Wronskian,Pfaffian and periodic wave solutions for a $$(2 + 1)$$ ( 2 + 1 ) -dimensional extended shallow water wave equation

Nonlinear Dynamics - Under investigation in this paper is a $$(2 + 1)$$ -dimensional extended shallow water wave equation. Bilinear form is obtained via the generalized dependent variable...     

New nonautonomous combined multi-wave solutions for ( $$\varvec{2+1}$$ 2 + 1 )-dimensional variable coefficients KdV equation

Nonlinear Dynamics - A variety of new types of nonautonomous combined multi-wave solutions of the ( $$2+1$$ )-dimensional variable coefficients KdV equation is derived by means of the generalized...     

Lump solutions of a ( $$2+1$$ 2 + 1 )-dimensional bSK equation

Nonlinear Dynamics - In this paper, we obtained a kind of lump solutions of ( $$2+1$$ )-dimensional bSK equation with the assistance of Mathematica by using the Hirota bilinear method. These lump...     

Degeneration of solitons for a ( $$3+1$$ 3 + 1 )-dimensional generalized nonlinear evolution equation for shallow water waves

Nonlinear Dynamics - A ( $$3+1$$ )-dimensional generalized shallow water waves equation is investigated with different methods. Based on symbolic computation and Hirota bilinear form, N-soliton...     

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

Nonlinear Dynamics - 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...     

Semi-rational solutions for a $$(2+1)$$ ( 2 + 1 ) -dimensional Davey–Stewartson system on the surface water waves of finite depth

Nonlinear Dynamics - Under investigation in this work is a $$(2+1)$$ -dimensional Davey–Stewartson system, which describes the surface water wave packets of finite depth. With respect to the...     

Riemann theta solutions and their asymptotic property for a (3 $$+$$ + 1)-dimensional water wave equation

Nonlinear Dynamics - In this paper, we consider the (3 $$+$$ 1)-dimensional water wave equation $$u_{yzt}+u_{xxxyz}-6u_{x}u_{xyz}-6u_{xy}u_{xz}=0.$$ Based on Bell polynomials, we obtain its Hirota...     

M-lump and interactive solutions to a (3 $${+}$$ + 1)-dimensional nonlinear system

Nonlinear Dynamics - This paper aims at computing M-lump solutions for the $$(3+1)$$ -dimensional nonlinear evolution equation. These solutions in all directions decline to an identical state...     

Multi-symplectic method for the generalized (2+1)-dimensional KdV-mKdV equation

In the present paper,a general solution involving three arbitrary functions for the generalized(2+1)dimensional KdV-mKdV equation,which is derived fromthe generalized(1+1)-dimensional KdV-mKdV equation,is first introduced by means of the Wiess,Tabor,Carnevale(WTC) truncation method.And then multisymplectic formulations with several conservation lawstaken into account are presented for the generalized(2+1)dimensional KdV-mKdV equation based on the multisymplectic theory of Bridges.Subsequently,in order tosimulate the periodic wave solutions in terms of rationalfunctions of the Jacobi elliptic functions derived from thegeneral solution,a semi-implicit multi-symplectic schemeis constructed that is equivalent to the Preissmann scheme.From the results of the numerical experiments,we can conclude that the multi-symplectic schemes can accurately simulate the periodic wave solutions of the generalized(2+1)dimensional KdV-mKdV equation while preserve approximately the conservation laws.     

Painlevé analysis,auto-Bäcklund transformation and analytic solutions of a (2 $$+$$ + 1)-dimensional generalized Burgers system with the variable coefficients in a fluid

Nonlinear Dynamics - Burgers-type equations are used to describe certain phenomena in gas dynamics, traffic flow, plasma astrophysics and ocean dynamics. In this paper, a (2 $$+$$ 1)-dimensional...     

Residual symmetry,Bäcklund transformation and CRE solvability of a ( $$\mathbf{2}{\varvec{+}}{} \mathbf{1}$$ 2 + 1 )-dimensional nonlinear system

Nonlinear Dynamics - In this paper, the truncated Painlevé expansion is employed to derive a Bäcklund transformation of a ( $$2+1$$ )-dimensional nonlinear system. This system can be...     

