Abstract. Some types of coupled Korteweg de-Vries (KdV) equations are derived from an atmospheric dynamical system. In the derivation procedure, an unreasonable yaverage trick (which is usually adopted in literature) is removed. The derived models are classified via Painlevé test. Three types of τ -function solutions and multiple soliton solutions of the models are explicitly given by means of the exact solutions of the usual KdV equation. It is also interesting that for a non-Painlevé integrable coupled KdV system there may be multiple soliton solutions.

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Exact solutions of the (n + 1)-dimensional sine-Gordon field equation are studied with help of those of the cubic nonlinear Klein-Gordon fields. The mapping relations among the sine-Gordon field equation and the cubic nonlinear Klein-Gordon fields are pure algebraic. By solving the cubic nonlinear Klein-Gordon equations, many new types of exact explicit solutions such as the periodic-periodic interaction waves, periodic-kink interaction waves, periodic perturbed "snake" shape solitary waves, etc., are displayed both analytically and graphically.

Abstract. Darboux transformation is developed to systematically find variable separation solutions for the Nizhnik-Novikov-Veselov equation. Starting from a seed solution with some arbitrary functions, the once Darboux transformation yields the variable separable solutions which can be obtained from the truncated Painlevé analysis and the twice Darboux transformation leads to some new variable separable solutions which are the generalization of the known results obtained by using a guess ansatz to solve the generalized trilinear equation.

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The consistent tanh expansion (CTE) method is developed for the combined KdVmKdV equation. The combined KdV-mKdV equation is proved to be CTE solvable. New exact interaction solutions such as soliton-cnoidal wave solutions, soliton-periodic wave solutions for the combined KdV-mKdV equation are given out analytically and graphically. Ó 2016 The Authors. Production and hosting by Elsevier B.V. on behalf of University of Bahrain. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Nonlocal symmetries are obtained for the coupled integrable dispersionless (CID) equation. The CID equation is proved to be consistent, tanh-expansion solvable. New, exact interaction excitations such as solitoncnoidal wave solutions, soliton-periodic wave solutions, and multiple resonant soliton solutions are discussed analytically and shown graphically.

The nonlocal symmetry is derived for an equation combining the modified Calogero–Bogoyavlenskii–Schiff equation with its negative-order form from the truncated Painlevé expansion method. The nonlocal symmetries are localized to the Lie point symmetry by introducing new auxiliary dependent variables. The finite symmetry transformation and the Lie point symmetry for the prolonged system are solved directly. Many new interaction solutions among soliton and other types of interaction solutions for the modified Calogero–Bogoyavlenskii–Schiff equation with its negative-order form can be obtained from the consistent condition of the consistent tanh expansion method by selecting the proper arbitrary constants.