$\phi ^4$ Solitary Waves in a Parabolic Potential: Existence, Stability, and Collisional Dynamics

Abstract

We explore a $\phi ^4$ model with an added external parabolic potential term. This term dramatically alters the spectral properties of the system. We identify single and multiple kink solutions and examine their stability features; importantly, all of these stationary structures turn out to be unstable. We complement these with a dynamical study of the evolution of a single kink in the trap, as well as of the scattering of kink and antikink solutions of the model. We observe that some of the key characteristics of kink-antikink collisions, such as the critical velocity and the multi-bounce windows, are sensitively dependent on the trap strength parameter, as well as the initial displacement of the kink and antikink.

Appendices

Appendix 1: Tables for n-Bounce Windows

We list the velocity intervals $[v_1,v_2]$ that result in multi-bounce windows when the initial velocity $v_\mathrm {in}$ is picked within these intervals. In Tables 10.1 and 10.2, we list those intervals for a small separation ($x_0=1.4$) and a big separation ($x_0=7$) with $\Omega =0.15$. In Tables 10.3 and 10.4, we list those intervals for bigger values of $\Omega$ (0.2 and 0.3 respectively) with fixed $x_0=2$.

Table 10.1 n-bounce windows for $x_0=1.4$. One-bounce windows occur for $v_\mathrm {in}>0.25845$

Table 10.2 n-bounce windows for $x_0=7$. One-bounce windows occur for $v_\mathrm {in}>0.74414$

Table 10.3 n-bounce windows for $x_0=2$, $\Omega =0.2$. One-bounce windows occur for $v_\mathrm {in}>0.28942$

Table 10.4 n-bounce windows for $x_0=2$, $\Omega =0.3$. One-bounce windows occur for $v_\mathrm {in}>0.36317$

Appendix 2: Derivation of the Coefficients in (10.14)

We define $\phi _{\pm }=\pm \phi _0(x\pm X(t))$ and $\chi _{\pm }=\pm \chi _{1}(x\pm X(t))$, $\phi '_{\pm }=\pm \phi _0'(x\pm X(t))$ and $\chi '_{\pm }=\pm \chi '_1(x\pm X(t))$. Then (10.13) becomes

$$\begin{aligned} u(x,t)=u_{\Omega }(\phi _+ + \phi _{-} -1) + A(\chi _{+}+\chi _{-}). \end{aligned}$$

(10.16)

Substituting (10.16) into (10.10) gives

$$\begin{aligned} \begin{aligned} L=&\int \left\{ \frac{1}{2}\left[ u_{\Omega }\left( \phi _{+}^{\prime }-\phi _{-}^{\prime }\right) \dot{X} +\dot{A}(\chi _{+}+\chi _{-})+A(\chi _{+}^{\prime }-\chi _{-}^{\prime })\dot{X} \right] ^{2} \right\} \mathrm {d}x\\ -&\int \left\{ \frac{1}{2}\left[ u_{\Omega }\left( \phi _{+}^{\prime }+\phi _{-}^{\prime }\right) +u_{\Omega }'(\phi _+ + \phi _{-} -1) +A(\chi _{+} ^{\prime }+\chi _{-}^{\prime }) \right] ^{2} \right\} \mathrm {d}x\\ -&\int V(u) \mathrm {d}x. \end{aligned} \end{aligned}$$

(10.17)

To handle the V(u) terms we first write $u =u _{a}+u _{b}$ where $u _{a}=u_{\Omega }(\phi _+ + \phi _{-} -1)$ and $u _{b}= A(\chi _{+}+\chi _{-})$. Then using the Taylor series expansion, we get

$$\begin{aligned} \begin{aligned} V(u )&= V(u _{a}+u _{b})\\&=V(u _{a})+V^{\prime }(u _{a})u _{b}+\frac{V^{\prime \prime }(u _{a})}{2!}u _{b}^{2}+\frac{V^{^{\prime \prime \prime }}(u _{a})}{3!}u _{b}^{3}+\frac{V^{(4)}(u _{a})}{4!}u _{b}^{4} \end{aligned} \end{aligned}$$

(10.18)

The corresponding reduced Lagrangian (ignoring higher order terms) that is used in our simulations is given by (10.14). Applying Euler–Lagrange equations, we obtain (10.15).

We will list the formulae of the coefficients below. Note that they are all functions of X(t). Since $\chi _{1}$ is not known explicitly, the coefficients presented are in the integral form. These coefficients are calculated numerically.

$$\begin{aligned} \begin{aligned} U(X)=&\int \frac{1}{2}\left( u_{\Omega } \left( \phi _{+}^{\prime }+\phi _{-}^{\prime }\right) +u_{\Omega }'(\phi _{+}+\phi _{-}-1)\right) ^{2}\,\mathrm {d}x \\&\qquad \quad \quad \,\,+\int V(u_{\Omega }(\phi _{+}+\phi _{-}-1))\,\mathrm {d}x \end{aligned} \end{aligned}$$

$$\begin{aligned} \begin{aligned} F(X)=-\frac{1}{2}\int&(u_{\Omega } \left( \phi _{+}^{\prime }+\phi _{-}^{\prime }\right) +u_{\Omega }'(\phi _{+}+\phi _{-}-1))(\chi _{+}^{\prime }+\chi _{-}^{\prime })\,\mathrm {d}x\\&-\frac{1}{2}\int V^{\prime }(u_{\Omega }(\phi _{+}+\phi _{-}-1))(\chi _{+}+\chi _{-})\,\mathrm {d}x \end{aligned} \end{aligned}$$

$$\begin{aligned} \begin{aligned} K(X)=&-\frac{1}{2}\int (\chi _{+}^{\prime }+\chi _{-}^{\prime })^{2}\,\mathrm {d}x -\frac{1}{2}\int V^{\prime \prime }(u_{\Omega }(\phi _{+}+\phi _{-}-1))(\chi _{+}+\chi _{-})^{2}\,\mathrm {d}x \end{aligned} \end{aligned}$$

$$\begin{aligned} I(X)=\frac{1}{2}\int u_{\Omega }^2\left( \phi _{+}^{\prime }-\phi _{-}^{\prime }\right) ^{2}\,\mathrm {d}x, Q(X)=\frac{1}{2}\int (\chi _{+}+\chi _{-})^{2}\,\mathrm {d}x, \end{aligned}$$

$$\begin{aligned} C(X)=\frac{1}{2}\int u_{\Omega }\left( \phi _{+}^{\prime }-\phi _{-}^{\prime }\right) (\chi _{+}+\chi _{-})\,\mathrm {d}x \end{aligned}$$