Explicit Peakon solutions to a family of wave-breaking equations

The singular traveling wave solutions of a general 4-parameter family equation which unifies the Camass-Holm equation, the Degasperis-Procesi equation and the Novikov equation are investigated in this paper. At first, we obtain the explicit peakon solutions for one of its specific case that $a=(p+2)c$, $b=(p+1)c$ and $c=1$, which is referred to a generalized Camassa-Holm-Novikov (CHN) equation, by reducing it to a second-order ordinary differential equation (ODE) and solving its associated first-order integrable ODE. By observing the characteristics of peakon solutions to the CHN equation, we construct the peakon solutions for the general 4-parameter breaking wave equation. It reveals that singularities of the peakon solutions come up only when the solutions attain singular points of the equation, which might be a universal principal for all singular traveling wave solutions for wave breaking equations.     

On a fractional sublinear elliptic equation with a variable coefficient

We study the existence and uniqueness of bounded weak solutions to a fractional sublinear elliptic equation with a variable coefficient, in the whole space. Existence is investigated in connection to a certain fractional linear equation, whereas the proof of uniqueness relies on uniqueness of weak solutions to an associated fractional porous medium equation with variable density.     

Self-similar solutions of a Burgers-type equation with quadratically cubic nonlinearity

Self-similar solutions are found for a quadratically cubic second-order partial differential equation governing the behavior of nonlinear waves in various distributed systems, for example, in some metamaterials. They are compared with self-similar solutions of the Burgers equation. One of them describing a single unipolar pulse is shown to satisfy both equations. The other self-similar solutions of the quadratically cubic equation behave differently from the solutions of the Burgers equation. They are constructed by matching the positive and negative branches of the solution, so that the function itself and its first derivative are continuous. One of these solutions corresponds to an asymmetric solitary N-wave of the sonic shock type. Self-similar solutions of a quadratically cubic equation describing the propagation of cylindrically symmetric waves are also found.     

A blow-up theorem for a discrete semilinear wave equation

In this paper, a discretization of a semilinear wave equation whose nonlinear term is a power function is investigated. It is shown that, when a condition on the initial value problem, similar to that governing the existence of blow-up solutions for the original continuous equation is met, the newly introduced difference equation has blow-up solutions with characteristics corresponding to those of the blow-up solutions for the original equation.     

On the Equation of a Rotating Film

We study positive periodic solutions to a nonautonomous nonlinear third-order ordinary differential equation of the theory of motion of a viscous incompressible fluid with free boundary. This equation describes the steady motions of a thin layer of a fluid film on the surface of a rotating horizontal cylinder in the gravity field. The linear operator on the left-hand side of the equation has a three-dimensional kernel. Moreover, the equation contains two nonnegative parameters proportional to the gravity acceleration and surface tension. Depending on these parameters the problem in question may have either two solutions or no solutions at all. We establish some qualitative properties of solutions to the problem: in particular, their asymptotic behavior at the extremal values of the parameters.     

Global existence and uniform decay of a damped Klein-Gordon equation in a noncylindrical domain

In this paper, we consider a damped Klein-Gordon equation in a noncylindrical domain. This work is devoted to proving the existence of global solutions and decay for the energy of solutions for a damped Klein-Gordon equation in a noncylindrical domain.     

On a certain class of a nonlinear evolution partial differential equations

An existence result and a priori bound for the solution of a second-order nonlinear parabolic equation are established. Also a generalized tanh-function method is used for constructing exact travelling wave solutions for the nonlinear diffusion equation of Fisher type originated from the considered partial differential equation. And new multiple soliton solutions are obtained.     

On the solvability of a functional equation

This article studies properties of solutions of a functional equation arising in dynamic programming of multistage decision processes. A sufficient condition for the existence, uniqueness and iterative approximation of solutions of the functional equation is established. A few other behaviours of solutions for certain functional equations which are particular cases of the functional equation are discussed. The results presented in this article extend, improve and unify some known results in literature.     

A Study of Separable Solutions of a Generalized Burgers Equation

This paper concerns the separable solutions of a generalized Burgers equation. Existence of separable solutions to the generalized Burgers equation is proved under certain conditions. A careful numerical study shows that these separable solutions of the generalized Burgers equation describe the large time asymptotic behavior of solutions of initial boundary value problems.     

Inverse coefficient problem for a wave equation in a bounded domain

The nonlinear inverse problem for a wave equation is investigated in a three-dimensional bounded domain subject to the Dirichlet boundary condition. Given a family of solutions to the equation defined on a closed surface within the original domain, it is required to reconstruct the coefficient determining the velocity of sound in the medium. The solutions used for this purpose correspond to the acoustic medium perturbations localized in the neighborhood of a certain closed surface. The inverse problem is reduced to a linear integral equation of the first kind, and the uniqueness of the solution to this equation is established. Numerical results are presented.     

