Search

A (2+1)-dimensional generalized potential Kadomtsev-Petviashvili (gpKP) equation which possesses a Hirota bilinear form is constructed. The lump waves are derived by using a positive quadratic function solution. By combining an exponential function with a quadratic function, an interaction solution between a lump and a one-kink soliton is obtained. Furthermore, an interaction solution between a lump and a two-kink soliton is presented by mixing two exponential functions with a quadratic function. This type of lump wave just appears to a line $k_2x+k_3y+k_4t+k_5 \sim 0$. We call this kind of lump wave is a special rogue wave. Some visual figures are depicted to explain the propagation phenomena of these interaction solutions.

Based on bilinear formulation of a (3 + 1)-dimensional soliton equation, lump solution and related interaction solutions are investigated. The lump solutions of the soliton equation are classified into three cases with nonsingularity conditions being given. The interaction solutions between lump and a stripe soliton are obtained in eight cases, which have interesting fusing and fission behaviors with changing time. The interaction solutions of the soliton equation between a lump and a resonant pair of stripe solitons are also given, and we find that the lump just exist for a finite period during the interaction process.

A nonlocal form of a two-layer fluid system is proposed by a simple symmetry reduction, then by applying multiple scale method to it a general nonlocal two place variable coefficient modified KdV (VCmKdV) equation with shifted space and delayed time reversal is derived. Various exact solutions of the VCmKdV equation, including elliptic periodic waves, solitary waves and interaction solutions between solitons and periodic waves are obtained and analyzed graphically. As an illustration, an approximate solution of the original nonlocal two-layer fluid system is also given.

A novel material parameter-dependent model is proposed in this work to investigate the nonlinear vibration of a beam under a moving load within a finite deformation framework. For the planar vibration problem, the Lagrange strain is adopted and the resulting model equations for the beam are established by the Hamilton principle. Under appropriate assumptions, the coupled model equations are simplified into a nonlinear integro-partial differential equation which incorporates a material parameter and a geometrical parameter. The dynamic response of the beam is determined with the help of the Galerkin method. The solutions show that both the material parameter and geometrical parameter have the effect of reducing the amplitude of the forced vibration, increasing the speed of the fluctuation and the critical velocity for the moving load. In comparison with small deformation formulations, the effect of finite deformation herein is to reduce the vibration amplitude, especially for slender beams. If the vibration amplitude is relatively small, the nonlinear model may be replaced by the corresponding higher-order linear model while preserving the main features of the vibration problem.

The bosonic supersymmetric modified KdV (BSmKdV) system is obtained by the bosonization approach. The nonlocal symmetry for the BSmKdV equation is obtained by the truncated Painleve method. By introducing multiple new fields, the finite symmetry transformation for the BSmKdV equation is derived by applying Lie’s first principle to the prolonged systems. The similarity reductions related to the nonlocal symmetry are studied. The interaction solutions among the solitons and other complicated waves, including Painleve II waves and periodic cnoidal waves, are presented through the reduction theorems. The concrete soliton-cnoidal interaction solutions are illustrated in detail by using the mapping and deformation method.

The residual symmetry of the (2+1)-dimensional Bogoyavlenskii equation is obtained and localized to a Lie point symmetry by introducing new dependent variables to enlarge the system, then the corresponding finite transformation is obtained by using Lie’s first theorem. Furthermore, by introducing more dependent variables, the linear superposition of arbitrary number of residual symmetries is also localized and the corresponding finite transformations which are just N th Backlund transformations of the (2+1)-dimensional Bogoyavlenskii equation are obtained in determinants form with some concrete solutions explicitly given.

In this paper, the (2 $$+$$ 1)-dimensional Boussinesq equation is studied by applying residual symmetry reduction method and consistent Riccati expansion (CRE) method, respectively. By introducing multiple new dependent variables to enlarge the (2 $$+$$ 1)-dimensional Boussinesq system, the residual symmetry is localized and the corresponding finite transformation is obtained by using Lie’s first theorem. The symmetry reduction solutions related to the residual symmetry of the (2 $$+$$ 1)-dimensional Boussinesq equation is obtained by using the standard Lie symmetry method, which includes complicated interaction models. Furthermore, the (2 $$+$$ 1)-dimensional Boussinesq equation is found to have CRE integrability, and new Backlund transformations (BTs) are consequently obtained. New interaction solutions are obtained from these BTs; particularly, the interaction solution between soliton and background cnoidal wave is given and analyzed explicitly.

The residual symmetry of the (3 + 1) -dimensional Burgers system is localized to a Lie point symmetry in a prolonged system and the corresponding finite transformation is obtained by using Lie’s first theorem. By further localize the linear superposition of multiple residual symmetries, the N -th Backlund transformations (BT) of the (3 + 1) -dimensional Burgers system are also got. By applying the standard Lie symmetry method to the prolonged system, not only the Lie symmetry group but also the symmetry reduction solutions are obtained, which include abundant interaction solutions between solitons and nonlinear waves. Furthermore, the (3 + 1) -dimensional Burgers system is proved to have consistent Riccati expansion (CRE) property, based on which some new BTs are given.

The consistent Riccati expansion (CRE) is used to the Drinfel’d–Sokolov–Wilson (DSW) equation. It demonstrates that the DSW equation is the CRE solvability system. The bilinear form of the DSW equation is constructed by the truncated Painleve method. One rogue wave including the dark and bright types is explicitly found via choosing a quadratic function in the bilinear form of the DSW equation. The interactions between rational solutions and other complicated waves are generated by mixing the polynomial function with an arbitrary function. These types of rational solutions are analyzed graphically by selecting appropriate parameters.

Through truncated Painleve expansion of the (2 + 1)-dimensional Burgers system the residual symmetry is obtained and localized to a local one in an enlarged system by introducing new dependent variables. Using Lie’s first theorem, the Backlund transformation related to the localized residual symmetry is derived. Furthermore, the N-th-Backlund transformation of the (2 + 1)-dimensional Burgers system, which is expressed by determinants in a compact form, related to the symmetry of linear superposition of multiple residual symmetries is obtained through localization procedure.

A (2[Formula: see text]+[Formula: see text]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.

Residual symmetry of the (3+1)-dimensional breaking soliton equation is obtained and localized to a Lie point symmetry in a properly prolonged system. The general form of Lie point symmetry group and the corresponding symmetry reduction solutions of the prolonged system are obtained by using the standard Lie symmetry method, which include various interaction solutions between solitons and nonlinear background waves of the (3+1)-dimensional breaking soliton equation. Furthermore, the (3+1)-dimensional breaking soliton equation is proved to be integrable in the sense of having consistent Riccati expansion. Based on this property, some new Backlund transformations of the (3+1)-dimensional breaking soliton equation are obtained, from which interaction solutions between solitions and cnoidal waves are explicitly given.

In this Letter, on the condition that the generalized Hulthen-type scalar potential is not less than its vector potential, the exact bound state solutions of the s-wave Klein–Gordon equation of the generalized Hulthen potential are obtained in terms of the super-symmetric quantum mechanics. The conclusions contain the standard Hulthen and Wood–Saxon potentials.