Camassa–Holm Cuspons, Solitons and Their Interactions via the Dressing Method

Abstract

A dressing method is applied to a matrix Lax pair for the Camassa–Holm equation, thereby allowing for the construction of several global solutions of the system. In particular, solutions of system of soliton and cuspon type are constructed explicitly. The interactions between soliton and cuspon solutions of the system are investigated. The geometric aspects of the Camassa–Holm equation are re-examined in terms of quantities which can be explicitly constructed via the inverse scattering method.

Appendix

In appendix, we provide some details on the derivation of the soliton–cuspon solution. Applying equation (3.2) to the dressing factor g as given by Eq. (4.54) ensures the matrix-valued residues satisfy the following

$$\begin{aligned} \begin{aligned}&A_{1,y}+h \sigma _3 A_1 - A_1 h_0\sigma _3 - i\omega _1 [J, A_1]=0, \\&B_{2,y}+h \sigma _3 B_2 - B_2 h_0\sigma _3 - \lambda _2 [J, B_2]=0. \end{aligned} \end{aligned}$$

(7.1)

Writing the rank one matrix solutions $A_1$ and $B_2$ in the form

$$\begin{aligned} \begin{aligned} A_1&= \left| {n}\right\rangle \left\langle {m}\right| , \qquad B_2 = \left| {N}\right\rangle \left\langle {M}\right| ,\\ \end{aligned} \end{aligned}$$

(7.2)

we deduce

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _y \left| {n}\right\rangle + (h\sigma _3-i\omega _1 J)\left| {n}\right\rangle =0, \qquad \partial _y\left\langle {m}\right| = \left\langle {m}\right| (h_0\sigma _3-i\omega _1 J)\\ \partial _y \left| {N}\right\rangle + (h\sigma _3-\lambda _2 J)\left| {N}\right\rangle =0, \qquad \partial _y\left\langle {M}\right| = \left\langle {M}\right| (h_0\sigma _3-\lambda _2 J). \end{array}\right. } \end{aligned}$$

(7.3)

The vectors $\left\langle {m}\right| ,\left\langle {M}\right|$ satisfy the bare equations and therefore are known in principle and have been obtained previously (see Sects. 4.3–4.4).

The dressing factor (4.54) at $\lambda =0$ is

$$\begin{aligned} g(y,t;0)= & {} \mathbb {1}- 2(A_1+B_2)=\mathrm {diag}(g_{11},g_{22})\nonumber \\= & {} \mathrm {diag}\left( \frac{i\omega _1 M_1 m_2 -\lambda _2 M_2 m_1}{i\omega _1 M_2 m_1-\lambda _2 M_1 m_2},\frac{i\omega _1 M_2 m_1-\lambda _2 M_1 m_2}{i\omega _1 M_1 m_2 -\lambda _2 M_2 m_1} \right) , \end{aligned}$$

(7.4)

while the differential equation for X(y, t) is

$$\begin{aligned} (\partial _y X) e^{X-2h_0 y-u_0 t} =g_{22}^2=\left( \frac{\lambda _1 M_2 m_1-\lambda _2 M_1 m_2}{\lambda _1 M_1 m_2 -\lambda _2 M_2 m_1}\right) ^2. \end{aligned}$$

(7.5)

Choosing $m_1, m_2$ as per the soliton solution cf. (Ivanov et al. 2017) and recalling $\Lambda _1=\sqrt{h_0^2-\omega _1^2}$, we then have

$$\begin{aligned} \begin{aligned} m_1&=\mu _1\sqrt{\frac{h_0+\Lambda _1}{2\Lambda _1}}e^{\Omega _1(y,t)}+\mu _2 \sqrt{\frac{h_0-\Lambda _1}{2\Lambda _1}}e^{-\Omega _1(y,t)},\\ m_2&=i\left( \mu _1 \sqrt{\frac{h_0-\Lambda _1}{2\Lambda _1}}e^{\Omega _1(y,t)}+\mu _2 \sqrt{\frac{h_0+\Lambda _1}{2\Lambda _1}}e^{-\Omega _1(y,t)}\right) , \end{aligned} \end{aligned}$$