A note on a symmetry analysis and exact solutions of a nonlinear fin equation

A similarity analysis of a nonlinear fin equation has been carried out by M. Pakdemirli and A.Z. Sahin [Similarity analysis of a nonlinear fin equation, Appl. Math. Lett. (2005) (in press)]. Here, we consider a further group theoretic analysis that leads to an alternative set of exact solutions or reduced equations with an emphasis on travelling wave solutions, steady state type solutions and solutions not appearing elsewhere.     

On some method for solving a nonlinear heat equation

Some exact solutions to a nonlinear heat equation are constructed. An initial-boundary value problem is examined for a nonlinear heat equation. To construct solutions, the problem for a partial differential equation of the second order is reduced to a similar problem for a first order partial differential equation.     

Symbolic computation and construction of soliton-like solutions for a breaking soliton equation

Based on the symbolic computation system––Maple and a Riccati equation, by introducing a new more general ansätz than the ansätz in the tanh method, extended tanh-function method, modified extended tanh-function method, generalized tanh method and generalized hyperbolic-function method, we propose a generalized Riccati equation expansion method for searching for exact soliton-like solutions of nonlinear evolution equations and implemented in computer symbolic system––Maple. Making use of our method, we study a typical breaking soliton equation and obtain new families of exact solutions, which include the nontravelling wave’ and coefficient function’ soliton-like solutions, singular soliton-like solutions and periodic solutions. The arbitrary functions of some solutions are taken to be some special constants or functions, the known solutions of this equation can be recovered.     

Analysis of the convected Helmholtz equation with a uniform mean flow in a waveguide with complete radiation boundary conditions

In this paper, we propose complete radiation boundary conditions (CRBCs) for solutions of the convected Helmholtz equation with a uniform mean flow in a waveguide. We first study CRBCs for the Helmholtz equation in a waveguide. Noting that the convected Helmholtz equation is associated with the Helmholtz equation via the Prandtl–Glauert transformation, CRBCs for the convected Helmholtz equation is derived from CRBCs for the Helmholtz equation. We analyse well-posedness and convergence of approximate solutions satisfying CRBCs for the convected Helmholtz equation. In addition, simple numerical experiments will be presented to confirm the theoretical results.     

Convex solutions of a general similarity boundary layer equation for power-law fluids

This paper is devoted to a general similarity boundary layer equation for power-law fluids, which includes many important similarity boundary layer problems such as the Falker-Skan equation and the magnetohydrodynamic boundary layer equation which arises in the study of self-similar solutions of the two-dimensional steady laminar boundary layer flow for an incompressible electrically conducting power-law fluids along an isolated surface in the presence of an exterior magnetic field orthogonal to the flow. By a rigorous mathematical analysis, the uniqueness, existence and nonexistence results for convex solutions, normal convex solutions and generalized convex solutions to the general similarity boundary layer equation are established. Also the asymptotic behavior of the normal convex solutions at the infinity are displayed.     

Conditional Symmetry and Exact Solutions of a Multidimensional Diffusion Equation

We investigate the conditional symmetry of a multidimensional nonlinear reaction–diffusion equation by its reduction to a radial equation. We construct exact solutions of this equation and infinite families of exact solutions for the corresponding one-dimensional diffusion equation.     

Solutions of two-dimensional Schrödinger-type equations in a magnetic field

We use the method of dressing by a linear operator of general form to construct new solutions of Schrödinger-type two-dimensional equations in a magnetic field. In the case of a nonunit metric, we integrate the class of solutions that admit a variable separation before dressing. In particular, we show that the ratio of the coefficients of the differential operators in the unit metric case satisfies the Hopf equation. We establish a relation between the solutions of the two-dimensional eikonal equation with the unit right-hand side and solutions of the Hopf equation.     

A Note on a Functional Differential Equation with State Dependent Argument

This paper is concerned with solutions of a functional differential equation.Using Krasnoselskii's fixed point theorem,the solutions can be obtained from periodic solutions of a companion equation.     

Different physical structures of solutions for a generalized Boussinesq wave equation

A technique based on the reduction of order for solving differential equations is employed to investigate a generalized nonlinear Boussinesq wave equation. The compacton solutions, solitons, solitary pattern solutions, periodic solutions and algebraic travelling wave solutions for the equation are expressed analytically under several circumstances. The qualitative change in the physical structures of the solutions is highlighted.