Solutions with separated variables and breather structures in the -dimensional nonlinear systems

New Exact Solutions of (1 + 1)-Dimensional Coupled Integrable Dispersionless System

“…where R ≡ R (x, y, z, t), α i ≡ α i (x, y, z, t), β i ≡ β i (x, y, z, t), γ i ≡ γ i (x, y, z, t), (i = −1, 0, 1), and ϕ satisfies Eq. ( 1) with its solutions ( 14)-( 16), (22), and (23).…”

“…Although the acquisition of the variable separation solutions to (2 + 1)-dimensional NLEEs is greatly successful, one thought that it is difficult to obtain the variable separation solutions of (1 + 1)-dimensional NLEEs in the past several years. However, recently the extension of the multilinear variable separation approach to (1 + 1)dimensional nonlinear models was presented by Zhang et al [22] In this paper, at first we use the exp-function method [11−17] to seek new exact solutions of the Riccati equation ( 1), then employ the Riccati equation ( 1) and its new exact solutions to find new and more general variable separation solutions of the coupled integrable dispersionless (CID) equations…”

Multi-linear variable separation approach to nonlinear systems

“…By using this method, many (2+1)-dimensional mathematical physical models have been solved, such as the NNV equation, the asymmetric NNV (ANNV) equation, the asymmetric DS (ADS) equation, the dispersive long wave equation (DLWE), the Broer-Kaup-Kupershmidt (BKK) system, the higher order BKK system, the nonintegrable (2+1)-dimensional KdV equation, the long wave-short wave interaction model (LWSWIM), the Maccari system, the Burgers equation, the 2DsG system, the general (N + M )-component AKNS system, and so on [16,19,[21][22][23][24][25][26][27][28] (and references therein). In addition, the method has also been applied to several (1+1)-dimensional systems such as the negative KdV hierarchy, the Ito system, the shallow water wave equations, the long-wave-short-wave resonant interaction equation [29], etc., (3+1)-dimensional systems such as the Burgers equation [30], the JM equation [31], etc., and differential difference systems such as a (2+1)-dimensional special Toda equation [32], a (2+1)-dimensional differential-difference asymmetric NNV equation [33], a (1+1)-dimensional differentialdifference Toda-like equation [34], etc. It is discovered that the MLVSS for differential-difference systems share a similar form (1) with the only difference that p (or q) being a difference function.…”

Solitary wave fission and fusion in the (2+1)-dimensional generalized Broer–Kaup system

“…As one of the effective methods in linear physics, the variable separation approach (VSA) has been successfully extended to nonlinear domains. The multilinear variable separation approach (MLVSA) has also been established for various (1+1)-dimensional [11], (2 + 1)-dimensional [12] and (3 + 1)-dimensional [13] models. Recently, along with linear variable separation idea, the extended tanh-function method (ETM) [14][15][16] and the general projective Riccati equation method [17] based on mapping method have also been successfully generalized to obtain variable separation solutions for many (1+1)-dimensional, (2 + 1)-dimensional and (3 + 1)-dimensional models.…”

“…(10). Therefore, W 0 , P 0 and Q can be derived by solving (10) and (11), respectively, while the function P is considered as arbitrary function of {x, t}. Therefore we obtain another special exact excitation…”