Liu-Qing Li

Bio: Liu-Qing Li is an academic researcher from National Laboratory for Computational Fluid Dynamics. The author has contributed to research in topics: Breather & Physics. The author has an hindex of 8, co-authored 12 publications receiving 298 citations.

Bilinear form, soliton, breather, lump and hybrid solutions for a ( $$\varvec{2+1}$$ 2 + 1 )-dimensional Sawada–Kotera equation

Abstract: In this paper, we investigate a ( $$2+1$$ )-dimensional Sawada–Kotera (SK) equation for the atmosphere, rivers, lakes, oceans, as well as the conformal field and two-dimensional quantum gravity gauge field. Bilinear form and N-soliton solutions, which are different from those in the existing literatures, are derived, where N is a positive integer. The higher-order breather, lump and hybrid solutions for the ( $$2+1$$ )-dimensional SK equation are also constructed based on the N-soliton solutions. Three kinds of the first-order breathers are obtained, and the higher-order breathers are constructed. The higher-order lump solutions are also derived via the long-wave limit method. Hybrid solutions composed of the solitons, breathers and lumps are worked out, and interaction between the waves is discussed graphically. Finally, similar solutions for a generalized form of the ( $$2+1$$ )-dimensional SK equation are given.

Bilinear form, solitons, breathers and lumps of a (3 + 1)-dimensional generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt equation in ocean dynamics, fluid mechanics and plasma physics

Abstract: A $$(3+1)$$ -dimensional generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt equation in ocean dynamics, fluid mechanics and plasma physics is investigated in this paper. Bilinear form, soliton and breather solutions are derived via the Hirota method. Lump solutions are also obtained. Amplitudes of the solitons are proportional to the coefficient $$h_1$$ , while inversely proportional to the coefficient $$h_2$$ . Velocities of the solitons are proportional to the coefficients $$h_1$$ , $$h_3$$ , $$h_4$$ , $$h_5$$ and $$h_9$$ . Elastic and inelastic interactions between the solitons are graphically illustrated. Based on the two-soliton solutions, breathers and periodic line waves are presented. We find that the lumps propagate along the straight lines affected by $$h_4$$ and $$h_9$$ . Both the amplitudes of the hump and valleys of the lump are proportional to $$h_4$$ , while inversely proportional to $$h_2$$ . It is also revealed that the amplitude of the hump of the lump is eight times as large as the amplitudes of the valleys of the lump. Graphical investigation indicates that the lump which consists of one hump and two valleys is localized in all directions and propagates stably.

On the quintic time-dependent coefficient derivative nonlinear Schrödinger equation in hydrodynamics or fiber optics

Abstract: Under investigation in this paper is a quintic time-dependent coefficient derivative nonlinear Schrodinger equation for certain hydrodynamic wave packets or a medium with the negative refractive index. A gauge transformation is found to obtain the equivalent form of the equation. With respect to the wave envelope for the free water surface displacement or envelope of the electric field, Painleve integrable condition, different from that in the existing literature, is derived, with which the bilinear forms and N-soliton solutions are constructed. Asymptotic analysis illustrates that the interactions between the bright and bound solitons as well as between the bright solitons and Kuznetsov–Ma breathers are elastic with certain conditions, while some other interactions are inelastic under other conditions. Propagation paths and velocities for the solitons are both affected by the dispersion coefficient function when the relations among the coefficients are linear, or affected by the dispersion coefficient, self-steepening coefficient and cubic nonlinearity functions when the relations among the coefficients are nonlinear. Under different conditions, bell-shaped solitons can evolve into the bound solitons or Kuznetsov–Ma breathers, respectively. Interactions between the bright and parabolic (or hyperbolic) solitons are related to the dispersion coefficient, self-steepening coefficient and cubic nonlinearity functions. Compression effect on the propagation paths of the solitons, caused by the dispersion coefficient, is observed.

Higher-order hybrid waves for the (2 + 1)-dimensional Boiti–Leon–Manna–Pempinelli equation for an irrotational incompressible fluid via the modified Pfaffian technique

Abstract: Fluids, as a phase of matter including liquids, gases and plasmas, are seen to be common in nature, the study of which helps the design in the related industries. In this paper, we optimize the Pfaffian technique and investigated the Boiti–Leon–Manna–Pempinelli equation for an irrotational incompressible fluid. Higher-order hybrid solutions consisting of the L lumps, M breathers and N solitons are constructed with L, M and N being positive integers. Relative extrema of the breather and lump are presented, respectively. Breather is found to be localized along the curve $$a_1 x+b_1 \varphi (y)+\omega _1 t+\xi _1=0$$ and periodic along the curve $$\alpha _1 x+\beta _1 \varphi (y)+\gamma _1 t+\theta _1=0$$ . Under the lump existence condition, higher-order rogue wave solutions do not exist. Hybrid solutions composed of breathers, lumps and solitons are illustrated graphically. It can be found that when certain parameters are chosen, the breather, lump and soliton included in the hybrid solutions possess the same properties as those of the breather and lump solutions.

