Generalized perturbation $(n, M)$-fold Darboux transformations and multi-rogue-wave structures for the modified self-steepening nonlinear Schrödinger equation

In this paper, a simple and constructive method is presented to find the generalized perturbation $(n,M)$-fold Darboux transformations (DTs) of the modified nonlinear Schrödinger (MNLS) equation in terms of fractional forms of determinants. In particular, we apply the generalized perturbation $(1,N-1)$-fold DTs to find its explicit multi-rogue-wave solutions. The wave structures of these rogue-wave solutions of the MNLS equation are discussed in detail for different parameters, which display abundant interesting wave structures, including the triangle and pentagon, etc., and may be useful to study the physical mechanism of multirogue waves in optics. The dynamical behaviors of these multi-rogue-wave solutions are illustrated using numerical simulations. The same Darboux matrix can also be used to investigate the Gerjikov-Ivanov equation such that its multi-rogue-wave solutions and their wave structures are also found. The method can also be extended to find multi-rogue-wave solutions of other nonlinear integrable equations.

Figure 1

The first-order rogue-wave solution ${\tilde{q}}_2(x,t)$ given by Eqs. (42) and (43) with $a=-1,c=1$. [(a) and (c)] $\rho =0$; [(b) and (d)] $\rho =2$.

Figure 2

The second-order rogue-wave solution given by Eqs. (48) and (52) with $a=-1,c=1,b_1=d_1=0$. [(a) and (c)] $\rho =0$; [(b) and (d)] $\rho =2$.

Figure 3

The second-order rogue-wave solution ${\tilde{q}}_3(x,t)$ given by Eq. (44) with $a=-1,c=1,\rho =2$. [(a) and (c)] $b_1=-{10}^3,d_1=0$; [(b) and (d)] $b_1=0,d_1={10}^3$.

Figure 4

The third-order rogue-wave solution ${\tilde{q}}_4(x,t)$ given by Eq. (55) with $a=-1,c=1$ and $b_1=b_2=d_1=d_2=0$. [(a) and (c)] $\rho =0$; [(b) and (d)] $\rho =2$.

Figure 5

The third-order rogue-wave solution ${\tilde{q}}_4(x,t)$ given by Eq. (55) with $a=-1,c=1,\rho =2$. [(a) and (c)] $b_1={10}^3,d_1=b_2=d_2=0$; [(b) and (d)] $b_1=-{10}^3,d_1=b_2=d_2=0$.

Figure 6

The third-order rogue-wave solution ${\tilde{q}}_4(x,t)$ given by Eq. (55) with $a=-1,c=1,\rho =2$. [(a) and (c)] $b_2={10}^4,d_1=b_1=d_2=0$; [(b) and (d)] $b_2=-{10}^4,d_1=b_1=d_2=0$.

Figure 7

The third-order rogue-wave solution ${\tilde{q}}_4(x,t)$ given by Eqs. (55) and (43) with $a=-1,c=1,\rho =2$. [(a) and (c)] $b_1=100,b_2=3000,d_1=d_2=0$; [(b) and (e)] $b_1=100,d_1=0,b_2=10,d_2={10}^4$; [(c) and (f)] $b_1=d_2=0,d_1=100,b_2={10}^4$.

Figure 8

The first-order rogue-wave solution ${\tilde{q}}_{21}(x,t)$ given by Eq. (42) with $a=-1,c=1,\rho =0$. (a) Exact solution, (b) time evolution using exact solution (42) as the initial condition, and (c) time evolution using exact solution (42) perturbated by a $2\%$ noise as the initial condition.

Figure 9

The first-order rogue-wave solution ${\tilde{q}}_{22}(x,t)$ given by Eq. (43) with $a=-1,c=1,\rho =2$. (a) Exact solution, (b) time evolution of the wave using exact solution (43) as the initial condition, and (c) time evolution of the wave using exact solution (43) perturbated by a $1\%$ noise as the initial condition.

Figure 10

The second-order interactive rogue-wave solution ${\tilde{q}}_{31}(x,t)$ given by Eq. (48) with $a=-1,c=1,\rho =0,b_1=d_1=0$. (a) Exact solution, (b) time evolution of the wave using exact solution (48) as the initial condition, and (c) time evolution of the wave using exact solution (48) perturbated by a $2\%$ noise as the initial condition.

Figure 11

The second-order interactive rogue-wave solution ${\tilde{q}}_{32}(x,t)$ given by Eq. (52) with $a=-1,c=1,\rho =2,b_1=d_1=0$. (a) Exact solution, (b) time evolution of the wave using exact solution (52) as the initial condition, and (c) time evolution of the wave using exact solution (52) perturbated by a $1\%$ noise as the initial condition.

Figure 12

The second-order separatable rogue-wave solution ${\tilde{q}}_3(x,t)$ given by Eq. (44) with $a=-1,c=1,\rho =2,b_1=0,d_1={10}^3$. (a) Exact solution, (b) time evolution of the wave using exact solution (44) as the initial condition, and (c) time evolution of the wave using exact solution (44) perturbated by a $0.5\%$ noise as the initial condition.

Figure 13

The third-order interactive rogue-wave solution ${\tilde{q}}_4(x,t)$ given by Eq. (55) with $a=-1,c=1,\rho =2,b_1=b_2=d_1=d_2=0$. (a) Exact solution, (b) time evolution of the wave using exact solution (55) as the initial condition, and (c) time evolution of the wave using exact solution (55) perturbated by a $1.5\%$ noise as the initial condition.

Figure 14

The third-order separatable rogue-wave solution ${\tilde{q}}_4(x,t)$ given by Eq. (55) with $a=-1,c=1,\rho =2,b_1={10}^3,b_2=d_1=d_2=0$. (a) Exact solution, (b) time evolution of the wave using exact solution (55) as the initial condition, (c) time evolution of the wave using exact solution (55) perturbated by a $0.1\%$ noise as the initial condition.

Figure 15

The third-order separatable rogue-wave solution ${\tilde{q}}_4(x,t)$ given by Eq. (55) with $a=-1,c=1,\rho =2,b_1={10}^4,b_2=d_1=d_2=0$. (a) Exact solution, (b) time evolution of the wave using exact solution (55) as the initial condition, (c) time evolution of the wave using exact solution (55) perturbated by a $0.1\%$ noise as the initial condition.

Figure 16

[(a) and (b)] The first-order rogue-wave solution ${\tilde{q}}_2(x,t)$ given by Eq. (68) with $a=0,c=2$.

Figure 17

The second-order rogue-wave solution ${\tilde{q}}_3(x,t)$ with $a=0,c=2$. (a), (c): $b_1=d_1=0$; (b), (d): $b_1=100,d_1=0$.

Figure 18

The third-order rogue-wave solution ${\tilde{q}}_4(x,t)$ with $a=0,c=2$. [(a) and (d)] $b_1=b_2=d_1=d_2=0$; [(b) and (e)] $b_1=1000,d_1=b_2=d_2=0$; [(c) and (f)] $b_2=1000,b_1=d_1=d_2=0$.