Narrow Results

In this paper, we define and explore the Darboux and binary Darboux transformations for a semi-discrete short pulse equation. By iterating the binary Darboux transformation, we obtain quasi-Grammian solutions. Furthermore, we derive explicit matrix solutions for the binary Darboux matrix and reduce them to the elementary Darboux matrix. Finally, we plot the dynamics of bright, dark, double breather, and peak soliton solutions.

Breather is an elementary excitation regarded as a bound state of a fluxon and an antifluxon in a long Josephson junction. In quantum-mechanical regime, the breather energy is quantized so that the breather can be considered as an artificial moving atom. We propose a new type of fluxon qubit that is constructed by quantum-mechanical superposition of the breather's states. We describe quantum logic gates of breather qubit required for constructing quantum computer. In addition, our qubit can move in the system so that transfer of quntum information is possible between mobile qubits as well as stationary qubits. Our talking qubits support the global information sharing in quantum information networks.

We present the detailed dynamics of the particles in the $\beta$-Fermi–Pasta–Ulam–Tsingou (FPUT) chain after the initiation of a localized nonlinear excitation (LNE) by squeezing a central bond in the monodispersed chain at time $t$ = 0 while all other particles remain in their original relaxed positions. In the absence of phonons in the system, the LNE appears to initiate its relaxation process by symmetrically emitting two very weak solitary waves. The next stage involves the spreading of the LNE and the formation of nonsolitary wave-like objects to broaden the excitation region until a stage is reached when many weak solitary wave-like objects can be emitted as the system begins its journey to quasi-equilibrium and then to equilibrium. In addition to being of fundamental interest, these systems may be realized using cantilever systems and could well hold the key to constructing the next generation of broadband energy harvesting systems.

The higher-order breather, periodic-wave, lump, rational soliton solutions and mixed solutions of the Hirota–Maccari (HM) system by virtue of the Kadomtsev–Petviashvili (KP) hierarchy reduction method are investigated in this work. Through analyzing the structural characteristics of periodic-wave solutions, we attain the quasi-periodic W(M)-shaped waves and two kinds of breathers. The mixed solutions that consist of the quasi-periodic W(M)-shaped waves and breathers are constructed. Further, by taking the long wave limit on the periodic-wave solutions, the semi-rational solutions are derived, which illustrate the interaction of the rational soliton, lump, quasi-periodic wave and breather. Characteristics of these mixed solutions are discussed graphically and the corresponding generating conditions are given. Especially, a new bound-state interaction composed of lump and breather is generated under the velocity resonance mechanism. This newfangled pattern is a beautiful phenomenon for the HM system.

In this paper, we consider the transformation of a breather type soliton in DNA into its demodulated version due to the strong impact of proteins associated with some sequences of DNA chain. These proteins are, for example, RNA polymerase responsible for the formation of m-RNA or initiation protein complex indispensable for chromosome replication.

In this paper, an extended nonlinear Schrödinger equation with higher-order odd (third-order) and even (fourth-order) terms is investigated in detail. The equation for the one-dimensional magnetic systems is integrable and admits exact solutions. It is more accurate than the nonlinear Schrödinger equation in describing wave propagation in the ocean and optical fibers. First, the modulation instability of solutions is analyzed in the presence of small perturbation. Second, breather and rogue wave solutions of this equation are constructed via the modified Darboux transformation method. The effects of the higher-order terms are investigated graphically. Specially, the interactions between two breathers are studied by adjusting the spectral parameters and the collisions between breather and rogue waves are also discussed.

In this paper, we study the nonlinear Schrödinger equation with self-consistent sources, and obtain the rogue wave solution, the breather solution and their interactions by the generalized Darboux transformation. The dynamics of the rogue wave solution, the breather solution and their interactions are analyzed.

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.

In this paper, we study the interaction solutions of the first BKP equation by using the Hirota direct method and taking long-wave limit. In order to obtain the interaction solutions, the multi-soliton solutions are firstly derived using the Hirota direct method, then the interaction solutions are successfully constructed by properly choosing appropriate parameters and taking long-wave limit on the soliton solutions. These parameters have great influences on the propagation directions, shapes as well as energy. Moreover, the dynamic properties of these obtained solutions are illustrated vividly by some graphs. The results in this work could be used to solve nonlinear problems in nonlinear optics and engineering field.