Breathers and rogue waves on the periodic background for the Gerdjikov-Ivanov equation for the Alfvén waves in an astrophysical plasma

Abstract: Alfven waves in the magnetized astrophysical plasmas are invoked to explain the heating of the stellar coronae, acceleration of the stellar winds, as well as the origin and formation of the galactic and extragalactic jets. Under investigation in this paper is the Gerdjikov-Ivanov equation which is considered as a plasma-physics model for the Alfven waves propagating parallel to the ambient magnetic field. Based on the existing N-th order analytic solutions for the Gerdjikov-Ivanov equation with N being a positive integer, we construct two types of the breathers on the periodic background, and the k-th order rogue waves on the periodic background in terms of the determinant expression for the transverse magnetic field perturbation are derived via the Taylor expansion when N = 2 k + 1 with k being a non-negative integer. We show that the breathers and rogue waves are located in the periodic background rather than in the constant background. We obtain four different structures, which are the so-called fundamental, triangular, ring and ring-triangular structures, for the higher-order rogue waves on the periodic background. We graphically demonstrate some solutions for the transverse magnetic field perturbation to analyse the characteristics of the four different structures.

Lump, mixed lump-stripe and rogue wave-stripe solutions of a (3+1)-dimensional nonlinear wave equation for a liquid with gas bubbles

Abstract: Liquids with gas bubbles are very common in science, engineering, physics, nature and life. Under investigation in this paper is a (3+1)-dimensional nonlinear wave equation for a liquid with gas bubbles. With respect to the velocity of the liquid-gas bubble mixture, we obtain the lump, rogue wave, mixed lump-stripe soliton and mixed rogue wave-stripe soliton solutions via the symbolic computation. Based on the mixed lump-stripe soliton solutions, we construct the rogue wave and investigate the fusion and fission phenomena between a lump and the one-stripe soliton. We graphically study the mixed lump-stripe soliton under the influence of the parameters β , γ , δ and ξ , which represent the dispersion, perturbed effect, disturbed wave velocities along the y and z (i.e., the two transverse) directions, respectively. With the decreasing value of β to -1, the graph from a lump and one-stripe soliton shows a soliton; with the increasing value of γ to -3, location of the lump moves along the negative x axis; with the value of δ increasing to 0.5, location of the lump moves along the positive x axis; with the increasing value of ξ to -3, location and range of the lump soliton keep unchanged. With respect to the velocity of the mixture, we obtain the interaction between a rogue wave and a pair of stripe solitons according to mixed rogue wave-stripe soliton solutions. A lump is provided.

Shallow water in an open sea or a wide channel: Auto- and non-auto-Bäcklund transformations with solitons for a generalized (2+1)-dimensional dispersive long-wave system

Abstract: Oceanic water waves are actively studied. Hereby, taking into account the nonlinear and dispersive long gravity waves in two horizontal directions on the shallow water of an open sea or a wide channel of finite depth, we investigate a generalized (2+1)-dimensional dispersive long-wave system. With symbolic computation while with respect to the horizontal velocity and wave elevation above the undisturbed water surface, we work out two non-auto-Backlund transformations and two auto-Backlund transformations with solitons. All of our results are dependent on the constant coefficients in the original system.

Lie group analysis, solitons, self-adjointness and conservation laws of the modified Zakharov-Kuznetsov equation in an electron-positron-ion magnetoplasma

Abstract: Electron-positron-ion plasmas are found in the primordial Universe, active galactic nuclei, surroundings of black holes and peripheries of neutron stars. We focus our attention on a modified Zakharov-Kuznetsov (mZK) equation which describes the ion acoustic drift solitary waves in an electron-positron-ion magnetoplasma. Lie symmetry generators and groups are presented by virtue of the Lie symmetry method. Optimal system of the one-dimensional subalgebras is presented, which is influenced via the ratio of the unperturbed ion density to electron density nio/neo, the ratio of the unperturbed positron density to electron density npo/neo, the ratio of the electron temperature to positron temperature Te/Tp and the normalized ion drift velocity v o * . Based on the optimal system, we construct the power-series, multi-soliton, breather-like and periodic-wave solutions. Two types of the elastic interactions, including the overtaking and head-on interactions between (among) two (three) solitons are discussed. We find that the amplitudes of the solitons and periodic waves are positively related to the electron Debye length λDe and negatively related to |ρi| with ρi as the ion Larmor radius. Besides, we find that the mZK equation is not only strictly self-adjoint but also nonlinearly self-adjoint. Condition for the nonlinear self-adjointness is related to nio/neo, npo/neo, Te/Tp and v o * . Based on the nonlinear self-adjointness of the mZK equation, conservation laws, which are related to nio/neo, npo/neo, Te/Tp, v o * , λDe, ρi and may be associated with the conservation of momentum and energy, are obtained.