The nonlinear dispersive Boussinesq-like B(2,2) equation utt+(u2)xx−(u2)xxxx=0, which exhibits single peak solitons, is investigated. Peakons, cuspons and smooth soliton solutions are obtained by setting the B(2,2) equation under inhomogeneous boundary condition. Asymptotic behavior and numerical simulations are provided for these three types of single peak soliton solutions of the B(2,2) equation.

1. Introduction

The B(2,2) 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
(1)mt+[m(u2-ux2)]x=0,m=u-uxx,
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 B(m,n) equation with nonlinear dispersion is given by
(2)utt+a(un)xx+b(um)xxxx=0,
where a, b∈R and m, n∈Z+. This equation is the generalized form of the Boussinesq equation, where, in particular, the case (m,n)=(1,2) leads to the Boussinesq equation. Equation (2), for ab>0, is the major equation for compactons (solitons with compact support). Abundant compactons [10–13] are developed by the Adominan decomposition method. For ab<0, exact solutions with solitary patterns of Boussinesq-like B(m,n) 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 B(m,n) equation (2) has nonsmooth solitons such as peakons or cuspons. The present paper focuses on the following Boussinesq-like B(2,2) equation:
(3)utt+(u2)xx-(u2)xxxx=0.
We give all possible single peak soliton solutions of (3) through setting the traveling wave solution under the inhomogeneous boundary condition u→A (A is a nonzero constant) as x→±∞. 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 B(2,2) 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 Ck(Ω) denote the set of all k times continuously differential functions on the open set Ω. Llocp(R) refers to the set of all functions whose restriction on any compact subset is Lp integrable. Hloc1(R) stands for Hloc1(R)={u∈Lloc2(R)∣u′∈Lloc2(R)}.

Let us consider the traveling wave solution of the B(2,2) equation (3) through the setting u(x,t)=U(x-ct), where c is the wave speed. Let ξ=x-ct; then u(x,t)=U(ξ). Substituting it into (3) yields
(4)c2U′′+(U2)′′-(U2)′′′′=0,
where “′” is the derivative with respect to ξ. Integrating (4) once and neglecting the integration constant, we have
(5)c2U′+(U2)′-(U2)′′′=0.
Integrating (5) once again, we obtain
(6)c2U+U2-2U′2-2UU′′=g1,
where g1 is an integration constant. Furthermore, we get
(7)U′2=U24+c2U3-g12+g2U2,
where g2 is also an integration constant.

To seek exact solutions with solitary patterns for (7), we impose the boundary condition
(8)limξ→±∞U(ξ)=A,
where A is a nonzero constant. Equation (7) can be cast into the following ordinary differential equation:
(9)U′2=(U-A)2[3U2+(6A+4c2)U+3A2+2Ac2]12U2.
The fact that both sides of (9) are nonnegative implies that 3U2+(6A+4c2)U+3A2+2Ac2≥0. If 3A+2c2≥0, then (9) reduces to
(10)U′2=(U-A)2(U-B1)(U-B2)4U2,
where
(11)B1=-3A+2c23+2c2(3A+2c2)3,B2=-3A+2c23-2c2(3A+2c2)3.
Obviously, B1≥B2.

Definition 1.

A function u(x,t)=U(x-ct) is said to be a single peak soliton solution for the B(2,2) equation (3) if U(ξ) satisfies the following conditions.

U(ξ) is continuous on R and has a unique peak point ξ0, where u(ξ) attains its global maximum or minimum value.

U(ξ)∈C4(R-{ξ0}) satisfies (8) on R-{ξ0}.

limξ→±∞U(ξ)=A.

Definition 2.

A wave function U is called peakon if U is smooth locally on either side of ξ0 and limξ↑ξ0U′(ξ)=-limξ↓ξ0U′(ξ)=a,a≠0,a≠±∞.

Definition 3.

A wave function U is called cuspon if U is smooth locally on either side of ξ0 and limξ↑ξ0U′(ξ)=-limξ↓ξ0U′(ξ)=±∞.

Without any loss of generality, we choose the peak point ξ0 as vanishing, ξ0=0.

Theorem 4.

Suppose that U(ξ) is a single peak soliton solution for the B(2,2) equation (3) at the peak point ξ0=0. Then one has the following.

If A<-2c2/3, then U(0)=0.

If A≥-2c2/3, then U(0)=0 or U(0)=B1 or U(0)=B2.

Proof.

If U(0)≠0, then U(ξ)≠0 for any ξ∈R since U(ξ)∈C4(R-{0}). Differentiating both sides of (9) yields U(ξ)∈C∞(R).

For A<-2c2/3, if U(0)≠0, then U∈C∞(R). By the definition of single peak soliton, we have U′(0)=0. However, by (9) we must have U(0)=A, which contradicts the fact that 0 is the unique peak point.

For A≥-2c2/3, if U(0)≠0, by (9) we know that U′(0) exists. According to the definition of peak point, we have U′(0)=0. Thus we obtain U(0)=B1 or U(0)=B2 from (10) since U(0)=A contradicts the fact that 0 is the unique peak point.

