Semi-rational Solutions Research Articles

Novel (3 + 1)-Dimensional Variable-Coefficients Boussinesq-Type Equation: Exploring Integrability, Wronskian, and Grammian Solutions

Abstract In this paper, we introduce a novel (3+1)-dimensional variable-coefficients Boussinesq-type equation. We
analyze its integrability using the Painlev´e test and the N -soliton solution, demonstrating that both tests yield identical
conditions. Using the Hirota bilinear form of the equation, we derive Wronskian and Grammian determinant solutions
utilizing Pl¨ucker relations and the Jacobi identity for determinants. In particular, we use elementary transformation
and long wave limit to get the determinant expression of mth-order lump solutions from the 2mth-order Wronskian
determinant solutions. Furthermore, we reveal a variety of novel semi-rational solutions using the Hirota method and
Grammian determinant techniques.

Breather–breather interactions, rational and semi-rational solutions of an integrable spin system in optical fibers

This paper investigates an integrable spin system (or Heisenberg ferromagnet-type equation or Nurshuak-IIA equation) which is an integrable generalization of the Heisenberg ferromagnet equation that raises from magneto-optical devices. The N-fold Darboux transformation of the Nurshuak-IIA equation will be constructed. With the selection of several parameters, a variety of the solutions will be produced, including breather–breather interactions, higher-order rogue waves and semi-rational solutions.

Dynamics of rational and semi-rational solutions of the general N-component nonlinear Schrödinger equations

We investigate an integrable general N-component nonlinear Schrödinger equations. The Darboux-dressing transformation for the equations is firstly constructed according to its Lax pair. Applying the idea of separating variables, we generate the solutions of the Lax pair equations with a nontrivial background. Then we derive the rational and semi-rational vector solutions of the equations in detail. The rational solutions act as the pure rogue waves. The semi-rational solutions represent a combination of different types of rogue waves, breathers, and soliton waves. Moreover, a constraint condition is given for the semi-rational solutions being reduced to unmixed breather or nth-rogue waves. To illustrate the corresponding dynamic behaviors of these solutions, we analyze two-component and four-component cases, showing that the multi-component system exhibits richer phenomena and novel behaviors compared to lower-component systems, and also presenting some novel phenomena.

Fusion and fission phenomena in a (2+1)-dimensional Sawada-Kotera type system

In this study, we extend the generalized multilinear variable separation approach to a fifth-order nonlinear evolution equation. By performing asymptotic analysis on the variable separation solution, which is composed of three lower-dimensional functions, we identify a resonant regime governing dromion-dromion/solitoff interactions. In the case of two-dromion interactions, elastic, inelastic, and completely inelastic collisions are possible, while for the dromion-solitoff interaction only inelastic and completely inelastic collisions are permitted. Furthermore, we derive two types of semi-rational solutions from the quadratic function ansatz. In particular, in the scenario of a completely resonant collision between a lump and a line-soliton pair, the lump separates from one line soliton and exists briefly before merging with the other soliton, forming a localized lump in both time and space dimensions. The fusion or fission phenomena between the dromion-dromion/solitoff interaction and the lump-line soliton interaction are shown graphically.

Taking into consideration a fifth-order nonlinear Schrödinger equation in an optical fiber

In this paper, symbolic computation on a fifth-order nonlinear Schrödinger equation is done, for the attosecond pulses propagation in an optical fiber. With respect to the complex amplitude of the optical pulse envelope, we work out a Lax pair and derive the modified generalized Darboux transformation. Then, we give the semirational solutions via the modified generalized Darboux transformation method. By means of such solutions, we graphically discuss the properties for three types of the degenerate solitons.

