu, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Conservation laws for a variable coefficient nonlinear diffusion–convection–reaction equation

We find the Lie point symmetries of a class of second-order nonlinear diffusion–convection–reaction equations containing an unspecified coefficient function of the independent variable t and determine the subclasses of these equations which are nonlinearly self-adjoint. By using a general theorem on conservation laws proved recently by N.H. Ibragimov we establish conservation laws corresponding to the aforementioned Lie point symmetries, one by one, for the simultaneous system of the original equation together with its adjoint equation through a formal Lagrangian. Particularly, for the nonlinearly self-adjoint subclasses, we construct conservation laws for the corresponding equations themselves.     

Singular limits for a parabolic–elliptic regularization of scalar conservation laws

We consider scalar hyperbolic conservation laws with a nonconvex flux, in one space dimension. Then, weak solutions of the associated initial value problems can contain undercompressive shock waves. We regularize the hyperbolic equation by a parabolic–elliptic system that produces undercompressive waves in the hyperbolic limit regime. Moreover we show that in another limit regime, called capillarity limit, we recover solutions of a diffusive–dispersive regularization, which is the standard regularization used to approximate undercompressive waves. In fact the new parabolic–elliptic system can be understood as a low-order approximation of the third-order diffusive–dispersive regularization, thus sharing some similarities with the relaxation approximations. A study of the traveling waves for the parabolic–elliptic system completes the paper.     

Symmetries of the complex Dirac-Kähler equation

We consider a complex version of a Dirac-Kähler-type equation, the eight-component complex Dirac-Kähler equation with a nonvanishing mass, which can be decomposed into two Dirac equations by only a nonunitary transformation. We also write an analogue of the complex Dirac-Kähler equation in five dimensions. We show that the complex Dirac-Kähler equation is a special case of a Bhabha-type equation and prove that this equation is invariant under the algebra of purely matrix transformations of the Pauli-Gürsey type and under two different representations of the Poincaré group, the fermionic (for a two-fermion system) and bosonic -representations. The complex Dirac-Kähler equation is also written in a manifestly covariant bosonic form as an equation for the system ( , , ) of irreducible self-dual tensor, scalar, and vector fields. We illustrate the relation between the complex Dirac-Kähler equation and the known 16-component Dirac-Kähler equation.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 143, No. 1, pp. 64–82, April, 2005.     

Second-order schemes for conservation laws with discontinuous flux modelling clarifier–thickener units

Continuously operated clarifier–thickener (CT) units can be modeled by a non-linear, scalar conservation law with a flux that involves two parameters that depend discontinuously on the space variable. This paper presents two numerical schemes for the solution of this equation that have formal second-order accuracy in both the time and space variable. One of the schemes is based on standard total variation diminishing (TVD) methods, and is addressed as a simple TVD (STVD) scheme, while the other scheme, the so-called flux-TVD (FTVD) scheme, is based on the property that due to the presence of the discontinuous parameters, the flux of the solution (rather than the solution itself) has the TVD property. The FTVD property is enforced by a new nonlocal limiter algorithm. We prove that the FTVD scheme converges to a BV _t solution of the conservation law with discontinuous flux. Numerical examples for both resulting schemes are presented. They produce comparable numerical errors, while the FTVD scheme is supported by convergence analysis. The accuracy of both schemes is superior to that of the monotone first-order scheme based on the adaptation of the Engquist–Osher scheme to the discontinuous flux setting of the CT model (Bürger, Karlsen and Towers in SIAM J Appl Math 65:882–940, 2005). In the CT application there is interest in modelling sediment compressibility by an additional strongly degenerate diffusion term. Second-order schemes for this extended equation are obtained by combining either the STVD or the FTVD scheme with a Crank–Nicolson discretization of the degenerate diffusion term in a Strang-type operator splitting procedure. Numerical examples illustrate the resulting schemes.     

Accuracy of classical conservation laws for Hamiltonian PDEs under Runge–Kutta discretizations

We investigate conservative properties of Runge–Kutta methods for Hamiltonian partial differential equations. It is shown that multi-symplecitic Runge–Kutta methods preserve precisely the norm square conservation law. Based on the study of accuracy of Runge–Kutta methods applied to ordinary and partial differential equations, we present some results on the numerical accuracy of conservation laws of energy and momentum for Hamiltonian PDEs under Runge–Kutta discretizations. J. Hong, S. Jiang and C. Li are supported by the Director Innovation Foundation of ICMSEC and AMSS, the Foundation of CAS, the NNSFC (No. 19971089, No. 10371128, No. 60771054) and the Special Funds for Major State Basic Research Projects of China 2005CB321701.     

N-soliton solutions,Bäcklund transformation and conservation laws for the integro-differential nonlinear Schröbinger equation from the isotropic inhomogeneous Heisenberg spin magnetic chain

Under investigation in this paper is an integro-differential nonlinear Schröbinger (IDNLS) equation, which is equivalent to the spin evolution equation of a classical in-homogeneous Heisenberg magnetic chain in the continuum limit. Based on the Hirota method, the bilinear form and N-soliton solution for the IDNLS equation are derived with the help of symbolic computation. Moreover, N-soliton solution for the IDNLS equation is expressed in terms of the double Wronskian and testified through the direct substitution into the bilinear form. Besides, the bilinear Bäcklund transformation and infinitely many conservation laws are also obtained for the IDNLS equation. Propagation characteristics and interaction behaviors of the solitons are discussed by analysis of such physical quantities as the soliton amplitude, width, velocity and initial phase. Interactions of the solitons are proved to be elastic through the asymptotic analysis. Effect of inhomogeneity on the interaction of the solitons is studied graphically.     

Nonlinear nonisospectral differential coverings for the hyper-CR equation of Einstein–Weyl structures and the Gibbons–Tsarev equation

We apply the technique based on twisted extensions of symmetry algebras to construct new nonlinear four-dimensional differential coverings for the hyper-CR equation of Einstein–Weyl structures and for the associated integrable hierarchy. We expose related multi-component three-dimensional covering. By the symmetry reduction of the hyper-CR equation of Einstein–Weyl structures we derive nonlinear three-dimensional differential covering for the Gibbons-Tsarev equation.     

Paracontrolled calculus and Funaki–Quastel approximation for the KPZ equation

In this paper, we consider the approximating KPZ equation introduced by Funaki and Quastel (2015), which is suitable for studying invariant measures. They showed that the stationary solution of the approximating equation converges to the Cole–Hopf solution of the KPZ equation with extra term $\frac{1}{24}t$. On the other hand, Gubinelli and Perkowski (2017) gave a pathwise meaning to the KPZ equation as an application of the paracontrolled calculus. We show that Funaki and Quastel’s result is extended to nonstationary solutions by using the paracontrolled calculus.     

Existence and uniqueness theorem for the Safronov–Dubovski coagulation equation

The solutions of the discrete Safronov–Dubovski coagulation equation are investigated. We prove global existence for a class of unbounded coagulation kernels. We also show that for sub-linear unbounded kernels, the mass conservation law holds. Finally, we show that for bounded kernels, this equation has a unique global solution that is continuously dependent on the initial data.