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In this work, we study the Chafee-Infante model with conformable fractional derivative. This model describes the energy balance between equator and pole of solar system, which transmit energy via heat diffusion. To explore the multi soliton solutions and their interaction, we implemented the new modified simple equation (NMSE) scheme. Under some conditions, the obtained solutions are trigonometric, hyperbolic, exponential and their combine form. Only the proposed technique can be provided the solution in terms of trigonometric and hyperbolic form together directly. The periodic, solitary wave and novel interaction of such solitary and sinusoidal solutions has also been established and discussed analytically. For the special values of the existing free parameter, some novel waveforms are existed for the proposed model including, periodic solution, double periodic wave solution, multi-kink solution. The behavior of the obtained solutions is presented in 3-D plot, density plot and counter plot with the help of computational software Maple 18.
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This paper investigates the soliton solutions and dynamical analysis of (2+1)-dimensional Heisenberg ferro-magnetic spin chains model with beta fractional derivative, which is transformed into the ordinary differential equation. By using the second-order complete discriminant system, the soliton solutions are presented. By utilizing the theory of planar dynamical system, the phase portraits of the dynamical system and its disturbance system are drawn. Moreover, three-dimensional, two-dimensional, and contour plots of soliton solutions for (2+1)-dimensional Heisenberg ferro-magnetic spin chains model with beta fractional derivative have also been plotted.
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In this article, we study the soliton solutions with a time-dependent variable coefficient to the Kolmogorov-Petrovsky-Piskunov (KPP) model. At first, this model was used as the genetics model for the spread of an advantageous gene through a population, but it has also been used as a number of physics, biological, and chemical models. The enhanced modified simple equation technique applies to get the time-dependent variable coefficient soliton solutions from the KPP model. The obtained solutions provide diverse, exact solutions for the different functions of the time-dependent variable coefficient. For the special value of the constants, we get the kink, anti-kink shape, the interaction of kink, anti-kink, and singularities, the interaction of instanton and kink shape, instanton shape, kink, and bell interaction, anti-kink and bell interaction, kink and singular solitons, anti-kink and singular solitons, the interaction of kink and singular, and the interaction of anti-kink and singular solutions to diverse nature wave functions as time-dependent variable coefficients. The presented phenomena are clarified in three-dimension, contour, and two-dimension plots. The obtained wave patterns are powerfully exaggerated by the variable coefficient wave transformation and connected variable parameters. The effect of second-order and third-order nonlinear dispersive coefficients is also explored in 2D plots.
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This paper mainly concentrates on obtaining solutions and other exact traveling wave solutions using the generalized G-expansion method. Some new exact solutions of the coupled nonlinear Schrödinger system using the mentioned method are extracted. This method is based on the general properties of the nonlinear model of expansion method with the support of the complete discrimination system for polynomial method and computer algebraic system (AS) such as Maple or Mathematica. The nonparaxial solitons with the propagation of ultra-broad nonparaxial pulses in a birefringent optical waveguide is studied. To attain this, an illustrative case of the coupled nonlinear Helmholtz (CNLH) system is given to illustrate the possibility and unwavering quality of the strategy utilized in this research. These solutions can be significant in the use of understanding the behavior of wave guides when studying Kerr medium, optical computing and optical beams in Kerr like nonlinear media. Physical meanings of solutions are simulated by various Figures in 2D and 3D along with density graphs. The constraint conditions of the existence of solutions are also reported in detail. Finally, the modulation instability analysis of the CNLH equation is presented in detail.
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This paper focuses on the mathematical study of the existence of solitary gravity waves (solitons) and their characteristics (amplitude, velocity, [Formula: see text]) generated by a piston wave maker lying upstream of a horizontal channel. The mathematical model requires both incompressibility condition, irrotational flow of no viscous fluid and Lagrange coordinates. By using both the inverse scattering method and a given initial potential [Formula: see text] we can transform the KdV equation into Sturm-Liouville spectral problem. The latter problem amounts to find negative discrete eigenvalues [Formula: see text] and associated eigenfunctions [Formula: see text], where each calculated eigenvalue [Formula: see text] gives a soliton and the profile of the free surface. For solving this problem, we can use the Runge-Kutta method. For illustration, two examples of the wave maker movement are proposed. The numerical simulations show that the perturbation of wave maker with hyperbolic tangent displacement under physical conditions affect the number of solitons emitted.