Periodic line wave, rogue waves and the interaction solutions of the (2+1)-dimensional integrable Kadomtsev–Petviashvili-based system

This paper mainly investigates multiple types of solutions for the (2+1)-dimensional integrable Kadomtsev–Petviashvili-based system, which can describe the complex nonlinear wave phenomena observed in many physical systems, such as fluid mechanics, plasma physics and ocean dynamics. The Hirota’s bilinear method and the perturbation expansion skill are used to derive the periodic line wave solution and the interaction solution composed of the breather and the periodic line wave. By choosing appropriate parameters and employing the long wave limit of the soliton solution, two kinds of elementary rogue waves (RWs) are generated, which are kink-shaped line RW and W-shaped line RW. The interaction solutions among a lump, two lumps and two kinds of line RWs are obtained. Furthermore, the semi-rational solutions of the KP-based system are yielded, which include six types, namely (1) a line RW on a line soliton background, (2) a line RW on two line solitons background, (3) the line RW on the breather background, (4) a lump on the background with the periodic line wave, (5) the line RW on the background of the lump and the breather and (6) two lumps on the background with the periodic line wave. An effective analytical method related to the characteristic lines is presented to analyze the dynamical behaviors of the rogue waves and interaction waves. The method can be further extended to investigate other complex wave structures for the high-dimensional nonlinear integrable equations.

Rogue wave excitations and hybrid wave structures of the Heisenberg ferromagnet equation with time-dependent inhomogeneous bilinear interaction and spin-transfer torque.

In this paper, we focus on the localized rational waves of the variable-coefficient Heisenberg spin chain equation, which models the local magnetization in ferromagnet with time-dependent inhomogeneous bilinear interaction and spin-transfer torque. First, we establish the iterative generalized (m,N-m)-fold Darboux transformation of the Heisenberg spin chain equation. Then, the novel localized rational solutions (LRSs), rogue waves (RWs), periodic waves, and hybrid wave structures on the periodic, zero, and nonzero constant backgrounds with the time-dependent coefficients α(t) and β(t) are obtained explicitly. Additionally, we provide the trajectory curves of magnetization and the variation of the magnetization direction for the obtained nonlinear waves at different times. These phenomena imply that the LRSs and RWs play the crucial roles in changing the circular motion of the magnetization. Finally, we also numerically simulate the wave propagations of some localized semi-rational solutions and RWs.

Localized waves and interaction solutions to an integrable variable coefficient Date-Jimbo-Kashiwara-Miwa equation

This paper initiates an exploration into the exact solutions of the variable coefficient Date-Jimbo-Kashiwara-Miwa equation, first utilizing the Painlevé analysis method to discuss the integrability of the equation. Subsequently, By employing the Hirota bilinear method, N-soliton solutions for the equation are constructed. The application of the Long wave limit method to these N-soliton solutions yields rational and semirational solutions. Various types of localized waves, encompassing solitons, lumps, breather waves, and others, emerge through the careful selection of specific parameters. By analyzing the image of the solutions, the evolution process and its dynamical behavior are studied.

Families of exact solutions of a Generalized (2+1)-dimensional Boussinesq type equation

In this paper, we study a Generalized (2+1)-dimensional Boussinesq-type equation. Using the Hirota bilinear method, we present the N-order bright soliton solutions and dark soliton solutions. For the one-soliton solution, the bright soliton solution and the dark soliton solution share the same limit line but have different extreme values. Building on the soliton solutions, we derive higher-order bright and dark breather solutions as well as mixed solutions. The dynamic behavior is characterized using visual representations. Furthermore, through the long-wave limit method, we obtain the bright and dark lump solutions. Notably, they share the same extreme points but have different extreme values. Additionally, we derive two semi-rational solutions as lump-soliton and lump-breather. It is found that the lump moves along the peak amplitude of the soliton wave.

Some new nonlinear wave solutions and dynamical behavior of the (3+1)-dimensional Ito equation

Some new nonlinear wave solutions of the (3+1)-dimensional Ito equation, such as three-wave type solutions, lump wave solutions, rogue wave solutions and hybrid solutions of different forms, are studied by the bilinear neural network method. Besides, three-wave solutions are investigated via the single-layer neural network ‘4-3-1’ model, and hybrid soliton solutions between lump-type and strip-like soliton in ‘4-5-1’ model are discussed. Furthermore, rogue wave solutions are studied by the double-layer neural network the ‘4-2-3-1’ and the ‘4-3-3-1’ models, and the interference phenomena between them are analyzed using visual images. We also employed the extended parameter limit method to investigate the collapse degradation behavior of both solutions. Lump solutions and Y-type semi-rational solutions are derived separately.

