Single Peak Solitons for the Boussinesq-Like Equation
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.
The interest inspired by the well-known Camassa-Holm (CH) equation and its singular peakon solutions  prompted search for other integrable equations with nonsmooth solitons. An integrable CH-type equation with cubic nonlinearity was derived independently by Fokas , by Fuchssteiner , by Olver and Rosenau , and by Qiao . 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 . In the work by Hone and Wang , 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  and Zhang et al.  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 open set . refers to the set of all functions whose restriction on any compact subset is integrable. stands for .
Let us consider the traveling wave solution of the equation (3) through the setting , where is the wave speed. Let ; then . Substituting it into (3) yields where “” is the derivative with respect to . Integrating (4) once and neglecting the integration constant, we have Integrating (5) once again, we obtain where is an integration constant. Furthermore, we get where is also an integration constant.
To seek exact solutions with solitary patterns for (7), we impose the boundary condition where is a nonzero constant. Equation (7) can be cast into the following ordinary differential equation: The fact that both sides of (9) are nonnegative implies that . If , then (9) reduces to where Obviously, .
Definition 1. A function is said to be a single peak soliton solution for the equation (3) if satisfies the following conditions.(A1) is continuous on and has a unique peak point , where attains its global maximum or minimum value.(A2) satisfies (8) on .(A3).
Definition 2. A wave function is called peakon if is smooth locally on either side of and .
Definition 3. A wave function is called cuspon if is smooth locally on either side of and .
Without any loss of generality, we choose the peak point as vanishing, .
Theorem 4. Suppose that is a single peak soliton solution for the equation (3) at the peak point . Then one has the following.(i) If , then .(ii) If , then or or .
Proof. If , then for any since . Differentiating both sides of (9) yields .(i) For , if , then . By the definition of single peak soliton, we have . However, by (9) we must have , which contradicts the fact that is the unique peak point.(ii) For , if , by (9) we know that exists. According to the definition of peak point, we have . Thus we obtain or from (10) since contradicts the fact that is the unique peak point.
Theorem 5. Suppose that is a single peak soliton solution for the equation (3) at the peak point . Then one has the following solutions classification and asymptotic behavior.(i) If , then is a smooth soliton solution.(ii) If and , then gives the peaked soliton solution .(iii) If and , then is a cusped soliton solution and where . Thus .
Proof. (i) If , then for any , and so is a smooth soliton solution.
(ii) If and , then (9) becomes Solving (13), we obtain the peaked soliton solution
(iii) If and , then by the definition of single peak soliton solution we have ; thus, does not contain the factor . From (9), we obtain Let ; then and Inserting into (16) and using the initial condition , we obtain Thus which implies that . Therefore, we have where . Thus .
3. Peakons, Cuspons, and Smooth Soliton Solutions
Theorem 5 gives a classification for all single peak soliton solutions for the equation (3). In this section, we will present all possible single peak soliton solutions. We should discuss three cases: , , and .
Case I (). By virtue of Theorems 4 and 5, any single peak soliton solution for the equation (3) must satisfy the following initial and boundary values problem: Equation (20) implies that Since , introducing the constant yields which implies that
From the standard phase analysis, we know that if is a single peak soliton solution of the equation (3), then
Taking the integration of both sides of (24) leads to where . When , that is, , we obtain with and is an integration constant. Thus we obtain the implicit solution defined by Obviously, So, for or , the constant is defined by and for , (1).
If , then From , we know that is strictly decreasing on , and has the inverse denoted by . gives a smooth soliton solution satisfying The profile of smooth soliton solution is shown in Figure 1(a).(2).
If , then and ; there is no single peak soliton solution.(3).
If , then and From , we know that is strictly increasing on , and gives a unique cusped soliton solution. Therefore, is the solution satisfying The profile of cuspon is shown in Figure 1(b).
Case II (). If , then the only possible single peak soliton solution is the peakon
The profile of peaked soliton is shown in Figure 1(c).
is strictly decreasing on the interval . Define Then where Since is a strictly decreasing function, we can solve for uniquely from (47) and obtain which satisfies Therefore, the solution defined by (49) is a cusped soliton solution for the equation (3). The profile of cuspon is shown in Figure 1(d).
Let us summarize our results in the following theorem.
Theorem 6. Suppose that is a single peak soliton solution for the equation (3) at the peak point , which satisfies the inhomogeneous boundary condition (8). Then one has the following.(1) For , let ; then(i) if , there is no soliton for the equation (3);(ii) if and , the equation (3) has the smooth soliton solution with the following properties: (iii) if , the equation (3) has the cusped soliton solution with the following properties: (iv) if and , the equation (3) has the cusped soliton solution with the following properties: (2)When , then is the peakon
with the following properties:
This work is supported by the National Natural Science Foundation of China (no. 11071222) and the Natural Science Foundation of Huzhou University (no. KX21061). The authors would like to thank the anonymous referees for their suggestions and comments which made the presentation of this work better. And the first author also wants to express her sincere gratitude to Professor Zhijun Qiao for his kind help.
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