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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]

In this paper, abundant interaction solutions for the (1 + 1)-dimensional generalized Drinfel'd–Sokolov–Wilson (gDSW) equation are presented. First, two groups of lump-soliton-type interaction solutions are constructed by using a combination of quadratic function and two exponential functions. Especially, lump-twin soliton-type solution therein can generate rogue wave, which rarely appears in (1 + 1)-dimensional nonlinear systems in comparison with (2 + 1)-dimensional nonlinear systems. When certain parameters are taken to zero, these exact solutions are reduced to corresponding lump, soliton and lumpoff solutions. Second, the lump-periodic wave-type and soliton-periodic wave-type interaction solutions are also found on the basis of the bilinear form of gDSW equation. In the concrete analysis, we use mathematical software Maple to deal with some complicated symbolic computations. Figures are presented to show the dynamical features of these solutions. [ABSTRACT FROM AUTHOR]

Nonlinear waves are seen in nature, such as the water waves and plasma waves. Investigated in this paper is a (3 + 1)-dimensional B-type Kadomtsev–Petviashvili–Boussinesq equation. Based on the bilinear method, we get the rational solutions, which are different from the published ones, semi-rational solutions and breather-type kink soliton solutions. Through the rational solutions, we observe two types of waves: the lump waves and line rogue waves. The semi-rational solutions depict two types of interactions: (1) The fusion or fission between the lump wave and soliton; (2) The interaction between the line rogue wave and soliton. During the interaction between the line rogue wave and soliton, the line rogue wave evolves with three different shapes: the bright rogue waves, bright–dark rogue waves and dark rogue waves. Via the breather-type kink soliton solutions, we observe the breather-soliton mixture. [ABSTRACT FROM AUTHOR]

A nonlinear vibration system behaviors always periodically; however, sometimes a vibrating structure, e.g. an empty oil tank, will be collapsed suddenly. This can be explained as a kind of rogue waves. This paper gives a general approach to search for rational breather waves to nonlinear equations including nonlinear vibration systems and dispersive wave systems. A breather type of rogue wave solution which contains two peaks whose amplitudes are two to three times higher than its surrounding waves and generally forms in a short time is constructed. Moreover, by using three-wave method, new breather-type multisolitary coherent structures are presented, the fusion interactions of localized structures are discussed and graphically investigated, showing some novel features and interesting behaviors. [ABSTRACT FROM AUTHOR]

The bilinear method is employed to construct the multiple lump solutions of the (3+1)-dimensional potential Yu–Toda–Sasa–Fukuyama equation in fluid dynamics. The 1-lump solutions, 3-lump solutions and 6-lump solutions are explicitly presented. The centers of the 3-lump wave have a triangular structure, and the 6-lump wave possesses a central peak and five peaks in a ring. The dynamic characteristics of the obtained solutions are analyzed with the aid of numerical simulation. [ABSTRACT FROM AUTHOR]

By means of Hirota bilinear method and symbolic computation, this paper studied two types of interaction solutions for the potential Yu–Toda–Sasa–Fukuyama equation. The first type of interaction between lump soliton and one stripe soliton generate the fusion and fission phenomenon, while the second type of interaction solution between lump soliton and twin stripe solitons generate the rogue wave. The corresponding dynamic plots of the above phenomena are graphically displayed respectively. [ABSTRACT FROM AUTHOR]

Hirota equation is a modified nonlinear Schrödinger (NLS) equation, which takes into account higher order dispersion and delay correction of cubic nonlinearity. The propagation of the waves in the ocean is described, and the optical fiber can be regarded as a more accurate approximation than the NLS equation. Using the algebraic reductions from the Lie algebra g l (n , ℂ) to its commutative subalgebra Z n , we construct the general Z n -Hirota systems. Considering the potential applications of two-mode nonlinear waves in nonlinear optical fibers, including its Lax pairs, we use the algebraic reductions of the Lie algebra g l (2 , ℂ) to its commutative subalgebra Z 2 = ℂ [ Γ ] / (Γ 2). Then, we construct Darboux transformation of the strongly coupled Hirota equation, which implies the new solutions of (q [ 1 ] ,   r [ 1 ]) generated from the known solution (q ,   r). The new solutions (q [ 1 ] ,   r [ 1 ]) furnish soliton solutions and breather solutions of the strongly coupled Hirota equation. Furthermore, using Taylor series expansion of the breather solutions, the rogue waves of the strongly coupled Hirota equation can be given demonstrably. It is obvious that different images can be obtained by choosing different parameters. [ABSTRACT FROM AUTHOR]