Theorem 5.

Suppose that U(ξ) is a single peak soliton solution for the B(2,2) equation (3) at the peak point ξ0=0. Then one has the following solutions classification and asymptotic behavior.

If U(0)≠0, then U(ξ) is a smooth soliton solution.

If U(0)=0 and A=-2c2/3, then U(ξ) gives the peaked soliton solution (-2c2/3)(1-e-|x-ct|/2).

If U(0)=0 and A≠-2c2/3, then U(ξ) is a cusped soliton solution and
(12)U(ξ)=μ|ξ|1/2+O(|ξ|),ξ⟶0,U′(ξ)=μ2|ξ|-1/2sgn(ξ)+O(1),ξ⟶0,

where μ=±(1/3)3|A|9A2+6Ac2. Thus U(ξ)∉Hloc1(R).
Proof.

(i) If U(0)≠0, then U(ξ)∈C∞(R) for any ξ∈R, and so U(ξ) is a smooth soliton solution.

(ii) If U(0)=0 and A=-2c2/3, then (9) becomes
(13)U′2=14(U+2c23)2.
Solving (13), we obtain the peaked soliton solution
(14)U(ξ)=-2c23(1-e-|x-ct|/2).

(iii) If U(0)=0 and A≠-2c2/3, then by the definition of single peak soliton solution we have A≠0; thus, 3U2+(6A+4c2)U+3A2+2Ac2 does not contain the factor U. From (9), we obtain
(15)U′=-sgn(A)U-AU×3U2+(6A+4c2)U+3A2+2Ac212sgn(ξ).
Let h(U)=23sgn(A)/(A-U)3U2+(6A+4c2)U+3A2+2Ac2; then h(0)=23/|A|3A2+2Ac2 and
(16)∫Uh(U)dU=∫sgn(ξ)dξ.
Inserting h(U)=h(0)+O(U) into (16) and using the initial condition U(0)=0, we obtain
(17)h(0)2U2(1+O(U))=|ξ|.
Thus
(18)U=±2h(0)|ξ|1/2(1+O(U))-1/2=±2h(0)|ξ|1/2(1+O(U)),
which implies that U=O(|ξ|1/2). Therefore, we have
(19)U(ξ)=μ|ξ|1/2+O(|ξ|),ξ⟶0,U′(ξ)=μ2|ξ|-1/2sgn(ξ)+O(1),ξ⟶0,
where μ=±(1/3)3|A|9A2+6Ac2. Thus U(ξ)∉Hloc1(R).

3. Peakons, Cuspons, and Smooth Soliton Solutions

Theorem 5 gives a classification for all single peak soliton solutions for the B(2,2) equation (3). In this section, we will present all possible single peak soliton solutions. We should discuss three cases: A>-2c2/3, A=-2c2/3, and A<-2c2/3.

Case I (A>-2c2/3). By virtue of Theorems 4 and 5, any single peak soliton solution for the B(2,2) equation (3) must satisfy the following initial and boundary values problem:
(20)U′2=(U-A)2(U-B1)(U-B2)4U2,U(0)∈{0,B1,B2},limξ→±∞U(ξ)=A.
Equation (20) implies that
(21)U≥B1,orU≤B2,(A-B1)(A-B2)≥0.
Since 3A+2c2≠0, introducing the constant α=(A/(3A+2c2)) yields
(22)α(α+1)≥0,
which implies that
(23)α<-1,α=-1,α=0,α>0.

From the standard phase analysis, we know that if U(ξ) is a single peak soliton solution of the B(2,2) equation (3), then
(24)U′=-sgn(A)U-A2U(U-B1)(U-B2)sgn(ξ),(25)U(0)={max(0,B1),ifU(0)isaminimum,min(0,B2),ifU(0)isamaximum.

Taking the integration of both sides of (24) leads to
(26)∫ϕ(U)dU=-|ξ|2,
where ϕ(U)=-sgn(A)(U/(U-A)(U-B1)(U-B2)). When α≠0,-1, that is, 2A(2A+c2)≠0, we obtain
(27)∫ϕ(U)dU=sgn(A)I1(U)+|A|4A2+2Ac2I2(U)-K=Φ(U)-K,
with
(28)I1(U)=ln|2U-B1-B2+(U-B1)(U-B2)|,I2(U)=ln(|((A-B1)(A-B2)(U-B1)(U-B2)(A-B1)(U-B2)+(A-B2)(U-B1)-2(A-B1)(A-B2)(U-B1)(U-B2))×(U-A)-1(A-B1)(A-B2)(U-B1)(U-B2)|),
and K is an integration constant. Thus we obtain the implicit solution U(ξ) defined by
(29)Φ(U)=-|ξ|2+K.
Obviously,
(30)I1(B1)=I1(B2)=ln|B1-B2|=ln6Ac2+4c43,I2(B1)=I2(B2)=ln|B1-B2|=ln6Ac2+4c43.
So, for U(0)=B1 or B2, the constant K0=Φ(U(0)) is defined by
(31)K0=sgn(A)ln6Ac2+4c43+|A|6Ac2+4c4×ln6Ac2+4c43,
and for U(0)=0,
(32)K0=I1(0)-2A4A2-2AcI2(0).

