On Solitary Waves and Wave-Breaking Phenomena for a Generalized Two-Component Integrable Dullin–Gottwald–Holm System

Abstract

We study here a generalized two-component integrable Dullin–Gottwald–Holm system, which can be derived from the Euler equation with constant vorticity in shallow water waves moving over a linear shear flow. We first derive this system in the shallow-water regime. We next classify all traveling wave solution of this system. Finally, we study the blow-up mechanism and give two sufficient conditions which can guarantee wave-breaking phenomena.

Appendix

In this section, we supplement the proof of Theorem 3.9.

Proof of Theorem 3.9

First from (3.24) and the decay of φ(x) at infinity, we know that solitary waves exist if condition (3.15) holds.

If c=A ₁, then (3.24) becomes

$$ \varphi_{x}^{2}=\frac{-\varphi^{3}(A_{1}-A_{2}-\varphi)}{(A_{1}-\varphi )(A_{1}+\gamma-\sigma\varphi)}:=F_{1}( \varphi). $$

(A.1)

(1) If γ>−A ₁, then we see that φ(x)<0 near −∞. Similarly as in the proof of Theorem 3.7, we can find some x ₀ sufficiently negative with φ(x ₀)=−ε<0 and φ _x (x ₀)<0, and we can construct a unique local solution φ(x) on [x ₀−L,x ₀+L] for some L>0.

If σ<0, we see that $\frac{1}{A_{1}+\gamma-\sigma\varphi}$ is decreasing when (A ₁+γ)/σ<φ≤0. Combining this with (3.18) we know that F ₁(φ) decreases for φ<0. Because φ _x (x ₀)<0,φ(x) decreases near x ₀, so that F ₁(φ) increases near x ₀. Hence from (A.1), φ _x (x) decreases near x ₀, then φ and φ _x both decreases on [x ₀−L,x ₀+L]. Since $\sqrt{F_{1}(\varphi)}$ is locally Lipschitz in φ for (A ₁+γ)/σ<φ≤0, we can easily continue the local solution to (−∞,x ₀−L] with φ(x)→0 as x→−∞. As for x≥x ₀+L, we can solve the initial valued problem

$$\left \{ \begin{array}{l} \psi_{x}=-\sqrt{F_{1}(\varphi)},\\ \psi(x_{0}+L)=\varphi(x_{0}+L)\\ \end{array} \right . $$

all the way until ψ=(A ₁+γ)/σ, which is a simple pole of F ₁(φ). By (3.27) and (3.28), we deduce that we can construct an anticusped solution with a cusp singularity at φ=(A ₁+γ)/σ.

If σ>0, then $F'_{1}(\varphi)<0$ for φ<0. A similar argument as Theorem 3.7 shows that there is no solitary wave in this case.

(2) If γ<−A ₁, then we see that φ(x)>0 near −∞. Similarly as in the proof of Theorem 3.7, we can find some x ₀ sufficiently negative with φ(x ₀)=ε>0 and φ _x (x ₀)>0, and we can construct a unique local solution φ(x) on [x ₀−L,x ₀+L] for some L>0.

If σ<0, then $\frac{1}{A_{1}+\gamma-\sigma\varphi}$ is decreasing when 0≤φ<(A ₁+γ)/σ. Using (3.18) it is easy to find that F ₁(φ) increases for φ>0. If (σ−1)A ₁<γ<−A ₁, then $\sqrt{F_{1}(\varphi)}$ is locally Lipschitz in φ for 0≤φ<(A ₁+γ)/σ. Similarly as in the proof of (1), we can construct a cusped solution with a cusp singularity at φ=(A ₁+γ)/σ.

If σ>0, we also see that there is no solitary wave by the similar proof of Theorem 3.7.

Similarly, we conclude that when c=A ₂, there is no solitary wave when σ>0. When σ<0 and −A ₂<γ<(σ−1)A ₂, there is an anticusped solution with a cusp singularity at (A ₂+γ)/σ. When σ<0 and γ<−A ₂, there is an cusped solution with a cusp singularity at (A ₂+γ)/σ.

For the case (3.16), 14 cases are there we will consider. we will only look at −c<−A ₁<γ. The other cases can be handled in a very similar way. Applying (3.24), we know that φ cannot oscillate around zero near infinity. Let us consider the following two cases.

Case 1. φ(x)>0 near −∞. Then there is some x ₀ sufficiently negative so that φ(x ₀)=ε>0, with ε sufficiently small, and φ _x (x ₀)>0.

(i) When σ≤1, $\sqrt{F(\varphi)}$ is locally Lipschitz in φ for 0≤φ<c−A ₁. Hence there is a local solution to

$$\left \{ \begin{array}{l} \varphi_{x}=\sqrt{F(\varphi)},\\ \varphi(x_{0})=\varepsilon\\ \end{array} \right . $$

on [x ₀−L,x ₀+L] for some L>0. Therefore by (3.25) and (3.26), we obtain a smooth solitary wave with maximum height φ=c−A ₁ and an exponential decay to zero at infinity

$$ \varphi(x)=\mathrm{O} \biggl(\exp \biggl(-\frac{\sqrt{c^{2}+Ac-1}}{\sqrt{c(c-\gamma )}}|x| \biggr) \biggr) \quad \text{as}\ |x|\rightarrow\infty. $$

(A.2)

(ii) When σ>1, $\sqrt{F(\varphi)}$ is locally Lipschitz in φ for $0\leq\varphi<\frac{c+\gamma}{\sigma}$. Thus if $c-A_{1}<\frac{c+\gamma}{\sigma}$, i.e., $A_{1}<c<\frac{-\gamma-\sigma A_{1}}{1-\sigma}$, it becomes the same as (i) and we can obtain smooth solitary waves with exponential decay.

If $c-A_{1}=\frac{c+\gamma}{\sigma}$, then the smooth solution can be constructed until $\varphi=c-A_{1}=\frac{c+\gamma}{\sigma}$. However, at $\varphi=c-A_{1}=\frac{c+\gamma}{\sigma}$ it can make a sudden turn and so give rise to a peak. Since φ=0 is still a double zero of F(φ), we still have the exponential decay here.

If $c-A_{1}>\frac{c+\gamma}{\sigma}$, then $\varphi=\frac{c+\gamma}{\sigma}$ becomes a pole of F(φ). Using (3.27) and (3.28), we obtain a solitary wave with a cusp at $\varphi=\frac{c+\gamma}{\sigma}$ and decays exponentially.

Case 2. φ(x)<0 near −∞. In this case we are solving

$$\left \{ \begin{array}{l} \varphi_{x}=-\sqrt{F(\varphi)},\\ \varphi(x_{0})=-\varepsilon\\ \end{array} \right . $$

for some x ₀ sufficiently negative and ε>0 sufficiently small.

When σ>0 we see that F′(φ)<0, for φ<0. Therefore in this case there is no solitary wave.

If σ<0, then φ=(c+γ)/σ<0 is a pole of F(φ). Arguing as before, we obtain an anticusped solitary wave with min _x∈ℝ=(c+γ)/σ, which decays exponentially.

Finally, by the standard ODE theory and the fact that Eq. (3.11) is invariant under the transformations x⟶−x, we conclude that the solitary waves obtained above are unique and unambiguous up to translations. □