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An improved homogeneous balance principle and an F-expansion technique are used to construct exact self-similar solutions to the cubic-quintic nonlinear Schrodinger equation. Such solutions exist under certain conditions, and impose constraints on the functions describing dispersion, nonlinearity, and the external potential. Some simple self-similar waves are presented.

With the help of an objective reduction approach (ORA), abundant exact solutions of (2+1)-dimensional higher-order Boussinesq system (including some hyperboloid function solutions, trigonometric function solutions, and a rational function solution) are obtained. It is shown that some novel soliton structures, like single linearity soliton structure, breath soliton structure, single linearity y-periodic solitary wave structure, libration dromion structure, and kink-like multisoliton structure with actual physical meaning exist in the (2+1)-dimensional higher-order Boussinesq system.

Starting from an improved mapping approach and a linear variable separation approach, a new family of exact solutions (including solitary wave solutions, periodic wave solutions and rational function solutions) with arbitrary functions for a general (2+1)-dimensional Korteweg de Vries system (GKdV) is derived. According to the derived solutions, we obtain some novel dromion-lattice solitons, complex wave excitations and chaotic patterns for the GKdV system.

Starting from the extended mapping approach and a linear variable separation method, we find new families of variable separation solutions with some arbitrary functions for the (3+1)-dimensional Burgers system. Then based on the derived exact solutions, some novel and interesting localized coherent excitations such as embedded-solitons, taper-like soliton, complex wave excitations in the periodic wave background are revealed by introducing appropriate boundary conditions and/or initial qualifications. The evolutional properties of the complex wave excitations are briefly investigated.

By means of an extended mapping approach and a linear variable separation approach, a new family of exact solutions of the (3+1)-dimensional Jimbo–Miwa system is derived. Based on the derived solitary wave solution, we obtain some special localized excitations and study the interactions between two solitary waves of the system.

Using the mapping approach via the projective Riccati equations, several types of variable separated solutions of the (2+1)-dimensional Nizhnik–Novikov–Veselov equation are obtained, including multiple-soliton solutions, periodic-soliton solutions, and Weierstrass function solutions. Based on a periodic-soliton solution, a new type of localized excitation, i.e., the four-dromion soliton, is constructed and some evolutional properties of this localized structure are briefly discussed.

In this paper, we present an object reduction for nonlinear partial differential equations. As a concrete example of its applications in physical problems, this method is applied to the (2+1)-dimensional Boiti–Leon–Pempinelli system, which has the extensive physics background, and an abundance of exact solutions is derived from some reduction equations. Based on the derived solutions, the localized structures under the periodic wave background are obtained.

Starting from an improved projective method and a linear variable separation approach, new families of variable separation solutions (including solitary wave solutions, periodic wave solutions and rational function solutions) with arbitrary functions for the (2+1)-dimensional generalized Broer–Kaup (GBK) system are derived. Usually, in terms of solitary wave solutions and/or rational function solutions, one can find abundant important localized excitations. However, based on the derived periodic wave solution in this paper, we reveal some complex wave excitations in the (2+1)-dimensional GBK system, which describe solitons moving on a periodic wave background. Some interesting evolutional properties for these solitary waves propagating on the periodic wave background are also briefly discussed.

With the help of an extended mapping approach and a linear variable separation method, new families of variable separation solutions with arbitrary functions for the (3+1)-dimensional Burgers system are derived. Based on the derived exact solutions, some novel and interesting localized coherent excitations such as embed-solitons are revealed by selecting appropriate boundary conditions and/or initial qualifications. The time evolutional properties of the novel localized excitation are also briefly investigated.

For a higher-dimensional integrable nonlinear dynamical system, there are abundant coherent soliton excitations. With the aid of an improved projective Riccati equation approach, the paper obtains several types of exact solutions to the (2+1)-dimensional dispersive long-wave equation, including multiple-soliton solutions, periodic soliton solutions, and Weierstrass function solutions. From these solutions, apart from several multisoliton excitations, we derive some novel features of wave structures by introducing some types of lower-dimensional patterns.

Using a special Painleve–Backlund transformation as well as the extended mapping approach and the linear superposition theorem, we obtain new families of variable separation solutions to the (2+1)-dimensional generalized Broer–Kaup (GBK) system. Based on the derived exact solution, we reveal some novel evolutional behaviors of localized excitations, i.e. fission and fusion phenomena in the (2+1)-dimensional GBK system.

In this paper, the extended projective approach, which was recently presented and successfully used in some continuous nonlinear physical systems, is generalized to nonlinear partial differential-difference systems (DDEs). As a concrete example, new families of exact solutions to the (2+1)-dimensional Toda lattice system are obtained by the extended projective approach.

In this paper, we extend the mapping approach to the N-order Schrodinger equation. In terms of the extended mapping approach, new families of variable separation solutions with some arbitrary functions are derived.

Starting from a special variable transformation and with the help of an extended mapping approach, the high-order Schrodinger equation (n = 3,4) is solved. A new family of variable separation solutions with arbitrary functions is derived.

