Doubly Localized Two-Dimensional Rogue Waves in the Davey–Stewartson I Equation

Abstract

Doubly localized two-dimensional rogue waves for the Davey–Stewartson I equation in the background of dark solitons or a constant, are investigated by employing the Kadomtsev–Petviashvili hierarchy reduction method in conjunction with the Hirota’s bilinear technique. These two-dimensional rogue waves, described by semi-rational type solutions, illustrate the resonant collisions between lumps or line rogue waves and dark solitons. Due to the resonant collisions, the line rogue waves and lumps in these semi-rational solutions become doubly localized in two-dimensional space and in time. Thus, they are called line segment rogue waves or lump-typed rogue waves. These waves arise from the background of dark solitons, then exist in the background of dark solitons for a very short period of time, and finally completely decay back to the background of dark solitons. In particular circumstances which are characterized by special parametric conditions, the dark solitons in the long wave component of the DSI equation can degenerate into the constant background. In this case, the rogue waves appear and disappear in a constant background.

Appendix A

In this Appendix we will derive the semi-rational solutions (12), which comprise two dark solitons and one rational solitary wave. We will construct these solutions by taking a long wavelength limit of the four-soliton solutions. The four-soliton solutions (Satsuma and Ablowitz 1979) of the DSI are given by:

$$\begin{aligned} A=\sqrt{2}\frac{g_{4so}}{f_{4so}},\,Q={\epsilon }-( {2 {\mathrm{log}}} f_{4so})_{xx}, \end{aligned}$$

(70)

where the functions $g_{4so}$ and $f_{4so}$ take form below:

$$\begin{aligned} f_{4so}&=1+\sum \limits _{k=1}^{4}e^{\theta _k}+\sum \limits _{1\le k<s\le 4}^{4}A_{ks}e^{\theta _k+\theta _s} +\sum \limits _{1\le k<l<s\le 4}^{4}A_{kl}A_{ks}A_{ls}e^{\theta _k+\theta _l+\theta _s}\nonumber \\&\quad +\prod \limits _{1\le k<s\le 4}^{4}A_{ks}\,e^{\theta _1+\theta _2+\theta _3+\theta _4},\nonumber \\ g_{4so}&=1+\sum \limits _{k=1}^{4}e^{\theta _k+i\chi _k}+\sum \limits _{1\le k<s\le 4}^{4}A_{ks}e^{\theta _k+\theta _s+i(\chi _k+\chi _s)}\nonumber \\&\quad +\sum \limits _{1\le k<l<s\le 4}^{4}A_{kl}A_{ks}A_{ls}e^{\theta _k+\theta _l+\theta _s+i(\chi _k+\chi _l+\chi _s)}\nonumber \\&\quad +\prod \limits _{1\le k<s\le 4}^{4}A_{ks}\,e^{\theta _1+\theta _2+\theta _3+\theta _4+i(\chi _1+\chi _2+\chi _3+\chi _4)},\nonumber \\ \theta _{j}&=\widehat{p}_jx+\widehat{q}_jy+\widehat{\omega }_j t+\theta _{j,0},\nonumber \\ A_{ls}&=-\frac{2\epsilon \cos (\phi _l-\phi _s)+(\widehat{p}_l-\widehat{p}_s)^2-(\widehat{q}_l-\widehat{q}_s)^2-2\epsilon }{2\epsilon \cos (\phi _l+\phi _s) +(\widehat{p}_l+\widehat{p}_s)^2-(\widehat{q}_l+\widehat{q}_s)^2-2\epsilon }, \nonumber \\ e^{i\chi _j}&=1-\frac{\widehat{p}_j^2-\widehat{q}_j^2}{2\epsilon }-i\frac{(\widehat{p}_j^2-\widehat{q}_j^2)\sqrt{-1+\frac{4\epsilon }{\widehat{p}_j^2-\widehat{q}_j^2}}}{2\epsilon }, \end{aligned}$$

