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Abstract: Using an object reduction approach (a special conditional reduction approach), new types of variable separation solutions with arbitrary functions in a (2+1)-dimensional Broer–Kaup system are obtained. In terms of the derived variable separation solutions, abundant nonpropagating and propagating solitons such as dromions, rings, peakons and compactons etc. are revealed by prescribing appropriate functions in this paper. [Copyright &y& Elsevier]

Abstract: With the help of an extended mapping method and a linear variable separation method, new types of variable separation solutions (including solitary wave solutions, periodic wave solutions and rational function solutions) with two arbitrary functions for (2+1)-dimensional Korteweg–de Vries system (KdV) are derived. Usually, in terms of solitary wave solutions and rational function solutions, one can find some important localized excitations. However, based on the derived periodic wave solution in this paper, we find that some novel and significant localized coherent excitations such as dromions, peakons, stochastic fractal patterns, regular fractal patterns, chaotic line soliton patterns as well as chaotic patterns exist in the KdV system as considering appropriate boundary conditions and/or initial qualifications. [Copyright &y& Elsevier]

Abstract: Starting from a special variable transformation and an extended mapping approach, the (1+1)-dimensional Korteweg–de Vries (KdV) type model related to nonlinear Schrödinger system is solved, and families of variable separation solutions with arbitrary functions are derived. Then based on the derived exact solution, we reveal some periodic loop solitons and chaotic solitons for the (1+1)-dimensional KdV type model. [Copyright &y& Elsevier]

Abstract: Starting from a quite universal formula, which is valid for some quite universal (2+1)-dimensional physical models, Bai et al. [Bai CL, Bai CJ, Zhao H. Chaos, Solitons & Fractals 2005;25:481] have claimed that the interactions among different types of solitary waves like peakons, dromions, and compactons are investigated both analytically and graphically. We show that all of the claims by Bai et al. are wrong since the adopted q-functions do not satisfy the differential equation. From this point of view, the interactions for peakon–antidromion, compacton–antidromion, and antipeakon–compacton say nothing of completely inelastic or not completely elastic. [Copyright &y& Elsevier]

Abstract: Starting from a special Bäcklund transformation, Bai and Zhao [Bai CL, Zhao H. Chaos, Solitons & Fractals. doi: 10.1016/j.chaos.2006.02.005] have claimed that a general variable separation solution of the (2+1)-dimensional nonlinear Schrödinger equation is derived. Based on the quite universal variable separation solution which is valid for some (2+1)-dimensional physical models and by selecting appropriate functions, they have asserted that new types of interactions between the multi-valued solitary waves and the single-valued solitary waves, i.e., the fractal semifolded localized structures, are investigated. We show that several serious wrongs exist in their paper, and say nothing of the conclusion that some new fractal semifolded localized structures for the (2+1)-dimensional nonlinear Schrödinger equation have been found. [Copyright &y& Elsevier]

Abstract: With the help of an extended mapping approach, a new type of variable separation solution with two arbitrary functions of the (2+1)-dimensional generalized Broer–Kaup (GBK) system is derived. Based on the derived solitary wave excitation, we reveal some regular fractal and stochastic fractal patterns in the (2+1)-dimensional GBK system. [Copyright &y& Elsevier]

Abstract: Starting from a special Painlevé–Bäcklund transformation, the nonlinear generalized Broer–Kaup(GBK) system in (2+1)-dimensions is reduced to a linear system. Then by means of the linear superposition theorem, a general variable separation excitation to the generalized Broer–Kaup system is obtained. Finally, based on the derived solution, a new type of localized structure, i.e., semifolded localized coherent structure is constructed and some evolution properties of the novel semifolded localized structure are briefly discussed. [Copyright &y& Elsevier]

Abstract: Using an extended homogeneous balance approach and a linear variable separation method, a general variable separation excitation of the (2+1)-dimensional variable coefficient Broer–Kaup (VCBK) system is derived. Based on the derived solution, we reveal soliton fission and fusion phenomena in the (2+1)-dimensional soliton system. [Copyright &y& Elsevier]

Abstract: By means 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 localized structures such as dromion, ring, peakon and foldon etc. are re-revealed by selecting appropriate functions in this paper. [Copyright &y& Elsevier]

Abstract: By means of a so-called generalizing Riccati equation mapping method, Zhu [Zhu S D, Chaos, Solitons & Fractals; 2006. doi:10.1016/j.chaos.2006.10.015] has claimed that abundant new solutions to the (2+1)-dimensional Boiti–Leon–Pempinelle (BLP) equation are derived. Based on the derived variable separation solution and by selecting appropriate functions, he has asserted that abundant new non-travelling waves are obtained. We show that the generalizing Riccati equation mapping method is equivalent to the usual mapping approach, and say nothing of the conclusion that many new non-travelling wave solutions have been found. [Copyright &y& Elsevier]

Abstract: The mapping approach is a kind of classic, efficient and well-developed method to solve nonlinear evolution equations, the remarkable characteristic of which is that we can have many different ansatzs and thus end up with the abundance of solutions. In this paper, with the help of the improved mapping approach, we obtain new exact solutions (including solitary wave solutions, periodic wave solutions and variable separation solutions) of the (3+1)-dimensional Jimbo–Miwa system. [Copyright &y& Elsevier]

Abstract: By means of an extended mapping method and a variable separation method, a series of solitary wave solutions, periodic wave solutions and variable separation solutions to the (2+1)-dimensional breaking soliton system is derived. [Copyright &y& Elsevier]

Abstract: In this paper, we present a method of special conditional similarity reductions for nonlinear partial differential equations. As an concrete example of its applications in physical problems, we apply this method to the (2+1)-dimensional Broer–Kaup–Kupershmidt system with variable coefficients (VCBKK), which has the extensive physics background, and obtain abundant exact solutions derived from some reduction equations. [Copyright &y& Elsevier]

Starting from the standard truncated Painleve´ expansion, a Ba¨cklund transformation for the (2 + 1)-dimensional breaking soliton equation is derived. Making use of the variable separation approach, a variable separation solution of this model is obtained. By selecting appropriate multi-valued functions, a new class of localized coherent structures for the model are found. These structures exhibit interesting novel features not found in one-dimensions. [Copyright &y& Elsevier]