The Hirota equation, a modified nonlinear Schrödinger (NLS) equation, takes into account higher-order dispersion and time-delay corrections to the cubic nonlinearity. Its wave propagation is like in the ocean and optical fibers can be viewed as an approximation which is more accurate than the NLS equation. By considering the potential application of two mode nonlinear waves in nonlinear fibers under a certain case, we use the algebraic reductions from the Lie algebra g l (2 , ℂ) to its commutative subalgebra Z 2 = ℂ [ Γ ] / (Γ 2) and Γ = (δ i , j + 1) i j to define a weakly coupled Hirota equation (called Frobenius Hirota equation) including its Lax pair, in this paper. Afterwards, Darboux transformation of the Frobenius Hirota equation is constructed. The Darboux transformation implies the new solutions of ( q [ 1 ] , r [ 1 ] ) generated from the known solution (q , r). The new solutions ( q [ 1 ] , r [ 1 ] ) provide soliton solutions, breather solutions of the Frobenius Hirota equation. Further, rogue waves of the Frobenius Hirota equation are given explicitly by a Taylor series expansion of the breather solutions. In particular, by choosing different parameter values for the rogue waves, we can get different images. [ABSTRACT FROM AUTHOR]

We study the linear stability of the Peregrine breather both numerically and with analytical arguments based on its derivation as the singular limit of a single-mode spatially periodic breather as the spatial period becomes infinite. By constructing solutions of the linearization of the nonlinear Schrödinger equation in terms of quadratic products of components of the eigenfunctions of the Zakharov–Shabat system, we show that the Peregrine breather is linearly unstable. A numerical study employing a highly accurate Chebychev pseudo-spectral integrator confirms exponential growth of random initial perturbations of the Peregrine breather. [ABSTRACT FROM AUTHOR]

We study a deformed Fokas–Lenells equation which is related to the integrable derivative nonlinear Schrödinger hierarchy with higher-order nonholonomic constraint. The baseband modulation instability as an origin of rogue waves is displayed. The explicit rogue wave solutions are obtained via the Darboux transformation. Typical rogue wave patterns such as the standard rogue wave, dark rogue wave and twisted rogue wave pair in three different components of the deformed Fokas–Lenells equation are presented. Besides, the state transitions between rogue waves and solitons are analytically found when the modulation instability growth rate tends to zero in the zero-frequency perturbation region. The explicit soliton solutions under the special parameter condition are given. The anti-dark and W-shaped solitons in their respective components are shown. [ABSTRACT FROM AUTHOR]

By means of the Hirota bilinear method, lump-type solution and two types of interaction solutions of the (2+1)-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada equation are obtained, respectively. Lump-type solution is constructed by assuming f in the corresponding bilinear equation as a ternary quadratic polynomial function. It is interesting that the rogue wave is aroused by the interaction between one-lump soliton and a pair of stripe solitons, and the fusion and fission phenomena are also found in the interaction between lump solitons and one-stripe soliton. The corresponding phenomena are vividly demonstrated by the graphs. [ABSTRACT FROM AUTHOR]

A (2 + 1)-dimensional high-order nonlinear evolution (HNE) equation is considered in this paper. A Hirota bilinear form of the HNE equation is constructed by the dependent variable function. Solitary waves are derived by solving the Hirota bilinear form of the HNE equation. Lump waves of the HNE equation are obtained by introducing a positive quadratic function. By mixing an exponential function or two exponential functions with a quadratic function, interaction solutions between a lump and a one-soliton, and between a lump and a two-soliton are presented. For the interaction solution between a lump and a two-soliton, this kind of solution can be considered as a special rogue wave. The propagation phenomena of these explicit solutions are illustrated by some graphs. [ABSTRACT FROM AUTHOR]

