The nonlinear dispersive Boussinesq-like equation , which exhibits single peak solitons, is investigated. Peakons, cuspons and smooth soliton solutions are obtained by setting the equation under inhomogeneous boundary condition. Asymptotic behavior and numerical simulations are provided for these three types of single peak soliton solutions of the equation. 1. Introduction The interest inspired by the well-known Camassa-Holm (CH) equation and its singular peakon solutions [1] prompted search for other integrable equations with nonsmooth solitons. An integrable CH-type equation with cubic nonlinearity was derived independently by Fokas [2], by Fuchssteiner [3], by Olver and Rosenau [4], and by Qiao [5]. It is shown in [5–7] that (1) admits Lax pair and bi-Hamiltonian structures and possesses the M/W-shape soliton solution and a new type of cusped soliton solution. Another peakon equation with cubic nonlinearity has been recently discovered by Novikov [8]. In the work by Hone and Wang [9], it is shown that Novikov's equation admits peakon solutions like the CH equation. Also, it has a Lax pair in matrix form and a bi-Hamiltonian structure. The Boussinesq-like equation with nonlinear dispersion is given by where , and , . This equation is the generalized form of the Boussinesq equation, where, in particular, the case leads to the Boussinesq equation. Equation (2), for , is the major equation for compactons (solitons with compact support). Abundant compactons [10–13] are developed by the Adominan decomposition method. For , exact solutions with solitary patterns of Boussinesq-like equations are obtained in the works by Shang [14] and Zhang et al. [15] by extending sinh-cosh method and by using the integral approach, respectively. A natural question is that whether the Boussinesq-like equation (2) has nonsmooth solitons such as peakons or cuspons. The present paper focuses on the following Boussinesq-like equation: We give all possible single peak soliton solutions of (3) through setting the traveling wave solution under the inhomogeneous boundary condition ( is a nonzero constant) as . New cusped soliton solutions, and smooth soliton solutions are obtained. Asymptotic analysis and numerical simulations are provided for peaked solitons, cusped solitons and smooth solitons of the equation. The method used here is based on the phase portrait analysis technique which is similar to that in [16–18]. 2. Asymptotic Behavior of Solutions In this section, we first introduce some notations. Let denote the set of all times continuously differential functions on the
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