Narrow Results

Under investigation in this paper is the variable-coefficient Kadomtsev–Petviashvili equation, which describes the long waves with small amplitude and slow dependence on the transverse coordinate in a single-layer shallow fluid. Employing the bilinear form and symbolic computation, we obtain the lump, mixed lump-stripe soliton and mixed rogue wave-stripe soliton solutions. Discussions indicate that the variable coefficients are related to both the lump soliton’s velocity and amplitude. Mixed lump-stripe soliton solutions display two different properties, fusion and fission. Mixed rogue wave-stripe soliton solutions show that a rogue wave arises from one of the stripe solitons and disappears into the other. When the time approaches 0, rogue wave’s energy reaches the maximum. Interactions between a lump soliton and one-stripe soliton, and between a rogue wave and a pair of stripe solitons, are shown graphically.

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, 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.

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.

In this paper, the investigation is made on a (3+1)-dimensional generalized Calogero–Bogoyavlenskii–Schiff equation. We obtain the lump, mixed lump-stripe, mixed rogue wave-stripe and breather wave solutions via the Hirota method and symbolic computation. Accordingly, we observe that the shape and amplitude of the lump keep unchanged, while the mixed lump-stripe solutions present that (1) the lump/stripe and one-stripe waves fuse into a one-stripe wave; (2) the one-stripe wave splits into a lump/stripe and a one-stripe wave. We graphically show the interaction between a rogue wave and a pair of stripe waves, and observe that the amplitude of the rogue wave reaches the maximum value at a certain position. Besides, the breather wave propagates steadily in a certain direction.

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.

Results of numerical study of lump solutions (2D solitons) of a generalised 2D Gardner equation are presented. In the moving coordinate frame the equation is:

We construct the general vortex solution in a fully-Higgsed, color-flavor locked vacuum of a non-Abelian gauge theory, where the gauge group is taken to be the product of an arbitrary simple group and U(1), with a Fayet-Iliopoulos term. The strict correspondence between vortices and lumps in the associated NLσM which arise in the limit of strong coupling is pointed out. The construction of the vortex moduli space is derived here as a consequence of this correspondence.

A 50-year-old woman complains of a lump down below.

• What other symptoms are important for reaching a diagnosis?
• What are the common conditions causing this symptom?

"A lump down below" is a common complaint in gynecology clinic. The lesion may be arising from the vulva, urethra, vagina, cervix, uterus or anus. Several symptoms may reveal or point to the diagnosis of the lesion.