Starting from an extended mapping approach, a new type of variable separation solution with arbitrary functions of generalized (2+1)-dimensional Broer–Kaup system (GBK) system is derived. Then based on the derived solitary wave solution, we obtain some specific chaotic solitons to the (2+1)-dimensional GBK system.

Starting from a Backlund transformation and taking a special ansatz for the function f, we can obtain a much more general expression of solution that includes some variable separated functions for the higher-order Broer–Kaup system. From this expression, we investigate the interactions of localized coherent structures such as the multi-solitonic excitations and find the novel phenomenon that their interactions have non-elastic behavior because the fission/fusion may occur after the interaction of each localized coherent structure.

Using an extended projective method, a new type of variable separation solution with two arbitrary functions of the (2+1)-dimensional generalized Broer-Kaup system (GBK) is derived. Based on the derived variable separation solution, some special localized coherent soliton excitations with or without elastic behaviors such as dromions, peakons, and foldons etc. are revealed by selecting appropriate functions in this paper.

The linear variable separation approach is successfully extended to (1+1)-dimensional Korteweg-de Vries (KdV) type models related to Schrodinger system. Some significant types of solitons such as compacton, peakon, and loop solutions with periodic behavior are simultaneously derived from the (1+1)-dimensional soliton system by entrancing appropriate piecewise smooth functions and multivalued functions.

By means of a special Painleve–Backlund transformation and a multilinear variable separation approach, an exact solution with arbitrary functions of the (2+1)-dimensional Boiti–Leon–Pempinelli system (BLP) is derived. Based on the derived variable separation solution, we obtain some special soliton fission and fusion solutions for the higher dimensional BLP system.

With the help of an extended mapping approach, a series of new types of exact excitations with two arbitrary functions of the (2+1)-dimensional Broer–Kaup–Kupershmidt (BKK) system is derived. Based on the derived solitary wave excitation, some specific soliton fission and fusion solutions of the higher-dimensional BKK system are also obtained.

With the help of an extended mapping approach, a series of new types of exact excitations with two arbitrary functions of the (2+1)-dimensional Broer-Kaup-Kupershmidt (BKK) system is derived. Based on the derived solitary wave excitation, some specific soliton fission and fusion solutions of the higher-dimensional BKK system are also obtained.

With the help of an extended mapping approach, a new type of variable separation excitation with three arbitrary functions of the (2+1)-dimensional dispersive long-water wave system (DLW) is derived. Based on the derived variable separation excitation, abundant non-propagating solitons such as dromion, ring, peakon, and compacton etc. are revealed by selecting appropriate functions in this paper.

In this work, an adaptation of the tanh /tan-method that is discussed usually in the nonlinear partial differential equations is presented to solve nonlinear polynomial differential-difference equations. As a concrete example, several solitary wave and periodic wave solutions for the chain which is related to the relativistic Toda lattice are derived. Some systems of the differential-difference equations that can be solved using our approach are listed and a discussion is given in conclusion.

A general mapping deformation method is applied to a generalized variable coefficient KdV equation. Many new types of exact solutions, including solitary wave solutions, periodic wave solutions, Jacobian and Weierstrass doubly periodic wave solutions and other exact excitations are obtained by the use of a simple algebraic transformation relation between the generalized variable coefficient KdV equation and a generalized cubic nonlinear Klein–Gordon equation.

By means of the standard truncated Painleve expansion and a special Backlund transformation, the higher-dimensional coupled Burgers system (HDCB) is reduced to a linear equation, and an exact multisoliton excitation is derived. The evolution properties of the multisoliton excitation are investigated and some novel features or interesting behaviors are revealed. The results show that after interactions for dromion-dromion, solitoff-solitoff, and solitoff-dromion, they are combined with some new types of localized structures, which are similar to classic particles with completely nonelastic behaviors.

An extended Jacobian elliptic function expansion method is presented and successfully applied to the nonlinear Schrodinger (NLS) equation and Zakharov equation. We obtain some new solutions besides Fu et al's results. The results show that our method is more powerful to construct Jacobian elliptic function and can be applied to other nonlinear physics systems.

A general mapping deformation method is presented and applied to a (2+1)-dimensional Boussinesq system. Many new types of explicit and exact travelling wave solutions, which contain solitary wave solutions, periodic wave solutions, Jacobian and Weierstrass doubly periodic wave solutions, and other exact excitations like polynomial solutions, exponential solutions, and rational solutions, etc., are obtained by a simple algebraic transformation relation between the (2+1)-dimensional Boussinesq equation and a generalized cubic nonlinear Klein–Gordon equation.

Considering that folded phenomena are rather universal in nature and some arbitrary functions can be included in the exact excitations of many (2+1)-dimensional soliton systems, we use adequate multivalued functions to construct folded solitary structures in multi-dimensions. Based on some interesting variable separation results in the literature, a common formula with arbitrary functions has been derived for suitable physical quantities of some significant (2+1)-dimensional soliton systems like the generalized Ablowitz?Kaup?Newell?Segur (GAKNS) model, the generalized Nizhnik?Novikov?Veselov (GNNV) system and the new (2+1)-dimensional long dispersive wave (NLDW) system. Then a new special type of two-dimensional solitary wave structure, i.e. the folded solitary wave and foldon, is obtained. The novel structure exhibits interesting features not found in the single valued solitary excitations.

