In this paper, we study fixed points of solutions of the differential equation f" + A1 (z) f' + A0 (z) f = 0, where Aj (z) ( ≡ ≠ 0) (j = 0,1) are transcendental meromorphic functions with finite order. Instead of looking at the zeros of f (z) - z, we proceed to a slight generalization by considering zeros of g (z) -φ(z), where g is a differential polynomial in f with polynomial coefficients,φ is a small meromorphic function relative to f, while the solution f is of infinite order.
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The complex method is systematic and powerful to build various kinds of exact meromorphic solutions for nonlinear partial differential equations on the complex plane C. By using the complex method, abundant new exact meromorphic solutions to the (2 + 1)-dimensional and the (3 + 1)-dimensional Boiti-Leon-Manna-Pempinelli equations and the (2 + 1)-dimension Kundu-Mukherjee-Naskar equation are investigated. Abundant new elliptic solutions, rational solutions and exponential solutions have been constructed.
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