Exact solutions with compact and noncompact structures for the one-dimensional generalized Benjamin–Bona–Mahony equation

This paper is devoted to analyzing the physical structures of nonlinear dispersive variants of the Benjamin–Bona–Mahony equation. It is found that these generalized forms give rise to compactons solutions: solitons with the absence of infinite tails, solitons: nonlinear localized waves of infinite support, solitary patterns solutions having infinite slopes or cusps, and plane periodic solutions. It is also found that the qualitative change in the physical structure of solutions depends strongly on whether the exponents of the wave function u(x, t) whether it is positive or negative, and on the speed c of the traveling wave as well.     

Travelling wave solutions for a class of generalized KdV equation

In this work, the K^∗(l,p) equation is investigated. The sine-cosine method, the tanh method and the extended tanh method are efficiently used for analytic study of this equation. New solitary patterns solutions and compactons solutions are formally derived. The proposed schemes are reliable and manageable.     

Exact special solitary solutions with compact support for the nonlinear dispersive K(m, n) equations

The nonlinear dispersive K(m, n) equations, u_t−(u^m)_x−(uⁿ)_xxx = 0 which exhibit compactons: solitons with compact support, are studied. New exact solitary solutions with compact support are found. The two special cases, K(2, 2) and K(3, 3), are chosen to illustrate the concrete features of the decomposition method in K(m, n) equations. General formulas for the solutions of K(m, n) equations are established.     

Bifurcations of travelling wave solutions for the generalized KP-BBM equation

By using the bifurcation theory of dynamical systems to the generalized Kadomtsov-Petviashvili-Benjamin-Bona-Mahony equation, the existence of solitary wave solutions, compactons solution, non-smooth periodic cusp wave solutions and uncountably infinite many smooth periodic wave solutions is obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. Some exact explicit parametric representations of the above waves are determined.     

New solitary-wave special solutions with compact support for the nonlinear dispersive K(m,n) equations

The genuinely nonlinear dispersive K(m,n) equation, u_t+(u^m)_x+(uⁿ)_xxx=0, which exhibits compactons: solitons with compact support, is investigated. New solitary-wave solutions with compact support are developed. The specific cases, K(2,2) and K(3,3), are used to illustrate the pertinent features of the proposed scheme. An entirely new general formula for the solution of the K(m,n) equation is established, and the existing general formula is modified as well.     

New exact solitary-wave special solutions for the nonlinear dispersive K(m, n) equations

In this paper, we study the nonlinear dispersive K(m, n) equations: u_t + (u^m)_x − (uⁿ)_xxx = 0 which exhibit solutions with solitary patterns. New exact solitary solutions are found. The two special cases, K(2, 2) and K(3, 3), are chosen to illustrate the concrete features of the decomposition method in K(m, n) equations. The nonlinear equations K(m, n) are studied for two different cases, namely when m = n being odd and even integers. General formulas for the solutions of K(m, n) equations are established.     

The tanh and the sine–cosine methods for a reliable treatment of the modified equal width equation and its variants

The modified equal width equation and two of its variants are investigated. The strategy here rests mainly on a sine–cosine ansatz and the tanh method. Both schemes work well and reveal exact solutions with distinct physical structures. The obtained solutions include compactons, solitons, solitary patterns, and periodic solutions.     

Stability of multi-compacton solutions and Backlund transformation in K(m,n,1)

We introduce a fifth-order K(m,n,1) equation with nonlinear dispersion to obtain multi-compacton solutions by Adomian decomposition method. Using the homogeneous balance (HB) method, we derive a Backlund transformation of a special equation K(2,2,1) to determine some solitary solutions of the equation. To study the stability of multi-compacton solutions in K(m,n,1) and to obtain some conservation laws, we present a similar fifth-order equation derived from Lagrangian. We finally show the linear stability of all obtained multi-compacton solutions.     

Bifurcations of travelling wave solutions for the mK(n, n) equation

Using the method of planar dynamical systems to the mK(n, n) equation, the existence of uncountably infinite many smooth and non-smooth periodic wave solutions, solitary wave solutions and kink and anti-kink wave solutions is proved. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. All possible exact explicit parametric representations of smooth and non-smooth travelling wave solutions are obtain.     

New compacton solutions of an extended Rosenau–Pikovsky equation

The K(cos ^m , cos ⁿ ) equation is proposed, which extends the Rosenau–Pikovsky K(cos) equation to the case of power-law dependence of nonlinearity and dispersion. The properties of compacton and kovaton solutions are numerically studied and compared with solutions of the K(2,2) and K(cos) equations. New types of peak-shaped compactons and kovatons of various amplitudes are found.     

Compacton and generalized kink wave solutions of the CH-DP equation

The CH-DP equation is investigated by using the bifurcation method of planar systems and simulation method of differential equations. The bifurcation phase portraits are drawn in different regions of parameter plane. The planar graphs of compactons and generalized kink waves are simulated by using mathematical software Maple. Exact explicit parameter expressions of compactons and implicit expressions of generalized kink wave solutions are given, and the dynamic characters of these solutions are investigated.     

