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This paper focuses on various properties of the non-integrable discrete focusing Hirota (Hirota +) equation, encompassing spatial structure, discrete solitons, and linear stability analysis. Through a planar nonlinear discrete dynamical map method, we construct the spatially periodic solutions of the non-integrable discrete stationary Hirota + equation under special conditions. From the area-preserving property of the two-dimensional map, the types of fixed points are classified based on the defined residue, which depends on the feature of the linearized map. It is emphasized that there is a great difference between the periodic solution of the non-integrable discrete focusing Hirota equation and that of the non-integrable discrete defocusing Hirota equation. We numerically analyze the influence of the distinct parameters and the initial points on general orbits of the map. In addition, the comparison between spatial properties of the non-integrable discrete focusing Hirota equation and that of the non-integrable discrete focusing nonlinear Schrödinger (NLS +) equation suggests that the former has more plentiful properties. It is worth mentioning that the more general period-2 solutions of the non-integrable discrete stationary Hirota + equation and the period-3 solutions of the non-integrable discrete stationary NLS + equation in specific situations are obtained for the first time. On the other hand, we also explore the stationary solitons and traveling wave solutions of the non-integrable discrete Hirota + equation using the discrete Fourier transformation and the Neumann iteration scheme. The effects of the parameters and the initial values on the shapes of the solitons are numerically investigated. It is revealed that the traveling solitons depend sensitively on both the parameters and the initial values. Finally, we elaborate the linear stability of the stationary solitary waves under small perturbation. Meanwhile, the corresponding results of the non-integrable discrete NLS + equation are compared numerically. • The spatially periodic solutions for the non-integrable discrete stationary Hirota + equation under the particular case are studied based on the defined residue, which differs from that for defocusing case. It is revealed that there is a great difference between the periodic solution of the non-integrable discrete Hirota + equation and that of the defocusing equation. • The more general period-2 solutions of the discrete stationary Hirota + equation are attained, which has not yet considered for the non-integrable discrete NLS + equation. The influence of the distinct parameters and the initial values on the general orbits of the map is analyzed numerically. • A special period-3 solution of the non-integrable discrete stationary NLS + equation is obtained first. • The traveling solitons and stationary solitons are investigated with the help of discrete Fourier transformation and the Neumann iteration scheme. It is shown that the shapes and peaks of discrete solitons depend sensitively on the parameters and the initial values. • The comparison between the spatial properties and the linear stability of the stationary solitons of the non-integrable discrete Hirota + equation and those of the non-integrable discrete NLS + equation shows the former equation has more plentiful properties. [ABSTRACT FROM AUTHOR]

Families of symmetric embedded solitary waves of a perturbed Fifth-order Korteweg–de Vries (FKdV) system were treated in Choudhury et al. (2022) using perturbative and reversible systems techniques. Here, the stability of those solutions, which was not considered in the earlier paper, is detailed. In addition, the results of Choudhury et al. (2022) are extended to the case of asymmetric solitary waves, as well as their stability. Finally, other novel multi-humped regular solitary waves of this system are derived using convergent infinite series solutions for the homoclinic orbits of the FKdV-traveling wave equation. • Embedded solitary waves of a fifth order Korteweg–de Vries equation (FKdV): existence and stability. • Asymmetric solitary waves of the FKdV: existence and stability. • Novel multi-humped regular solitary waves of the FKdV using infinite series. [ABSTRACT FROM AUTHOR]

In this paper, we develop a Riemann–Hilbert (RH) approach to the inverse scattering transform for the coupled modified complex short pulse (cmcSP) equation, which appeared as a reduction of the four-component system of short pulse type introduced by Popowicz in 2017. This approach allows us to present the general soliton solutions of the cmcSP equation in parametric form. Compared with the early results for the scalar modified complex short pulse equation, the cmcSP equation possesses richer soliton solutions, which include smooth solitons, cuspons, breathers and their various interactions. • A coupled modified complex short pulse (cmcSP) equation, whose spectral problem is of 4x4 matrix type, is considered. • A Riemann–Hilbert approach to the inverse scattering transform is developed for the cmcSP equation. • The general soliton solution formula of the cmcSP equation is obtained. • Rich solutions, such as smooth solitons, cuspons, breathers and their interactions, are presented for the cmsSP equation. [ABSTRACT FROM AUTHOR]

