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# Dynamical behaviors and oblique resonant nonlinear waves with dual-power law nonlinearity and conformable temporal evolution

• In this article, the oblique resonant traveling waves and dynamical behaviors of (2+1)-dimensional Nonlinear Schrödinger equation along with dual-power law nonlinearity, and fractal conformable temporal evolution are reported. The considered equation is converted to an ordinary differential equation by taking the traveling variable wave transform and properties of Khalil's conformable derivative into account. The modified Kudryashov method is implemented to divulge the oblique resonant traveling wave of such an equation. It is found that the obliqueness is only affected on width, but not on amplitude and phase patriots of resonant nonlinear propagating wave dynamics. The research outcomes are very helpful for analyzing the obliquely propagating nonlinear resonant wave phenomena and their dynamical behaviors in several nonlinear systems having Madelung fluids and optical bullets.

Mathematics Subject Classification: Primary: 35C07, 35Q60; Secondary: 74J35, 82B23.

 Citation: • • Figure 1.  Resonant kink-shaped structures of $|\Psi|$ for (a) $\lambda = 0.1$ (red) and $\lambda = 0.9$ (orange), with $t = 50$ and $\theta = 30^0$, (b) $t = 1$ (red) and $t = 200$ (orange) with $\lambda = 0.5$ and $\theta = 30^0$, (c) with respect $x$ and $\theta$ keeping $z$-axis constant ($\theta = 30^0$, $\lambda = 0.5$, $t = 10$), and (d) with respect $z$ and $\theta$ keeping x-axis constant ($\theta = 30^0$, $\lambda = 0.5$, $t = 10$). The remaining parameters are selected as $\eta = -1$, $\sigma = 0.5$, $N = 1$, $\delta = 2$, $\rho = -0.5$, $k = 0.5$ and $d = 1$

Figure 2.  Resonant kink-shaped structures of (a) $|\Psi|$ for $\theta = 5^0$ (red) and $\theta = 30^0$ (orange), (b) $|\Psi|$ for $\theta = 55^0$ (orange) and $\theta = 85^0$ (red), (c) real part of $|\Psi|$ with $\theta = 30^0$ and (d) imaginary part of $|\Psi|$ with $\theta = 30^0$. The remaining parameters are selected as $\lambda = 0.5$, $\eta = -1$, $\sigma = 0.5$, $N = 1$, $\delta = 2$, $\rho = -0.5$, $k = 0.5$, $t = 10$ and $d = 1$

Figure 3.  Resonant periodic wave structures of (a) $|\Psi|$ for $\theta = 45^0$, (b) $|\Psi|$ for $\theta = 80^0$ and (c) $|\Psi|$ for $\theta = 45^0$ (red) and $\theta = 80^0$ (orange). The remaining parameters are selected as $\eta = -0.1$, $\sigma = 0.5$, $N = 1.5$, $\delta = 1$, $\rho = 0.09$, $k = 1$, $t = 1$ and $d = 1$

Figure 4.  The phase portraits and its vector fields of nonlinear dynamical system as mentioned in Eq. (12) by assuming different values of $\rho = 0.09$ (Figs. 4(a) and (b)), $\rho = 0.01$ (Figs. 4(c) and (d)) and $\rho = -0.09$ (Figs. 4(e) and (f)) with $N = 0.5$, $\omega = 2$, $\eta = 0.1$, $k = 0.4$, $\delta = 1$ and $\sigma = 0.5$

Figure 5.  The (a) phase portrait and its (b) vector fields of the nonlinear dynamical system as mentioned in Eq.(12) by assuming for the values of parameters $N = 0.5$, $\omega = -2$, $\eta = 0.1$, $k = 0.4$, $\delta = 1$, $\sigma = 0.5$, and $\rho = 0.09$

Figure 6.  The (a) phase portrait and its (b) vector fields of the nonlinear dynamical system as mentioned in Eq.(12) by assuming for the values of parameters $N = 1.5$, $\omega = 1$, $\eta = 0.1$, $k = 0.4$, $\delta = 1$, $\sigma = 0.5$, and $\rho = 0.09$

Figure 7.  The (a) phase portrait and its (b) vector fields of the nonlinear dynamical system as mentioned in Eq.(12) by assuming for the values of parameters $N = 3$, $\omega = 1$, $\eta = 0.1$, $k = 0.4$, $\delta = 1$, $\sigma = 0.5$, and $\rho = 0.09$

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