Starting from the known variable separation excitations of a (2+1)-dimensional generalized Ablowitz–Kaup–Newell–Segur system, rich coherent structures can be derived. The interactions among different types of solitary waves like peakons, dromions, and compactons are investigated and some novel features or interesting behaviors are revealed. The results show that the interactions for peakon-dromion, compacton-dromion, and peakon-compacton may be completely nonelastic or completely elastic.

By means of the standard truncated Painleve expansion and a variable separation approach, a general variable separation solution of the generalized Burgers system is derived. In addition to the usual localized coherent soliton excitations like dromions, lumps, rings, breathers, instantons, oscillating soliton excitations, peakons, foldons, and previously revealed chaotic and fractal localized solutions, some new types of excitations — compacton and Jacobi periodic wave solutions are obtained by introducing appropriate lower dimensional piecewise smooth functions and Jacobi elliptic functions.

Starting from a special Backlund transform and a variable separation approach, a quite general variable separation solution of the generalized (2+1)-dimensional perturbed nonlinear Schrodinger system is obtained. In addition to the single-valued localized coherent soliton excitations like dromions, breathers, instantons, peakons, and previously revealed chaotic localized solution, a new type of multi-valued (folded) localized excitation is derived by introducing some appropriate lower-dimensional multiple valued functions.

In this work, we reveal a novel phenomenon that the localized coherent structures of some (2+1)-dimensional physical models possess chaotic and fractal behaviors. To clarify these interesting phenomena, we take the (2+1)-dimensional modified dispersive water-wave system as a concrete example. Starting from a variable separation approach, a general variable separation solution of this system is derived. Besides the stable localized coherent soliton excitations like dromions, lumps, rings, peakons, and oscillating soliton excitations, some new excitations with chaotic and fractal behaviors are derived by introducing some types of lower dimensional chaotic and fractal patterns.

Starting from the standard truncated Painleve expansion and a variable separation approach, a general variable separation solution of the breaking soliton system is derived. In addition to the usual localized coherent soliton excitations like dromions, lumps, rings, breathers, instantons, oscillating soliton excitations, and previously revealed chaotic and fractal localized solutions, some new types of excitations, peakons and foldons, are obtained by introducing appropriate lower-dimensional piecewise smooth functions and multiple valued functions.

In the previous Letter (Zheng C L and Zhang J F 2002, Chin. Phys. Lett., 19 1399), a localized excitation of the generalized Ablowitz-Kaup-Newell-Segur (GAKNS) system was obtained via the standard Painleve truncated expansion and a special variable separation approach. In this work, starting from a new variable separation approach, a more general variable separation excitation of this system is derived. The abundance of the localized coherent soliton excitations like dromions, lumps, rings, peakons and oscillating soliton excitations can be constructed by introducing appropriate lower-dimensional soliton patterns. Meanwhile we discuss two kinds of interactions of solitons. One is the interaction between the travelling peakon type soliton excitations, which is not completely elastic. The other is the interaction between the travelling ring type soliton excitations, which is completely elastic.

With the help of variable separation approach, a quite general excitation of a new (2+1)-dimensional long dispersive wave system is derived. The chaotic behaviour, such as chaotic line soliton patterns, chaotic dromion patterns, chaotic-period patterns, and chaotic-chaotic patterns, in some new localized excitations are found by selecting appropriate functions.

By means of variable separation approach, quite a general excitation of the new (2 + 1)-dimensional long dispersive wave system: is derived. Some types of the usual localized excitations such as dromions, lumps, rings, and oscillating soliton excitations can be easily constructed by selecting the arbitrary functions appropriately. Besides these usual localized structures, some new localized excitations like fractal-dromion, fractal-lump, and multi-peakon excitations of this new system are found by selecting appropriate functions.

Considering that there are abundant coherent soliton excitations in high dimensions, we reveal a novel phenomenon that the localized excitations possess chaotic and fractal behaviour in some (2+1)-dimensional soliton systems. To clarify the interesting phenomenon, we take the generalized (2+1)-dimensional Nizhnik-Novikov-Vesselov system as a concrete example. A quite general variable separation solutions of this system is derived via a variable separation approach first, then some new excitations like chaos and fractals are derived by introducing some types of lower-dimensional chaotic and fractal patterns.

A simple and direct method is applied to solving the (2+1)-dimensional perturbed Ablowitz–Kaup–Newell–Segur system (PAKNS). Starting from a special Backlund transformation and the variable separation approach, we convert the PAKNS system into the simple forms, which are four variable separation equations, then obtain a quite general solution. Some special localized coherent structures like fractal dromions and fractal lumps of this model are constructed by selecting some types of lower-dimensional fractal patterns.