In this paper, the two-dimensional Leznov lattice equation is studied by Hirota's bilinear method and KP-Toda hierarchy reduction approach. We present solutions of the lattice equation in the ratio of τ functions given by the N × N Gram-type determinants. Solitons, breathers and rational solutions on constant and periodic backgrounds are derived. In particular, we find three types of breather solutions that are periodic in x -direction, y -direction and a general oblique line, respectively. Algebraic solitons are obtained by introducing two differential operators applied to the soliton solutions. We investigate the asymptotic behavior and interactions of algebraic soliton. The interaction of two algebraic solitons are analyzed and proved to be completely elastic. We demonstrate that the asymptotic algebraic solitons in the straight lines well approach the exact ones. Multiple algebraic solitons on the constant and periodic backgrounds are derived with higher-order determinant size. We introduce Schur polynomials to generate explicit expression for fundamental and higher-order lump solutions. Dynamics of lump solutions are analyzed with plots. • Gram-type determinant solutions for the 2D Leznov lattice equation are presented. • Solitons, breathers and rational solutions on constant and periodic backgrounds are analyzed respectively. • Algebraic solitons are constructed and asymptotic behaviors and interactions are studied. • Schur polynomials are introduced to express fundamental and higher-order lump solutions. [ABSTRACT FROM AUTHOR]

The Darboux transformation (DT) for the coupled complex short pulse (CCSP) equation is constructed through the loop group method. The DT is then utilized to construct various exact solutions including bright-soliton, dark-soliton, breather and rogue wave solutions to the CCSP equation. In case of vanishing boundary condition (VBC), we perform the inverse scattering analysis. Breather and rogue wave solutions are constructed under non-vanishing boundary condition (NVBC). Moreover, we conduct a modulational instability (MI) analysis based on the method of squared eigenfunctions, whose result confirms the condition for the existence of rogue wave solution. • Darboux transformation for the coupled complex short pulse equation is derived. • The inverse scattering analysis for the vanishing background is given. • The modulational instability analysis for the plane wave is obtained. • The multi-hump solitons, high order solitons and rogue waves are constructed. [ABSTRACT FROM AUTHOR]

This paper aims to constructing the Darboux transformation and soliton solutions of the focusing and defocusing spatial discrete coupled complex short pulse (sd-CCSP) equation. With the Darboux transformation, we derived three types of soliton solutions for the focusing sd-CCSP equation, which include smooth-, cuspon- and loop-type. These solutions converge to exact solutions for the focusing CCSP equation in the continuum limit. The interaction of two solitons and their asymptotic behavior are analyzed. We also obtain three types of soliton solutions for the defocusing sd-CCSP equation, as well as the rational solutions for the focusing sd-CCSP equation. We find that there exist new properties of soliton solutions for the sd-CCSP equation which do not exist in the sd-CSP equation, and there is a big difference between sd-CCSP equation and spatial discrete coupled nonlinear Schrödinger (sd-CNLS) equation. • We construct the Darboux transformation for the spatial discrete coupled complex short pulse (sd-CCSP) equation. • We derive soliton, rational and rational-soliton solutions for the sd-CCSP equation. • The interaction of two solitons and their asymptotic behavior for the sd-CCSP equation are analyzed. [ABSTRACT FROM AUTHOR]

In this work, we derive a family of high order rational solitons of the 3-wave resonant interaction (3WRI) equation by utilizing generalized Darboux-dressing transformation. We observe that breathers in the three components display self-similar behaviors in the process of propagation. For the first order rational soliton, it is interesting that the dark soliton component bifurcates so as to generate a peak whose amplitude is three times higher than the surrounding background itself with an infinite life time. It would provide us a way to generate 'eternal' large amplitude wave in weakly nonlinear dispersive media. Furthermore, the interactions of two rational solitons are studied. • High-order rational solitons of 3WRI are derived by using DT. • Dark soliton bifurcates a peak whose amplitude is three times. • The technique provides a way to generate 'eternal' large amplitude wave. • The interactions of two rational solitons are studied. [ABSTRACT FROM AUTHOR]

In this paper, we derive the determinant representation of higher-order algebraic soliton solutions of the Gerdjikov–Ivanov equation by using the Darboux transformation and some limit technique. Based on the asymptotic balance between different algebraic terms, we obtain the asymptotic expressions of algebraic soliton solutions with the order 2 ≤ N ≤ 4. It turns out that all the asymptotic solitons have the same amplitudes, most of them are located in the parabolic curves and thus have the varying velocities with the rate O ( | t | − 1 2 ) (except that one pair of asymptotic solitons are located in the straight lines for the odd-order cases), and they exhibit the elastic interactions of the attractive type. In addition, we reveal that the transient rogue waves are generated in the soliton-interaction region and the peak value is exactly N times the amplitude of individual soliton. • The determinant representation of higher-order algebraic soliton solutions is obtained for the Gerdjikov–Ivanov equation. • The asymptotic analysis method is developed on the balance between different algebraic terms up to the subdominant level. • The asymptotic solitons exhibit the attractive interactions and most of them are located in the parabolic curves. • The transient rogue waves can be generated in the soliton-interaction region. [ABSTRACT FROM AUTHOR]

