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In this paper, we investigate rational solitons and rogue wave (RW) solutions of the nonlocal Lakshmanan-Porsezian-Daniel (LPD) equation by the generalized Darboux transformation method. The exact rational solutions are presented in terms of determinants whose entries are given by the initial eigenfunction and continuous-wave solution. Their dynamics investigated reveals the interesting patterns such as rational solitons, singular and nonsingular RWs depending on the selection of the free parameters. The compression effects of the RWs with respect to the strength of the higher-order nonlinear effects parameter γ are discussed and graphically illustrated. • Construction the rational soliton and higher-order RW solutions for the nonlocal LPD equation by the generalized DT. • Dynamic behavior analysis for the singular and nonsingular rational solutions. • The growing compressed effects of the nonsingular RW solutions are analyzed through numerical plots. [ABSTRACT FROM AUTHOR]

This study presents recent advancements in computational techniques for discovering new and precise solitary wave solutions in the (1 + 1)-dimensional Mikhailov-Novikov-Wang (MNW) integrable equation. Solitary wave solutions are crucial in understanding complex nonlinear phenomena and have practical applications in fields such as fluid dynamics, nonlinear optics, and plasma physics. The proposed computational techniques utilize powerful numerical methods, including the extended simplest equation (ESE) method, modified Riccati expansion MRE method, and He's variational iteration (HVI) method. These techniques provide robust and accurate ways to compute solitary wave solutions, taking advantage of the integrability properties of the MNW equation. Compared to traditional computational methods, these approaches offer significant improvements in efficiency, accuracy, and the ability to explore a broader range of solitary wave solutions. The study demonstrates the effectiveness of these techniques through extensive numerical simulations and comparisons with other computational approaches. The results highlight the superior performance of the proposed methods in identifying new solitary wave solutions, accurately characterizing their properties, and providing deeper insights into the behavior of the model. These computational techniques contribute to advancing our understanding of the underlying dynamics of the MNW equation and have potential applications across various scientific disciplines. • The unified solver method is used to solve the (1+1)-dimensional Mikhailov-Novikov-Wang Integrable Equation. • Innovative rational, trigonometry and hyperbolic stochastic solutions are given for the model problem. • The dynamical features of obtained results are demonstrated through exciting plots. [ABSTRACT FROM AUTHOR]

In this article the question of the integrability of the so-called A ± equation is considered. Two powerful approaches, the Painlevé property and Hirota three-soliton test, are used as the integrability predictors. The A ± equation is a new soliton equation with the MKdV-type soliton superposition formulae, but with richer nonlinear dynamics and richer soliton families. Also, explicit expressions for the multiple bell-shape and breather solitons are derived based on the Hirota substitution involved. • The new A ± soliton equation is investigating for the integrability. • It is shown that the equation possesses the Painlevé property. • The Hirota three-soliton test is satisfied. • The expressions for the multiple bell-shape solitons are obtained. • The real value, nonsingular breather solutions are found. [ABSTRACT FROM AUTHOR]

Based on the bifurcation theory, a new (3+1)-dimensional Kadomtsev-Petviashvili (KP) equation is studied. The phase portraits of this equation are drawn, and some exact solutions, including solitons and periodic wave solutions, are derived. The (G ′ / G) -expansion method is applied to obtain many exact solutions, such as bright soliton, singular wave, and periodic wave solutions. Moreover, the interesting physical structures of these obtained solutions are described. • A new (3+1)-dimensional Kadomtsev-Petviashvili equation is investigated. • The phase portraits of this equation by bifurcation theory are drawn. • The model is solved using the (G'/G)-expansion method. • Some soliton, singular and periodic wave solutions are derived. [ABSTRACT FROM AUTHOR]

In this paper, based on the nonlinear fractional equations proposed by Ablowitz, Been, and Carr in the sense of Riesz fractional derivative, we explore the fractional coupled Hirota equation and give its explicit form. Unlike the previous nonlinear fractional equations, this type of nonlinear fractional equation is integrable. Therefore, we obtain the fractional n -soliton solutions of the fractional coupled Hirota equation by inverse scattering transformation in the reflectionless case. In particular, we analyze the one- and two-soliton solutions and prove that the fractional two-soliton can also be regarded as a linear superposition of two fractional single solitons as | t | → ∞. Moreover, we obtain the nondegenerate fractional soliton solutions and give a simple analysis for them. • Extend the fractional integrable system to the coupled system for the first time. • Give the explicit form of the fractional coupled Hirota equation. • Construct the fractional n-soliton solutions. • Obtain the nondegenerate fractional soliton solutions. [ABSTRACT FROM AUTHOR]