(71)

and the parameters $\widehat{p}_j,\widehat{q}_j$ and ${\widehat{\omega }}_j$ satisfy the dispersion relation

$$\begin{aligned} (\widehat{p}_j^2-\widehat{q}_j^2){\widehat{\omega }}_j^2+(\widehat{p}_j^2-\widehat{q}_j^2-4\epsilon )(\widehat{p}_j^2+\widehat{q}_j^2)^2=0,\; j=1,2,3,4. \end{aligned}$$

(72)

To construct the semi-rational solution given by Eq. (12), we first let $\theta _{s,0}\rightarrow \delta {\widehat{\theta }}_{s,0}+i\pi ,\widehat{p}_s\rightarrow \delta \widehat{p}_s,\widehat{q}_s\rightarrow \delta \widehat{q}_s$ ($s=1,2$) and then take the limit as $\delta \rightarrow 0$. One obtain

$$\begin{aligned} \begin{aligned} e^{\theta _s}&={-1}-\delta {\widehat{\theta }}_s+ \vartheta \left( {\delta ^2}\right) ,\\ e^{\theta _s+i\chi _s}&=-\delta ({\widehat{\theta }}_s+b_s)+ \vartheta \left( {\delta ^2}\right) ,\\ A_{12}&=1+a_{12}\delta ^2+ \vartheta \left( \delta ^4\right) ,A_{sj}=1+a_{sj}\delta + \vartheta ({\delta ^2}),\; j=3, 4, \end{aligned} \end{aligned}$$

(73)

where

$$\begin{aligned} {\widehat{\theta }}_s&=\widehat{p}_sx+\widehat{q}_sy+{2\gamma _s\left( \widehat{p}_s^2+\widehat{q}_s^2\right) \sqrt{\frac{\epsilon }{\widehat{p}_s^2-\widehat{q}_s^2}}}t+\theta _{s,0},\nonumber \\ a_{12}&=\frac{|\widehat{p}_1^2-\widehat{q}_1^2|^2}{2\gamma _1\gamma _2|\widehat{p}_1^2-\widehat{q}_1^2|-2\epsilon \left( |\widehat{p}_1|^2-|\widehat{q}_1|^2\right) },\nonumber \\ a_{sj}&=\frac{\epsilon (\widehat{p}_s^2-\widehat{q}_s^2)\left( \widehat{p}_j^2-\widehat{q}_j^2\right) }{\gamma _s(\widehat{p}_s^2-\widehat{q}_s^2)(\widehat{p}_j^2-\widehat{q}_j^2) \sqrt{\left( -1+\frac{4\epsilon }{\widehat{p}_j^2-\widehat{q}_j^2}\right) \frac{\epsilon }{\widehat{p}_s^2-\widehat{q}_s^2}}-2(\widehat{p}_s\widehat{p}_j-\widehat{q}_s\widehat{q}_j)},\nonumber \\ b_s&=-i\epsilon \gamma _s\left( \widehat{p}_s^2-\widehat{q}_s^2\right) \sqrt{\frac{\epsilon }{\widehat{p}_s^2-\widehat{q}_s^2}}, \gamma _s=\pm 1. \end{aligned}$$

(74)

Then, by implementing the above limit procedure, the functions $f_{4so}$ and $g_{4so}$ in Eq. (71) can be rewritten in the following form:

$$\begin{aligned} \begin{aligned} f&={\widehat{\theta }}_1{\widehat{\theta }}_2+a_{12}+(({\widehat{\theta }}_1+a_{13})({\widehat{\theta }}_2+a_{23})+a_{12})e^{\theta _3}\\&\quad +(({\widehat{\theta }}_1+a_{14})({\widehat{\theta }}_2+a_{24})+a_{12})e^{\theta _4}+\\&\quad a_{34}(({\widehat{\theta }}_1+a_{13}+a_{14})({\widehat{\theta }}_2+a_{23}+a_{24})+a_{12})e^{\theta _3+\theta _4},\\ g&=({\widehat{\theta }}_1+b_1)({\widehat{\theta }}_2+b_2)+a_{12}+(({\widehat{\theta }}_1+b_1+a_{13})({\widehat{\theta }}_2+b_2+a_{23})+a_{12})e^{\theta _3+\chi _3}\\&\qquad +(({\widehat{\theta }}_1+b_1+a_{14})({\widehat{\theta }}_2+b_2+a_{24})+a_{12})e^{\theta _4+i\chi _4}+\\&\quad a_{34}(({\widehat{\theta }}_1+b_1+a_{13}+a_{14})({\widehat{\theta }}_2+b_2+a_{23}+a_{24})+a_{12})e^{\theta _3+\theta _4+i(\chi _3+\chi _4)}.\nonumber \end{aligned}\\ \end{aligned}$$