Various types of discrete exact localized wave solutions and dynamical analysis for the discrete complex modified Korteweg-de Vries equation

Under consideration is the second-order integrable discretization of complex modified Korteweg-de Vries (mKdV) equation which is regarded as the discrete counterpart of the mKdV equation having an essential role in describing the propagation behavior of water waves and acoustic waves in nonlinear media. First of all, based on the known linear spectral problem, the discrete generalized (n,N−n)-fold Darboux transformation is constructed to derive various types of discrete exact localized wave solutions, including soliton and semi-rational soliton solutions on vanishing background, breather, rogue wave and hybrid interaction solutions on plane wave background, and rational soliton solution on constant background, and the relevant evolution structures are studied graphically. Secondly, the asymptotic analysis is used to discuss the elastic interaction for two-soliton solutions and limit states for rational soliton solutions. Finally, the numerical simulation is utilized to investigate the dynamical behavior and propagation stability of some exact solutions. The findings presented in this paper may contribute to explaining physical phenomena described by the mKdV equation.

Collision dynamics of high-order localized waves in a novel [formula omitted]-dimensional integrable Boussinesq model

The pursuit of finding precise solutions for complex nonlinear systems in high dimensions has long been a significant goal in the fields of mathematics and physics. This paper focuses on investigating a novel equation, referred to as the (3+1)-dimensional Boussinesq equation, which accurately describes the behavior of gravity waves on the surface of water. Using the Hirota bilinear method, the study successfully derives multiple-soliton and multi-breather solutions for the Boussinesq equation. Moreover, it presents mixed solutions that combine solitons and breathers under specific parameter conditions. By considering the long wave limit for these solitons, rational and semi-rational solutions for the equation are obtained. These rational solutions encompass line rogue waves and lump waves, while the semi-rational solutions emerge from the interaction between line rogue waves and solitons. Additionally, employing symbolic computation, the paper introduces a class of multiple lumps for the (3+1)-d Boussinesq equation, showcasing intriguing triangular formations and circular patterns. Overall, the findings of this research enhance our understanding of unique nonlinear phenomena and offer valuable theoretical support for future experimental investigations.

Rational and semi-rational solutions to the nonlocal Davey–Stewartson III equation

We utilize the bilinear method and Kadomtsev–Petviashvili (KP) hierarchy reduction technique to obtain solutions for the nonlocal DS III equation. These solutions are elegantly expressed in terms of an N×N determinant, incorporating various parameter reduction constraints. It is important to note that these solutions are nontrivial and do not necessarily satisfy the local DS III equation. In the case of an even value of N, we derive solutions on constant backgrounds, while for odd values of N, solutions on periodic backgrounds are generated. To gain a deeper understanding of the dynamics involved, we conduct a thorough analysis of the derived solutions, accompanied by informative plots. Overall, our findings showcase the power of the bilinear method and KP hierarchy reduction technique in obtaining meaningful solutions for the nonlocal DS III equation, shedding light on its complex behavior and enriching our understanding of the system.

Vector rogue waves and their dynamics in the nonlocal three-component Manakov system

In this work, by combining the Darboux transformation and variable separation technique, we generate and discuss a semirational vector solution to the nonlocal three-component Manakov system. The semirational solution is expressed in separation-of-variables form. The semirational vector solution exhibits breathers and rogue waves on a bright-dark soliton background. Moreover, the dynamic behaviors of the semirational vector solutions are discussed with some graphics. Our results may contribute to explaining and enriching the corresponding rogue wave phenomena emerging in nonlocal wave modes.