• A revised Riemann-Hilbert problem and a revised inverse problem for Kundu equation are provided. • The integral factor involved in solutions is removed. • The explicit formula of N -th order soliton is displayed. • The explicit soliton and positon solutions are provided. We apply Riemann-Hilbert problem (RHP) to Kundu equation with zero boundary condition. By solving the revised RHP related to N pairs of simple pole points, a pair of differential equations are solved from the asymptotic behaviors, and the explicit formula of N -th order soliton is obtained. Unlike the known representation of N -th order soliton in many literatures, our formula does not include unsatisfactory integral due to using revised RHP. Based on this new formula, the explicit first order soliton is obtained directly, and the explicit second order bound-state soliton (BSS) is also obtained by applying limit technique. Additionally, higher-order soliton and bound-state soliton (BSS) solutions are discussed, and the approximate trajectories of second order BSS solution are studied. [ABSTRACT FROM AUTHOR]

The main result of this paper is to reveal the linear structure for the Sharma-Tasso-Olver-Burgers (STOB) equation by the Cole-Hopf transformation. Then a general form of the multi-wave solution is given by the superposition principle and the method of separation of variables, and all the existing exact solutions to this equation are contained in this general form. Especially, the kink-kink and dark-bright soliton molecules are derived, and the fission and fusion phenomena among the kink-kink soliton molecule are then revealed. Moreover, the trajectory to each of the soliton molecule is given, and the corresponding global phase portraits are also shown to establish the existences of soliton molecules via the bifurcation method. • The linear structure of Sharma-Tasso-Olver-Burgers equation is presented. • A general form of the multi-wave solution is given. • Soliton molecules in traveling wave form are given. [ABSTRACT FROM AUTHOR]

Advanced mathematics has analyzed complex systems, a key area of research in nonlinear sciences, including fluid mechanics, the theory of solitons, hydrodynamics, optical fibers, and chaos theory. Nonlocal integrable mKdV equations could help understand dispersive waves in nonlinear and complex media. In this manuscript, we derive bright one and two soliton solutions for a nonlocal nonlinear integrable KdV equation using an improved Hirota bilinear method (HBM). We use MATLAB to visualize the obtained results in 3D space by choosing the appropriate parameter values. It is demonstrated that these solutions have novel characteristics that differ from those of the mKdV equation. • The novel nonlocal nonlinear integrable KdV equation is considered. • The bright one and two soliton solutions for the proposed problem are presented using an improved Hirota bilinear method. • The obtained results are given in 3D figures by choosing the appropriate parameter values. [ABSTRACT FROM AUTHOR]

We study the integrable three-level coupled Maxwell–Bloch equations with the mixed focusing-defocusing case. The n -fold Darboux transformation is constructed on basis of a linear 3 × 3 matrix eigenvalue problem. As an application, the n -dark-dark soliton solution in a simple determinant form is presented and the asymptotic behavior of the n -dark-dark soliton solution is rigorously given. In particular, under the resonant mechanism, the unusual dark-dark soliton molecules which represent the soliton bound states are analytically demonstrated. The double- and triple-dark-dark soliton molecules that are composed of two and three solitons propagating with identical velocities are graphically shown, respectively. The elastic collisions between a double- or triple-dark-dark soliton molecule and a usual dark-dark soliton, and the elastic collision of two double-dark-dark soliton molecules are discussed via the standard asymptotic analysis method. • The n-dark-dark soliton solution is derived. • The unusual dark-dark soliton molecules are analytically demonstrated. • The elastic collisions are discussed via the asymptotic analysis method. [ABSTRACT FROM AUTHOR]

• All possible integrable shifted nonlocal NLS and MKdV equations are given. • One- and two-soliton solutions of these equations are obtained by Hirota method. • One-soliton solutions of shifted and standard nonlocal equations are compared. • The relations between solutions of shifted and standard nonlocal equations are discussed. We find one- and two-soliton solutions of shifted nonlocal NLS and MKdV equations. We discuss the singular structures of these soliton solutions and present some of the graphs of them. [ABSTRACT FROM AUTHOR]

• Dual-wave dynamics of resonant NLS equation. • Propagation of two different wave modes flow. • The hydrodynamic mathematical methods. In this work, a new dual-mode nonlinear Schrödinger's equation with Bohm potential is studied with three different forms of nonlinearities. This model has many applications in nonlinear physics and fiber optics. The new model introduces three different physical parameters such as nonlinearity, phase velocity and dispersive factor. This model interprets the simultaneous propagation of two-mode waves instead of a single wave. The singular dual-mode wave solutions are retrieved by versatile exp(- ϕ)-expansion method. The necessary conditions on dissipative nonlinearity parameters that guarantee the existence of soliton solutions are determined. The impact of phase velocity on propagation dynamics of these dual waves is comprehensively studied with the aid of 3D graphs. [ABSTRACT FROM AUTHOR]