Periodic structures based on variable separation solution of the (2+1)-dimensional Boiti–Leon–Pempinelli equation

In this paper, firstly, a new mapping method is used to obtain the variable separation solutions, with two arbitrary functions, of the (2+1)-dimensional Boiti–Leon–Pempinelli equation. From the variable separation solution and by selecting appropriate functions, some novel Jacobian elliptic wave structure and periodic wave evolutional behaviors are investigated.     

Non-Lie symmetry groups and new exact solutions of a (2+1)-dimensional generalized Broer–Kaup system

By the modified CK’s direct method, the symmetry groups theorem of a (2+1)-dimensional generalized Broer–Kaup system is derived. Based upon the results, Lie point symmetry groups and new exact solutions of a (2+1)-dimensional generalized Broer–Kaup system are obtained.     

The non-traveling wave solutions and novel fractal soliton for the (2 + 1)-dimensional Broer–Kaup equations with variable coefficients

A method is proposed by extending the linear traveling wave transformation into the nonlinear transformation with the (G′/G)-expansion method. The non-traveling wave solutions with variable separation can be constructed for the (2 + 1)-dimensional Broer–Kaup equations with variable coefficients via the method. A novel class of fractal soliton, namely, the cross-like fractal soliton is observed by selecting appropriately the arbitrary functions in the solutions.     

New variable separation solutions and nonlinear phenomena for the (2+1)-dimensional modified Korteweg–de Vries equation

Variable separation approach, which is a powerful approach in the linear science, has been successfully generalized to the nonlinear science as nonlinear variable separation methods. The (2 + 1)-dimensional modified Korteweg–de Vries (mKdV) equation is hereby investigated, and new variable separation solutions are obtained by the truncated Painlevé expansion method and the extended tanh-function method. By choosing appropriate functions for the solution involving three low-dimensional arbitrary functions, which is derived by the truncated Painlevé expansion method, two kinds of nonlinear phenomena, namely, dromion reconstruction and soliton fission phenomena, are discussed.     

Obtaining multisoliton solutions of the (2+1)-dimensional Camassa–Holm system using Darboux transformations

Theoretical and Mathematical Physics - We construct and study Darboux transformations for the $$(2{+}1)$$ -dimensional Camassa–Holm system. We apply a reciprocal transformation that relates...     

A generalized Broer–Kaup (GBK) hierarchy and its expanding integrable system

A new Lax pair is first constructed. By making use of Tu scheme, a Lax integrable system is engendered. Since it can reduce to a generalized Broer–Kaup (GBK) system, we call it GBK hierarchy. Second, both Darboux transformations of the GBK system are obtained, which can generate new solutions. At last, an expanding integrable system of the GBK hierarchy, which is also an integrable coupling, is presented by using the direct sum relations and isomorphic relations between two subalgebras of a high order loop algebra $\stackrel{\sim }{G}$.     

Multiple lump solutions of the (2+1)-dimensional Konopelchenko–Dubrovsky equation

In this paper, multiple lump solutions of the (2+1)-dimensional Konopelchenko–Dubrovsky equation are obtained by means of the Hirota bilinear method. With the aid of positive quartic-quadratic-functions, we can get the 1-lump solutions, 3-lump solutions, and 6-lump solutions. Via the density plots and three-dimensional plots, the dynamic properties of multiple lump solutions are discussed by choosing the appropriate parameters. It is expected that our results are valuable for revealing the high-dimensional dynamic phenomenon of the nonlinear evolution equations.     

Differential form method for finding symmetries of a (2 + 1)-dimensional Camassa–Holm system based on its Lax pair

In this paper, we use the differential form method to seek Lie point symmetries of a (2 + 1)-dimensional Camassa–Holm (CH) system based on its Lax pair. Then we reduce both the system and its Lax pair with the obtained symmetries, as a result some reduced (1 + 1)-dimensional equations with their new Lax pairs are presented. At last, the conservation laws for the CH system are derived from a direct method.     

Solitary-wave solutions to a dual equation of the Kaup–Boussinesq system

In this paper, we employ the bifurcation theory of planar dynamical systems to investigate the travelling-wave solutions to a dual equation of the Kaup–Boussinesq system. The expressions for smooth solitary-wave solutions are obtained.     

On the Whitham system for the (2+1)-dimensional nonlinear Schrödinger equation

Whitham modulation equations are derived for the nonlinear Schrödinger equation in the plane ((2+1)-dimensional nonlinear Schrödinger [2d NLS]) with small dispersion. The modulation equations are obtained in terms of both physical and Riemann-type variables; the latter yields equations of hydrodynamic type. The complete 2d NLS Whitham system consists of six dynamical equations in evolutionary form and two constraints. As an application, we determine the linear stability of one-dimensional traveling waves. In both the elliptic and hyperbolic cases, the traveling waves are found to be unstable. This result is consistent with previous investigations of stability by other methods and is supported by direct numerical calculations.     

Dynamics and interaction scenarios of localized wave structures in the Kadomtsev–Petviashvili-based system

In this letter, we investigate the dynamics and various interaction scenarios of localized wave structures in the Kadomtsev–Petviashvili (KP)-based system. By using a combination of the Hirota’s bilinear method and the KP hierarchy reduction method, new families of determinant semi-rational solutions of the KP-based system are derived, including lump solitons and rogue-wave solitons. The generic interaction scenarios between distinct types of localized wave solutions are investigated. Our detailed study reveals different types of interaction phenomena: fusion of lumps and line solitons into line solitons, fission of line solitons into lumps and line solitons, a mixture of fission and fusion processes of lumps and line solitons, and the inelastic collision of line rogue waves and line solitons.     

Existence of positive solution for a sixth–order differential system with variable parameters

This paper investigates the existence of positive solutions for a sixth-order differential system with three variable parameters. Using a fixed point theorem and an operator spectral theorem we give some new existence results.     

New (2 + 1)-dimensional Sine–Gordon equation with self-consistent sources derived by the nonlinear variable separation method

By using the Hirota’s bilinear transformation method and direct variable separation assumption, a new (2 + 1)-dimensional Sine–Gordon equation with self-consistent sources is derived for the first time. Correspondingly, a nonlinear variable separation solution included two lower-dimensional arbitrary functions is obtained.