where $\mu _k$ are positive constants. Ignoring an irrelevant overall constant of $\sqrt{\mu _1 \mu _2}(2\Lambda _1)^{-1/2}$ (see Sect. 4.3) and changing the definition of $\Omega _1(y,t)$ by an additive constant, as given by

$$\begin{aligned} \Omega _1(y,t)=\Lambda _{1}\left( y-\frac{t}{2h_0}\left( u_0+\frac{1}{2\omega _1^2}\right) \right) + \ln \sqrt{\frac{\mu _1}{\mu _2}}, \end{aligned}$$

(7.6)

we obtain the simplified expressions

$$\begin{aligned} \begin{aligned} m_1&=\sqrt{h_0+\Lambda _1}e^{\Omega _1(y,t)}+\sqrt{h_0-\Lambda _1}e^{-\Omega _1(y,t)},\\ m_2&=i\left( \sqrt{h_0-\Lambda _1}e^{\Omega _1(y,t)}+\sqrt{h_0+\Lambda _1}e^{-\Omega _1(y,t)}\right) . \end{aligned} \end{aligned}$$

(7.7)

Similarly, as per the cuspon solution, we define the constant vector $\left\langle {M_{(0)}}\right| V_2= (\nu _1, \nu _2)$ with $\nu _1, \nu _2$ real and positive, thereby ensuring

$$\begin{aligned} {\left\{ \begin{array}{ll} &{}M_1=\sqrt{\Lambda _2+h_0}e^{\Omega _2(y,t)}-\sqrt{\Lambda _2-h_0}e^{-\Omega _2(y,t)},\\ &{}M_2=\sqrt{\Lambda _2-h_0}e^{\Omega _2(y,t)}+\sqrt{\Lambda _2+h_0}e^{-\Omega _2(y,t)},\\ &{}\Omega _2(y,t)=\Lambda _{2}\left( y-\frac{t}{2h_0}\left( u_0-\frac{1}{2\lambda _2^2}\right) \right) + \ln \sqrt{\frac{\nu _1}{\nu _2}} , \\ &{}\Lambda _2=\sqrt{h_0^2+\lambda _2^2}. \end{array}\right. } \end{aligned}$$

(7.8)

The expression

$$\begin{aligned} g_{22}=\frac{i\omega _1 M_2 m_1-\lambda _2 M_1 m_2}{i\omega _1 M_1 m_2 -\lambda _2 M_2 m_1}=\frac{\mathcal {T}_{CS}}{\mathcal {B}_{CS}}, \end{aligned}$$

(7.9)

whose explicit form may be deduced from Eqs. (7.7)–(7.8), has denominator

$$\begin{aligned} \begin{aligned} \mathcal {B}_{CS}=&-\omega _1 \lambda _2 \left( \frac{\Lambda _2 - \Lambda _1}{\sqrt{(h_0-\Lambda _1)(\Lambda _2-h_0)}}e^{\Omega _1+\Omega _2}+ \frac{\Lambda _2 - \Lambda _1}{\sqrt{(h_0+\Lambda _1)(\Lambda _2+h_0)}}e^{-\Omega _1-\Omega _2} \right. \\&\quad + \left. \frac{\Lambda _1 + \Lambda _2}{\sqrt{(h_0-\Lambda _1)(\Lambda _2+h_0)}}e^{\Omega _1-\Omega _2} + \frac{\Lambda _1 + \Lambda _2}{\sqrt{(h_0+\Lambda _1)(\Lambda _2-h_0)}}e^{-\Omega _1+\Omega _2} \right) , \end{aligned} \end{aligned}$$

(7.10)

where we note that $\Lambda _2> h_0 > \Lambda _1$, thus ensuring $\Lambda _2 - \Lambda _1 >0$.