α<-1.

If α<-1, then
(33)A<B2<0<B1,U(0)=B2,A<U≤B2.
From ϕ(U)<0, we know that Φ(U) is strictly decreasing on (A,B2], and
(34)Φ1(U)=Φ∣(A,B2](U)
has the inverse denoted by U1(ξ)=Φ1-1(-(|ξ|/2)+K0). U1(ξ) gives a smooth soliton solution satisfying
(35)U1(0)=B2,limξ→±∞U1(ξ)=A,U1′(0)=0.
The profile of smooth soliton solution is shown in Figure 1(a).

If α=-1, then A=B2=-c2/2 and B1=c2/6; there is no single peak soliton solution.

α=0.

In case A=0, (9) becomes
(36)U′=-12U(U+4c43)sgn(ξ),U(±∞)=0.
Thus there is no single peak soliton solution for the previous boundary condition (8).

α>0.

If α>0, then A>0 and
(37)B2<B1<0<A,U(0)=0,0≤U<A.
From ϕ(U)>0, we know that Φ(U) is strictly increasing on [0,A), and
(38)Φ2(U)=Φ∣[0,A)(U)
gives a unique cusped soliton solution. Therefore, U2(ξ)=Φ2-1(-(|ξ|/2)+K0) is the solution satisfying
(39)U2(0)=0,limξ→±∞U2(ξ)=A,U2′(0+)=+∞,U2′(0-)=-∞.
The profile of cuspon is shown in Figure 1(b).

Case II (A=-2c2/3). If A=-2c2/3, then the only possible single peak soliton solution is the peakon
(40)U(ξ)=-2c23(1-e-|x-ct|/2).

The profile of peaked soliton is shown in Figure 1(c).

Case III (A<-2c2/3). In this case, according to Theorem 4 and standard phase portrait analytical technique, we have U(0)=0, A<U≤0, and
(41)U′=U-AU3U2+(6A+4c2)U+3A2+2Ac212sgn(ξ).
Let
(42)X=U+a1,a1=3A+2c23,a22=-29(3A+2c2),
and then (41) becomes
(43)φ(X)dX=X-a1(X-a1-A)X2+a22dX=12sgn(ξ)dξ.
Integration of both sides of (43) gives
(44)Ψ(X)=|ξ|2+K,
where
(45)Ψ(X)=J1(X)+A(a1+A)2+a22J2(X),J1(X)=ln|X+X2+a22|,J2(X)=ln|2[a22+(a1+A)X-(a1+A)2+a22X2+a22]X-a1-A|.

Ψ(X) is strictly decreasing on the interval [a1,a1+A). Define
(46)Ψ1(X)=Ψ∣[a1,a1+A)(X).
Then
(47)Ψ1(X)=|ξ|2+K0,
where
(48)K0=J1(a1)+A(a1+A)2+a22J2(a1).
Since Ψ1 is a strictly decreasing function, we can solve for X uniquely from (47) and obtain
(49)U(ξ)=Ψ1-1(|ξ|2+K0)-a1,
which satisfies
(50)U(0)=0,limξ→±∞U(ξ)=A,U′(0+)=-∞,U′(0-)=+∞.
Therefore, the solution U defined by (49) is a cusped soliton solution for the B(2,2) 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 U(x-ct) is a single peak soliton solution for the B(2,2) equation (3) at the peak point ξ0=0, which satisfies the inhomogeneous boundary condition (8). Then one has the following.

For 3A+2c2≠0, let α=A/(3A+2c2); then

if -1≤α≤0, there is no soliton for the B(2,2) equation (3);

if α<0 and A<0, the B(2,2) equation (3) has the smooth soliton solution
(51)U(ξ)=Φ1-1(-|ξ|2+K0),

with the following properties:
(52)U(0)=B2,U(±∞)=A,U′(0)=0;

if α>0, the B(2,2) equation (3) has the cusped soliton solution
(53)U(x,t)=Φ2-1(-|ξ|2+K0),

with the following properties:
(54)U(0)=0,U(±∞)=A,U′(0+)=+∞,U′(0-)=-∞;

if α<0 and A>0, the B(2,2) equation (3) has the cusped soliton solution
(55)U(x,t)=Ψ1-1(|ξ|2+K0)-a1,

with the following properties:
(56)U(0)=0,U(±∞)=A,U′(0+)=-∞,U′(0-)=+∞.

When 3A+2c2=0, then U is the peakon
(57)U(x,t)=-2c23(1-e-|x-ct|/2),

with the following properties: (58)U(0)=0,U(±∞)=A,U′(0+)=-c23,U′(0-)=c23.

Acknowledgments

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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