Localized coherent structures of the (2+1)-dimensional higher order Broer-Kaup equations

ＩｎｔｒｏｄｕｃｔｉｏｎＳｏｌｉｔｏｎｉｓａｃｏｍｐｌｉｃａｔｅｄｍａｔｈｅｍａｔｉｃａｌｓｔｒｕｃｔｕｒｅｂａｓｅｄｏｎｔｈｅｎｏｎｌｉｎｅａｒｅｖｏｌｕｔｉｏｎｅｑｕａｔｉｏｎ[1].Ｔｈｏｕｇｈｓｏｌｉｔｏｎｓｔｒｕｃｔｕｒｅｓａｎｄｐｒｏｐｅｒｔｉｅｓｏｆｔｈｅ ( 1 + 1 )＿ｄｉｍｅｎｓｉｏｎａｌｎｏｎｌｉｎｅａｒｐｈｙｓｉｃａｌｍｏｄｅｌｓｈａｖｅｂｅｅｎｓｔｕｄｉｅｄｗｅｌｌａｎｄｕｎｄｅｒｓｔｏｏｄｆｕｒｔｈｅｒ,ｔｈｅｓｏｌｉｔｏｎｓｔｒｕｃｔｕｒｅｓｉｎｈｉｇｈｅｒｓｐａｔｉａｌｄｉ…     

State transition of lump-type waves for the (2+1)-dimensional generalized KdV equation

Nonlinear Dynamics - In this paper, we are mainly concerned with the (2+1)-dimensional generalized Korteweg–de Vries equation in fluid dynamics. Based on the translation transformation and...     

Bäcklund transformation,multiple wave solutions and lump solutions to a (3 + 1)-dimensional nonlinear evolution equation

Nonlinear Dynamics - In this paper, a $$(3+1)$$ -dimensional nonlinear evolution equation is cast into Hirota bilinear form with a dependent variable transformation. A bilinear Bäcklund...     

Pseudopotentials for non-linear evolution equations in 2 + 1 dimensions

In this paper the concept of pseudopotential is generalized to non-linear evolution equations in 2 + 1 dimensions. If the equations satisfied by the pseudopotential are of a Riccati-type in the x-variable, it is shown how to obtain both the generalized AKNS system and the auto-Bäcklund transformation for the corresponding non-linear evolution equation. Several examples are given: Kadomtsev-Petviashvili, modified Kadomtsev-Petviashvili, (2 + 1 dim.)-Harry-Dym, and (2 + 1 dim.)-Sawada-Kotera equations.     

Bright soliton solutions of the (2+1)-dimensional generalized coupled nonlinear Schrödinger equation with the four-wave mixing term

Nonlinear Dynamics - The (2+1)-dimensional generalized coupled nonlinear Schrödinger equation with the four-wave mixing (FWM) term is studied in this paper, which describes the optical...     

Topological Conjugation Classes of Tightly Transitive Subgroups of $$\text {Homeo}_{+}(\mathbb {S}^1)$$ Homeo + ( S 1 )

Journal of Dynamics and Differential Equations - Let $$\text {Homeo}_{+}(\mathbb {S}^1)$$ denote the group of orientation preserving homeomorphisms of the circle $$\mathbb {S}^1$$ . A subgroup G of...     

Nonlinear interaction of solitary waves described by multi-rational wave solutions of the (2 $$+$$ + 1)-dimensional Kadomtsev–Petviashvili equation with variable coefficients

Nonlinear Dynamics - In this paper, the generalized unified method is used to construct multi-rational wave solutions of the ( $$2 + 1$$ )-dimensional Kadomtsev–Petviashvili equation with...     

A new method to the (2+1)-dimensional modified KP equation

By means of the auxiliary ordinary differential equation method, we have obtained many solitary wave solutions, periodic wave solutions and variable separation solutions for the (2+1)-dimensional KP equation. Using a mixed method, many exact solutions have been obtained.     

Solving the $$\mathbf{(3+1) }$$ ( 3 + 1 ) -dimensional KP–Boussinesq and BKP–Boussinesq equations by the simplified Hirota’s method

Nonlinear Dynamics - We study two (3 $$+$$ 1)-dimensional generalized equations, namely the Kadomtsev–Petviashvili–Boussinesq equation and the B-type...