Introducing the constants

$$\begin{aligned} n_1=\root 4 \of {\frac{h_0+\Lambda _1}{h_0-\Lambda _1}},\qquad n_2=\root 4 \of {\frac{\Lambda _2+h_0}{\Lambda _2-h_0}}, \end{aligned}$$

(7.11)

we rewrite this denominator according to

$$\begin{aligned} \mathcal {B}_{CS}= & {} -\sqrt{\omega _1 \lambda _2} (\Lambda _1^2-\Lambda _1^2) \nonumber \\&\times \left( n_1 n_2\frac{e^{\Omega _1+\Omega _2}}{\Lambda _1+\Lambda _2} +\frac{1}{n_1 n_2}\frac{e^{-\Omega _1-\Omega _2}}{\Lambda _1+\Lambda _2} +\frac{n_1}{n_2}\frac{e^{\Omega _1-\Omega _2}}{\Lambda _2-\Lambda _1}+ \frac{n_2}{n_1}\frac{e^{-\Omega _1+\Omega _2}}{\Lambda _2-\Lambda _1} \right) .\nonumber \\ \end{aligned}$$

(7.12)

Similarly, it is found that the numerator assumes the form

$$\begin{aligned} \mathcal {T}_{CS}= & {} i\sqrt{\omega _1 \lambda _2} (\Lambda _1^2-\Lambda _2^2)\nonumber \\&\times \left( \frac{1}{n_1 n_2}\frac{e^{\Omega _1+\Omega _2}}{\Lambda _1+\Lambda _2} -n_1 n_2\frac{e^{-\Omega _1-\Omega _2}}{\Lambda _1+\Lambda _2} -\frac{n_2}{n_1}\frac{e^{\Omega _1-\Omega _2}}{\Lambda _2-\Lambda _1}+ \frac{n_1}{n_2}\frac{e^{-\Omega _1+\Omega _2}}{\Lambda _2-\Lambda _1} \right) .\nonumber \\ \end{aligned}$$

(7.13)

As with the two-cuspon solutions, we seek a solution of Eq. (4.32) in the form of Eq. (4.13) with

$$\begin{aligned} \mathcal {A}_{CS}=\alpha _1 e^{\Omega _1+\Omega _2}+\alpha _2 e^{-\Omega _1-\Omega _2}+\alpha _3 e^{\Omega _1-\Omega _2}+\alpha _4 e^{-\Omega _1+\Omega _2}, \end{aligned}$$

(7.14)

where the constants $\left\{ \alpha _l\right\} _{l=1}^{4}$ are as yet unknown. Equation (4.32) ensures that

$$\begin{aligned} 2h_0\mathcal {A}_{CS}\mathcal {B}_{CS}+\mathcal {B}_{CS}\partial _y\mathcal {A}_{CS}-\mathcal {A}_{CS}\partial _y\mathcal {A}_{CS}=2h_0\mathcal {T}_{CS}^2 \end{aligned}$$

(7.15)

has a solution given by

$$\begin{aligned} \mathcal {A}_{CS}= & {} \sqrt{\omega _1 \lambda _2} (\Lambda _2^2-\Lambda _1^2)\nonumber \\&\times \left( \frac{1}{n_1^3 n_2^3}\frac{e^{\Omega _1+\Omega _2}}{\Lambda _1+\Lambda _2}+n_1^3 n_2^3\frac{e^{-\Omega _1-\Omega _2}}{\Lambda _1+\Lambda _2} +\frac{n_2^3}{n_1^3}\frac{e^{\Omega _1-\Omega _2}}{\Lambda _2-\Lambda _1}+ \frac{n_1^3}{n_2^3}\frac{e^{-\Omega _1+\Omega _2}}{\Lambda _2-\Lambda _1} \right) .\nonumber \\ \end{aligned}$$