In this paper, an extended (3+1)-dimensional Kadomtsev–Petviashvili (KP) equation is studied via the Hirota bilinear derivative method. Soliton, breather, lump and rogue waves, which are four types of localized waves, are obtained. N-soliton solution is derived by employing bilinear method. Then, line or general breathers, two-order line or general breathers, interaction solutions between soliton and line or general breathers are constructed by complex conjugate approach. These breathers own different dynamic behaviors in different planes. Taking the long wave limit method on the multi-soliton solutions under special parameter constraints, lumps, two- and three-lump and interaction solutions between dark soliton and dark lump are constructed, respectively. Finally, dark rogue waves, dark two-order rogue waves and related interaction solutions between dark soliton and dark rogue waves or dark lump are also demonstrated. Moreover, dynamical characteristics of these localized waves and interaction solutions are further vividly demonstrated through lots of three-dimensional graphs.

Utilizing the Hirota bilinear method, the lump solutions, the interaction solutions with the lump and the stripe solitons, the breathers and the rogue waves for a (3+1)-dimensional Kudryashov–Sinelshchikov equation are constructed. Two types of interaction solutions between the lumps and the stripe solitons are exhibited. Some different breathers are given by choosing special parameters in the expressions of the solitons. Through a long wave limit of breathers, the lumps and rogue waves are derived.

The present paper is devoted to discussion of (3+1)-dimensional generalized B-type Kadomtsev–Petviashvili (gBKP) equation from point of view of their multi-soliton solutions and localized solutions associated with multi-soliton solutions. Firstly, taking advantage of the Bell-polynomial approach, we construct the Hirota bilinear form of (1.1). Based on that, the multi-soliton solutions are also singled out. Subsequently, the (3+1)-dimensional gBKP equation is also found to allow fruitful localized solutions, including breather, lump, rogue wave, and hybrid solutions. These results obtained in this work adequately illustrate the effectiveness of the long wave limit method and complex conjugate technique, which are expected to be employed to obtain more abundant exact solutions.

In this paper, we study the resonant collisions among different types of localized solitary waves in the (2+1)-dimensional Hirota–Satsuma–Ito equation, which are described by N-soliton solutions constructed using bilinear method. Through the asymptotic analysis and limit treatment of the phase shift of these localized waves, the elastic collisions among different localized waves can be transformed into resonant collisions. Hereby, we study the resonant collision between a breather/ lump and a bright line soliton and find two collision situations: (i) the breather is semi-localized in space and the shape of the breather is not localized during the propagation and (ii) the lump wave generates from the bright line wave. At the same time, we investigate the resonant collision between a breather/lump and two bright line solitons. In these evolution processes, we also gain two dynamical behaviors: (iii) the breather is always localized in space and the shape of the breather is not localized during the propagation, and (iv) the lump wave appears from a bright line soliton and then disappears into the other bright line soliton. Localized wave and interaction solutions of the nonlinear wave models have a great impact on oceanography and physics. The results may be useful in researching the physical phenomena in shallow water waves and nonlinear optics.

The Kuralay equation, as a Heisenberg ferromagnet-type equation, is of great significance for describing nonlinear phenomena in magnets. For the Kuralay-IIA equation, based on the constructed $N$-fold Darboux transformation, we derive its one-fold and two-fold rogue wave solutions, one-breathers and two-breathers including the Akhmediev two-breathers. By adding shifts on the time variable, we further separate the Akhmediev two-breathers into two one-breathers. In addition, the semi-rational solutions consisting of the rogue wave and one-breather solutions are also derived. The diagrams of the obtained solutions are plotted to analyze their dynamic features.

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.

Using the Inverse Scattering Method with a nonvanishing boundary condition, we obtain an explicit breather solution with nonzero vacuum parameter b of the focusing modified Korteweg–de Vries (mKdV) equation. Moreover, taking the limiting case of zero frequency, we obtain a generalization of the double pole solution introduced by M. Wadati et al.

Breather is an elementary excitation regarded as a bound state of a fluxon and an antifluxon in a long Josephson junction. In quantum-mechanical regime, the breather energy is quantized so that the breather can be considered as an artificial moving atom. We propose a new type of fluxon qubit that is constructed by quantum-mechanical superposition of the breather's states. We describe quantum logic gates of breather qubit required for constructing quantum computer. In addition, our qubit can move in the system so that transfer of quntum information is possible between mobile qubits as well as stationary qubits. Our talking qubits support the global information sharing in quantum information networks.