Fractional Soliton and Semirational Solutions of a Fractional Two-Component Generalized Hirota Equation

Fractional Soliton and Semirational Solutions of a Fractional Two-Component Generalized Hirota Equation

General Soliton and (Semi-)Rational Solutions of a (2+1)-Dimensional Sinh-Gordon Equation

In this paper, we investigate solutions of a (2+1)-dimensional sinh-Gordon equation. General solitons and (semi-)rational solutions are derived by the combination of Hirota’s bilinear method and Kadomtsev-Petviashvili hierarchy reduction approach. General solutions are expressed as Ntimes N Gram-type determinants. When the determinant size N is even, we generate solitons, line breathers, and (semi-)rational solutions located on constant backgrounds. In particular, through the asymptotic analysis we prove that the collision of solitons are completely elastic. When N is odd, we derive exact solutions on periodic backgrounds. The dynamical behaviors of those derived solutions are analyzed with plots. For rational solutions, we display the interaction of lumps. For semi-rational solutions, we find the interaction solutions between lumps and solitons.

Multiple nonlinear wave solutions of a generalized Heisenberg ferromagnet model and their interactions

Under investigation in this paper is a generalized Heisenberg ferromagnet (HF) equation which is named the Zhanbota-IIA equation. It is one of the integrable generalizations of the HF equation that plays an important role in nonlinear magnetization dynamics. Through the establishment of the N-fold Darboux transformation, a series of solutions will be obtained, including multi-solitons, one- and two-breathers, first- and higher-order rogue waves. Dynamic behaviors of those solutions will be analyzed, including several structures of rogue waves such as fundamental structure, triangular structure, ring structure and ring-fundamental structure, the coexistence of rogue waves and breathers, i.e. semi-rational solution and the interaction of two breathers.

Semirational rogue waves of the three coupled higher-order nonlinear Schrödinger equations

Under investigation in this paper is the three coupled Hirota equations in the long distance communication or ultrafast signal routing system. Semirational solutions are derived by virtue of the Darboux dressing transformation. Based on the discussion of the parameters in solutions, we see that the semirational rogue waves are formed by the interaction between the rogue wave and the dark–dark–bright/breather soliton.

Localized waves and interaction solutions to an integrable variable coefficients Jimbo-Miwa equation

In this paper, the reduced variable coefficients Jimbo-Miwa (vcJM) equation is studied. Firstly, the integrability of the reduced vcJM equation is verified by Painlevé analysis. Based on the Hirota bilinear method and the long wave limit method, the N-soliton solutions, rational and semirational solutions of the vcJM equation are obtained. By choosing different parameters and coefficient functions, some of different kinds of local waves, including of solition, breather wave and lumps, of the equation are obtained. Furthermore, the interaction solutions between different local waves are obtained. The dynamical behavior of the interaction between different local waves is studied by modifying the time parameters and the process is displayed by figures.

Diverse soliton solutions and dynamical analysis of the discrete coupled mKdV equation with 4×4 Lax pair

Under consideration in this study is the discrete coupled modified Korteweg–de Vries (mKdV) equation with 4 × 4 Lax pair. Firstly, through using continuous limit technique, this discrete equation can be mapped to the coupled KdV and mKdV equations, which may depict the development of shallow water waves, the optical soliton propagation in cubic nonlinear media and the Alfven wave in a cold collision-free plasma. Secondly, the discrete generalized (r, N–r)-fold Darboux transformation is constructed and extended to solve this discrete coupled equation with the fourth-order linear spectral problem, from which diverse exact solutions including usual multi-soliton and semi-rational soliton solutions on the vanishing background, higher-order rational soliton and mixed hyperbolic-rational soliton solutions on the non-vanishing background are derived, and the limit states of some soliton and rational soliton solutions are analyzed by the asymptotic analysis technique. Finally, the numerical simulations are used to explore the dynamical behaviors of some exact soliton solutions. These results may be helpful for understanding some physical phenomena in fields of shallow water wave, optics, and plasma physics.