New exact solution profiles of the nonlinear dispersion Drinfel’d–Sokolov (D(m, n)) system

In this paper, to understand the role of nonlinear dispersion in the coupled systems, the nonlinear dispersion Drinfel’d–Sokolov system (called D(m, n) system) is investigated. As a consequence, many types of compacton and solitary pattern solutions are obtained. Moreover, some solitary wave solutions are also deduced for differential parameters m, n. When n = 1, the D(m, 1) system with linear dispersion is shown to possess also compacton and solitary pattern solutions, which contain the known results. Moreover, some rational solutions of D(m, n) system are also deduced.     

Exact compacton and generalized kink wave solutions of the extended reduced Ostrovsky equation

The extended reduced Ostrovsky equation (EX-ROE) are investigated by using the bifurcation method of planar systems and simulation method of differential equations. The bifurcation phase portraits are drawn in different regions of parameter plane. The planar graphs of the compactons and the generalized kink waves are simulated by using software Maple. Exact explicit parameter expressions of the compactons and implicit expressions of the generalized kink wave solutions are given. The dynamic behavior of these solutions are also investigated.     

On the functional equationS(m,n)F[n/(m,n)] =F(n)h[n/(m,n)]

In this paper, we discuss the pairs (f, h) of arithmetical functions satisfying the functional equation in the title, whereF is the product off andh under the Dirichlet convolution; that is,F(n) = Σ _d|n ?(d)h(n/d) andS(m n) = Σd|(m, n) ?(d)h(n/d). The well-known Hölder's identity is a special case of this functional equation (?(n) =n, h(n) = μ(n)). We also generalize the functional equation in the title to any arbitrary regular arithmetical convolution and discuss the pairs of solutions (f, h) of the generalized functional equation and pose some problems relating to the characterization of all pairs of solutions.     

Bifurcations of travelling wave solutions for the combined kdv–mkdv equation

By using the theory of bifurcations of dynamical systems to the combined k dv–mk dv equation, the existence of solitary wave, kink and anti-kink wave solutions and uncountably infinite many smooth periodic wave solutions are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given.     

Smooth and non-smooth travelling waves in a nonlinearly dispersive Boussinesq equation

The dynamical behavior and special exact solutions of nonlinear dispersive Boussinesq equation (B(m,n) equation), u_tt−u_xx−a(uⁿ)_xx+b(u^m)_xxxx=0, is studied by using bifurcation theory of dynamical system. As a result, all possible phase portraits in the parametric space for the travelling wave system, solitary wave, kink and anti-kink wave solutions and uncountably infinite many smooth and non-smooth periodic wave solutions are obtained. It can be shown that the existence of singular straight line in the travelling wave system is the reason why smooth waves converge to cusp waves, finally. When parameter are varied, under different parametric conditions, various sufficient conditions guarantee the existence of the above solutions are given.     

Some new solutions for the Derrida–Lebowitz–Speer–Spohn equation

The well-known Derrida–Lebowitz–Speer–Spohn equation is investigated. By using specific ansätze and the classical symmetries of the equation, several families of new exact solutions have been found. In particular, there appear traveling waves that include compactons and soliton–compactons. Some other solutions conserve the mass and exhibit diffusion and convection processes from an instantaneous source and localized peakons.     

New compacton-like and solitary patterns-like solutions to nonlinear wave equations with linear dispersion terms

It has been shown that many fully nonlinear wave equations with nonlinear dispersion terms possess compacton solutions and solitary patterns solutions. In this paper, with the aid of Maple, the mKdV equation, the equation with a source term, the five order KdV-like equation and the KdV–mKdV equation are investigated using some new, generalized transformations. As a consequence, it is shown that these equations with linear dispersion terms admit new compacton-like solutions and solitary patterns-like solutions. These transformations can be also extended to other nonlinear wave equations with nonlinear dispersion terms to seek new compacton-like solutions and solitary patterns-like solutions.     

Analytic study for fifth-order KdV-type equations with arbitrary power nonlinearities

In this work we use the sine–cosine and the tanh methods for analytic study for higher-order KdV-type equations with arbitrary power nonlinearities. Exact solutions of distinct physical structures: solitons, compactons, periodic and solitary patterns, are derived. The study shows the power of the two schemes and presents a comparison between these two methods.     

Global existence of solutions to the compressible Navier-Stokes equation around parallel flows

The initial boundary value problem for the compressible Navier-Stokes equation is considered in an infinite layer of Rn. It is proved that if n?3, then strong solutions to the compressible Navier-Stokes equation around parallel flows exist globally in time for sufficiently small initial perturbations, provided that the Reynolds and Mach numbers are sufficiently small. The proof is given by a variant of the Matsumura-Nishida energy method based on a decomposition of solutions associated with a spectral property of the linearized operator.