(7.16)

The ratio $\mathcal {A}_{CS}/\mathcal {B}_{CS}$ may now be written as

$$\begin{aligned} \frac{\mathcal {A}_{CS}}{\mathcal {B}_{CS}}=n_1^4 n_2^4\frac{1+ \frac{1}{n_1^6 }\frac{\Lambda _1+\Lambda _2}{\Lambda _2-\Lambda _1}e^{2\Omega _1}+ \frac{1}{n_2^6 }\frac{\Lambda _1+\Lambda _2}{\Lambda _2-\Lambda _1}e^{2\Omega _2} +\frac{e^{2\Omega _1+2\Omega _2}}{n_1^6n_2^6} }{1+ n_1^2\frac{\Lambda _1+\Lambda _2}{\Lambda _2-\Lambda _1}e^{2\Omega _1}+ n_2^2 \frac{\Lambda _1+\Lambda _2}{\Lambda _2-\Lambda _1}e^{2\Omega _2}+ n_1^2n_2^2e^{2\Omega _1+2\Omega _2} } \end{aligned}$$

(7.17)

which we simplify by means of the following re-definitions:

$$\begin{aligned} \begin{aligned} \Omega _1(y,t)&=\Lambda _{1}\left( y-\frac{t}{2h_0}\left( u_0+\frac{1}{2\omega _1^2}\right) \right) + \ln \sqrt{\frac{\mu _1}{\mu _2}}-\ln n_1 + \frac{1}{2}\ln \frac{\Lambda _1+\Lambda _2}{b}{\Lambda _2 - \Lambda _1},\\ \Omega _2(y,t)&=\Lambda _{2}\left( y-\frac{t}{2h_0}\left( u_0-\frac{1}{2\lambda _2^2}\right) \right) + \ln \sqrt{\frac{\nu _1}{\nu _2}}-\ln n_2 + \frac{1}{2}\ln \frac{\Lambda _1+\Lambda _2}{\Lambda _2 - \Lambda _1}. \end{aligned} \end{aligned}$$

(7.18)

These re-definitions are valid since $\Lambda _2> \Lambda _1$, as was previously noted. Alternatively, these may be simply written as

$$\begin{aligned} {\begin{matrix} \Omega _1(y,t)&{}=\Lambda _{1}\left( y-\frac{t}{2h_0}\left( u_0+\frac{1}{2\omega _1^2}\right) \right) +\xi _1\\ \Omega _2(y,t)&{}=\Lambda _{2}\left( y-\frac{t}{2h_0}\left( u_0-\frac{1}{2\lambda _2^2}\right) \right) +\xi _2 \end{matrix}} \end{aligned}$$

(7.19)

for some constants $\left\{ \xi _k\right\} _{k=1}^{2}$ related to the initial separation of the cuspon and soliton. This allows the expression (7.17) to be written as

$$\begin{aligned} \frac{\mathcal {A}_{CS}}{\mathcal {B}_{CS}}=n_1^4 n_2^4\frac{1+ \frac{1}{n_1^4 }e^{2\Omega _1}+ \frac{1}{n_2^4 }e^{2\Omega _2} +\left( \frac{\Lambda _1-\Lambda _2}{\Lambda _1+\Lambda _2}\right) ^2\frac{e^{2\Omega _1+2\Omega _2}}{n_1^4n_2^4} }{1+ n_1^4e^{2\Omega _1}+ n_2^4e^{2\Omega _2} + \left( \frac{\Lambda _1-\Lambda _2}{\Lambda _1+\Lambda _2}\right) ^2n_1^4n_2^4e^{2\Omega _1+2\Omega _2} } \end{aligned}$$

(7.20)

with

$$\begin{aligned} X(y,t)=\frac{y}{\sqrt{u_0}}+u_0t+\ln \left| \frac{\mathcal {A}_{CS}}{\mathcal {B}_{CS}}\right| . \end{aligned}$$