(75)

Here we have denoted $A_{34}=a_{34}$ to be consistent with the expressions of the function f and g in Eq. (13). To keep the function f real, we take the following parametric constraint:

$$\begin{aligned} \widehat{p}_2=\widehat{p}_1^*,\widehat{q}_2=\widehat{q}_1^*,\theta _{2,0}=\theta _{1,0}^*,\gamma _1\gamma _2=1, \end{aligned}$$

(76)

when $p_{1R}\ne 0$ or $q_{1R}\ne 0$; or the parametric condition

$$\begin{aligned} \widehat{p}_2=\widehat{p}_1,\widehat{q}_2=\widehat{q}_1,\theta _{2,0}=\theta _{1,0}^*,\gamma _1\gamma _2=-1, \end{aligned}$$

(77)

when $\widehat{p}_1,\widehat{q}_1$ are real parameters and $\frac{\epsilon }{\widehat{p}_1^2-\widehat{q}_1^2}<0$. Then we can obtain

$$\begin{aligned} {\widehat{\theta }}_2={\widehat{\theta }}_1^*,b_2=-b_1^*,a_{12}^*=a_{12},a_{23}=a_{13}^{*},a_{24}=a_{14}^{*}. \end{aligned}$$

(78)

For simplicity, we take $\widehat{p}_s,\widehat{q}_s$ as

$$\begin{aligned} \begin{aligned} \widehat{p}_1&=\frac{1}{2}\left( p-\frac{\epsilon }{p}\right) , \widehat{p}_2=\frac{1}{2}\left( p^{*}-\frac{\epsilon }{p^{*}}\right) ,\\ \widehat{q}_1&=\frac{1}{2}\left( p+\frac{\epsilon }{p}\right) , \widehat{q}_2=\frac{1}{2}\left( p^{*}+\frac{\epsilon }{p^{*}}\right) , \\ \widehat{p}_3&=\frac{1}{2}\left( \widetilde{p}_3+\widetilde{p}_3^*+\frac{\epsilon }{\widetilde{p}_3}+\frac{\epsilon }{{\widetilde{p}_3}^*}\right) , \widehat{q}_3=\frac{1}{2}\left( \widetilde{p}_3+\widetilde{p}_3^*-\frac{\epsilon }{\widetilde{p}_3}-\frac{\epsilon }{{\widetilde{p}_3}^*}\right) ,\\ \widehat{p}_4&=\frac{1}{2}\left( \widetilde{p}_4+\widetilde{p}_4^*+\frac{\epsilon }{\widetilde{p}_4}+\frac{\epsilon }{{\widetilde{p}_4}^*}\right) ,\\ \widehat{q}_4&=\frac{1}{2}\left( \widetilde{p}_4+\widetilde{p}_4^*-\frac{\epsilon }{\widetilde{p}_4}-\frac{\epsilon }{{\widetilde{p}_4}^*}\right) , \gamma _1=-1,\gamma _2=1. \end{aligned} \end{aligned}$$

(79)

Then the functions f and g in Eq. (75) become the functions $\widehat{f}$ and $\widehat{g}$ in Eq. (13). This generates the semi-rational solution A and Q expressed by Eq. (12). It is noted that Ablowitz and Satsuma were first to use the long wavelength limit procedure to construct rational N-lump solutions of the DS equations from exponential 2N-soliton solutions (Satsuma and Ablowitz 1979). In this appendix, we mainly use the long wavelength limit procedure to obtain from four dark solitons certain mixed solutions consisting of a rational solitary wave and two dark solitons.