In this work, a variable coefficient nonlinear Schrödinger (vc-NLS) equation is under investigation, which can describe the amplification or absorption of pulses propagating in an optical fiber with distributed dispersion and nonlinearity. By means of similarity reductions, a similar transformation helps us to relate certain class of solutions of the standard NLS equation to the solutions of integrable vc-NLS equation. Furthermore, we analytically consider nonautonomous breather wave, rogue wave solutions and their interactions in the vc-NLS equation, which possess complicated wave propagation in time and differ from the usual breather waves and rogue waves. Finally, the main characteristics of the rational solutions are graphically discussed. The parameters in the solutions can be used to control the shape, amplitude and scale of the rogue waves. [ABSTRACT FROM AUTHOR]

Abstract Based on a 4 × 4 matrix Lax pair, we propose a negative matrix AKNS system with a Hermitian symmetric space. A Darboux transformation is constructed by setting a restrictive condition and using the loop group method. The restrictive condition can guarantee the evolution relations of the potential matrices. Using this Darboux transformation and different seed solutions and free parameters, we obtain different types of spatial–temporal distribution structures for various explicit solutions of the negative matrix AKNS system with a Hermitian symmetric space, including the rogue wave, Ma breather, the interaction of two Ma breathers, and parabolic-type soliton solutions. [ABSTRACT FROM AUTHOR]

Abstract A novel type of exact rogue wave is found for the (1+1)-dimensional Ito equation, which is generated by the interaction solution between an algebraic localized soliton (named "lump") and an exponentially localized twin soliton. In addition, the interaction solution among triangular periodic wave and twin soliton is also proposed. Three special interaction phenomenons are displayed by some visual figures, respectively. [ABSTRACT FROM AUTHOR]

In this paper, a (3 + 1)-dimensional nonlinear evolution equation is studied via the Hirota method. Soliton, lump, breather and rogue wave, as four types of localized waves, are derived. The obtained N-soliton solutions are dark solitons with some constrained parameters. General breathers, line breathers, two-order breathers, interaction solutions between the dark soliton and general breather or line breather are constructed by choosing suitable parameters on the soliton solution. By the long wave limit method on the soliton solution, some new lump and rogue wave solutions are obtained. In particular, dark lumps, interaction solutions between dark soliton and dark lump, two dark lumps are exhibited. In addition, three types of solutions related with rogue waves are also exhibited including line rogue wave, two-order line rogue waves, interaction solutions between dark soliton and dark lump or line rogue wave. [ABSTRACT FROM AUTHOR]

Abstract Based on Hirota bilinear method, four kinds of localized waves, solitons, breathers, lumps and rogue waves of the extended (3+1)-dimensional Jimbo–Miwa equation are constructed. Breathers are obtained through choosing appropriate parameters on soliton solutions, while lumps and rogue waves are derived via the long wave limit on the soliton solutions. The energy, phase shift, shape, and propagation direction of these localized waves can be influenced and controlled by parameters. Considering mixed cases of the above four types of solutions, we also give many kinds of interaction solutions in the same plane with different parameters or different planes with the same parameters. Dynamical characteristics of these localized waves and interaction solutions are further analyzed and vividly demonstrated through figures. [ABSTRACT FROM AUTHOR]

A generalized nonlocal nonlinear Hirota (GNNH) equation with variable coefficients is presented, which can be reduced into the nonlocal Hirota equation with the self-induced PT-symmetric potential. Especially, the nonlocal Gross–Pitaevskii (NGP) equation with the self-induced PT-symmetric potential is derived from the GNNH equation. Then, we obtain some novel non-autonomous breather solutions and rogue waves of GNNH equation via similarity and Hirota methods, and consider some controllable behaviors of these non-autonomous wave solutions. Furthermore, some properties of the non-autonomous rational (NR) waves are investigated analytically for the NGP equation. The trajectories of peaks and depressions of the non-autonomous rogue waves are produced by means of analytical method, and the dynamical stabilities of the NR solution are derived through the numerical method. The obtained results are different from the solutions of the local nonlinear equations. Some different propagation phenomena can also be produced through manipulating non-autonomous rogue waves, which can present the potential applications for the rogue wave phenomena in nonlocal wave models. [ABSTRACT FROM AUTHOR]