We consider the half-wave maps (HWM) equation which is a continuum limit of the classical version of the Haldane–Shastry spin chain. In particular, we explore a many-body dynamical system arising from the HWM equation under the pole ansatz. The system is shown to be completely integrable by demonstrating that it exhibits a Lax pair and relevant conservation lows. Subsequently, the analytical multisoliton solutions of the HWM equation are constructed by means of the pole expansion method. The properties of the one- and two-soliton solutions are then investigated in detail as well as their pole dynamics. Last, an asymptotic analysis of the N -soliton solution reveals that no phase shifts appear after the collision of solitons. This intriguing feature is worth noting since it is the first example observed in the head-on collision of rational solitons. A number of problems remain open for the HWM equation, some of which are discussed in concluding remarks. • A Lax pair for a many-body dynamical system associated with the half-wave maps equation. • Integrability of the many-body dynamical system. • The N -soliton solution of the half-wave maps equation. [ABSTRACT FROM AUTHOR]

In this paper, the dimensionless n -component Hirota (alias the n -Hirota) equation is investigated, which describes the wave propagations of n ultrashort optical fields in a fiber. Starting from the modified Darboux transform and its Lax pair with initial non-zero plane-wave conditions, we find the novel multi-parametric families of rational vector rogue wave (RW) solutions for the n -Hirota equation. Furthermore, some weak and strong interactions of rational vector RWs are exhibited for the n -Hirota equation with n = 2 , 3 , 4 , 5 , 6 in detail. In particular, we also deduce the rational vector W -shaped dark and bright solitons of the n -component complex mKdV equation, whose representative wave structures are illustrated for n = 2 , 3 , 4 , 5. Finally, the effect of a small non-integrable deformation of the 3-Hirota equation is explored numerically on the excitation of vector RWs in terms of the Fourier spectral method. These obtained rational vector RW and W-shaped soliton solutions will be useful to further explore the related nonlinear wave phenomena in the sense of multi-component physical systems with non-zero backgrounds. • The novel rational vector RWs are found for the n-Hirota equation. • Some weak and strong interactions of rational vector RWs are exhibited. • Rational vector W -shaped solitons are obtained for the n-cmKdV equation. • Effect of a small deformation of 3-Hirota system is explored on the RW excitations. [ABSTRACT FROM AUTHOR]

In this paper, we mainly analyze the long-time asymptotics of high-order soliton for the Hirota equation. Two different Riemann–Hilbert representations of Darboux matrices with high-order soliton are given to establish the relationships between inverse scattering method and Darboux transformation. The asymptotic analysis with single spectral parameter is derived through the formulas of determinant directly. Furthermore, the long-time asymptotics with k spectral parameters is given by combining the iterated Darboux matrices and the result of high-order soliton with single spectral parameter, which discloses the structure of high-order soliton clearly and is possible to be utilized in the optic experiments. • Two kinds of Riemann–Hilbert problem for high-order solitons are derived. • The long time asymptotics of high order soliton for Hirota equation is provided. • The approach for asymptotic analysis of general high-order solitons is presented. [ABSTRACT FROM AUTHOR]

We present a detailed numerical study of solutions to the Zakharov–Kuznetsov equation in three spatial dimensions. The equation is a three-dimensional generalization of the Korteweg–de Vries equation, though, not completely integrable. This equation is L 2 -subcritical, and thus, solutions exist globally, for example, in the H 1 energy space. We first study stability of solitons with various perturbations in sizes and symmetry, and show asymptotic stability and formation of radiation, confirming the asymptotic stability result in Farah et al. (0000) for a larger class of initial data. We then investigate the solution behavior for different localizations and rates of decay including exponential and algebraic decays, and give positive confirmation toward the soliton resolution conjecture in this equation. Finally, we investigate soliton interactions in various settings and show that there are both a quasi-elastic interaction and a strong interaction when two solitons merge into one, in all cases always emitting radiation in the conic-type region of the negative x -direction. • 3D numerical simulations for the quadratic Zakharov–Kuznetsov equation. • Soliton interaction. • Soliton resolution. [ABSTRACT FROM AUTHOR]