Multicomponent long-wave–short-wave resonance interaction system: Bright solitons, energy-sharing collisions, and resonant solitons

We consider a general multicomponent (2+1)-dimensional long-wave–short-wave resonance interaction (LSRI) system with arbitrary nonlinearity coefficients, which describes the nonlinear resonance interaction of multiple short waves with a long wave in two spatial dimensions. The general multicomponent LSRI system is shown to be integrable by performing the Painlevé analysis. Then we construct the exact bright multisoliton solutions by applying the Hirota's bilinearization method and study the propagation and collision dynamics of bright solitons in detail. Particularly, we investigate the head-on and overtaking collisions of bright solitons and explore two types of energy-sharing collisions as well as standard elastic collision. We have also corroborated the obtained analytical one-soliton solution by direct numerical simulation. Also, we discuss the formation and dynamics of resonant solitons. Interestingly, we demonstrate the formation of resonant solitons admitting breather-like (localized periodic pulse train) structure and also large amplitude localized structures akin to rogue waves coexisting with solitons. For completeness, we have also obtained dark one- and two-soliton solutions and studied their dynamics briefly.

Figure 1

Propagation of bright one-soliton of 2-LSRI system with positive nonlinearity coefficients ($c_{\ell }>0$) in (x-y) plane for $t=1$ (top panels) and in (x-t) plane for $y=1$ (bottom panels). The parametric choice is $k_1=1+i$, ${\omega }_1=-0.5+i$, $c_1=1.5$, $c_2=1$, ${\alpha }_1^{\left(1\right)}=1$, and ${\alpha }_1^{\left(2\right)}=0.5$.

Figure 2

Direct numerical simulation of bright one-soliton propagation in 2-LSRI system in (x-y) plane for $t=1$ (top panels) and in (x-t) plane for $y=1$ (bottom panels).

Figure 3

Type-I energy-sharing collision of two bright solitons in 2-LSRI system for positive nonlinearity coefficients ($c_1,c_2>0$). Head-on collision of solitons in the (x-y) plane at $t=1$ (top panels) and overtaking collision of solitons in the (x-t) plane at $y=1$ (bottom panels). The parameters are $c_1=2$, $c_2=1$, $k_1=1-1.5i$, $k_2=1.5-0.25i$, ${\omega }_1=-1-i$, ${\omega }_2=-2.5-0.5i$, ${\alpha }_1^{\left(1\right)}=2$, ${\alpha }_2^{\left(1\right)}=1.1$, ${\alpha }_1^{\left(2\right)}=0.6$, ${\alpha }_2^{\left(2\right)}=0.5$. In Figs. 3, 4, 5, 6, 7, 8, $s_{jb}$ and $s_{ja}$ refer to the soliton $s_j,j=1,2,$ before and after collision, respectively.

Figure 4

Elastic collision of two bright solitons in 2-LSRI system for positive nonlinearity coefficients.

Figure 5

Type-II energy-sharing collision of two bright solitons of 2-LSRI system for mixed-type nonlinearity coefficients. Soliton collision in the (x-y) plane (top panels) for $t=1$ and in the (x-t) plane (bottom panels) for $y=2$. Here the parameters are $c_1=2$, $c_2=-1$, $k_1=1-1.5i$, $k_2=1.5-0.25i$, ${\omega }_1=-1-i$, ${\omega }_2=-2.5-0.5i$, ${\alpha }_1^{\left(1\right)}=0.6$, ${\alpha }_2^{\left(1\right)}=0.7$, ${\alpha }_1^{\left(2\right)}=0.5$, and ${\alpha }_2^{\left(2\right)}=0.3$.

Figure 6

Resonant solitons (breathers) in the 2-LSRI system for ${\omega }_1={\omega }_2=-2$, ${\alpha }_1^{\left(1\right)}={\alpha }_1^{\left(2\right)}={\alpha }_2^{\left(1\right)}={\alpha }_2^{\left(2\right)}=1$, and $c_1=c_2=2$. Zero-amplitude resonant soliton in the SW components (a, b) and a general breather-like structure in the LW component (c) for $k_1=1-0.5$ and $k_2=2.5-0.5i$, resonant-breather localized in “$x$” for $k_1=1$ and $k_2=2.5$ (d), resonant-breather localized in “$y$” for $k_1=1+1.5i$, $k_2=1-1.5i$ (e), and a localized structure in both $x$ and $y$ coordinate coexisting with solitons for $k_1=1+0.05i$, $k_2=1.25+0.05i$ (f).

Figure 7

Quasiresonant breathers in the 2-LSRI system for ${\omega }_1=-2$, ${\omega }_2=-1.9999999$, ${\alpha }_1^{\left(1\right)}={\alpha }_1^{\left(2\right)}={\alpha }_2^{\left(1\right)}={\alpha }_2^{\left(2\right)}=1$, and $c_1=c_2=2$. QRB for $k_1=1-0.5$ and $k_2=2.5-0.5i$ (a), the QRB localized in “$x$” for $k_1=1$ and $k_2=2.5$ (b), the QRB localized in “$y$” (c), and the zero-amplitude resonant soliton in the SW component (d) for $k_1=1+1.5i$, $k_2=1-1.5i$.

Figure 8

Resonant solitons in the (x-t) plane of the LW component of the 2-LSRI system with oscillating sidebands (a–d) and without oscillations (e, f) at $y=-1$. The parameters are $c_1=c_2=2$ with $k_1=1+0.5i$, $k_2=1.5-2i$, ${\omega }_1={\omega }_2=-2$ (a), $k_1=-1+0.5i$, $k_2=-1.5-2i$, ${\omega }_1={\omega }_2=2$ (b), $k_1=0.75-1.5i$, $k_2=2-1.5i$, ${\omega }_1={\omega }_2=-2$ (e), and $k_1=-0.75-1.5i$, $k_2=-2-1.5i$, ${\omega }_1={\omega }_2=2$ (f). The 2D plots (c) and (d) clearly show the sideband oscillations corresponding to panels (a) and (b), respectively, at $t=12$ and $t=-12$.

Figure 9

Propagation of bright ($\left|L\right|$), dark ($|S^{\left(1\right)}{|}^2$) and gray ($|S^{\left(2\right)}{|}^2$) solitons in the 2-LSRI system in the (x-y) plane for $t=1$ (top panels) and in the (x-t) plane for $y=0.5$ (bottom panels).

Figure 10

Elastic collision of bright-bright, dark-dark, and dark-gray solitons, respectively, in the $L$, $S^{\left(1\right)}$, and $S^{\left(2\right)}$ components of 2-LSRI system. Soliton collisions in the (x-y) plane for $t=1$ and in the (x-t) plane for $y=0.5$ with other parameters as $\delta =1$, $c_1=2$, $c_2=1$, $k_1=3$, $k_2=3,p_1=-2.5$, and $p_2=2.5$, $a_1=1.5$, $a_2=-1$, $b_1=1.5$, $b_2=2$, ${\tau }_1=2$, and ${\tau }_2=1$.