A note on "New kink-shaped solutions and periodic wave solutions for the (2 + 1)-dimensional Sine-Gordon equation"

Exact solutions of the Nizhnik-Novikov-Veselov equation by Li [New kink-shaped solutions and periodic wave solutions for the (2+1)-dimensional Sine-Gordon equation, Appl. Math. Comput. 215 (2009). 3777-3781 ] are analyzed. We have observed that fourteen solutions by Li from thirty do not satisfy the equation. The other sixteen exact solutions by Li can be found from the general solutions of the well – known solution of the equation for the Weierstrass elliptic function. In the paper [1], Li looked for the exact solutions of the nonlinear evolution equation ut + uxxx + uyyy + 3(u∂−1 y ux)x + 3(u∂ −1 x uy)y = 0. (1) Author believes that Eq. (1) is the (2+1) dimensional Sine-Gordon equation. However, Eq. (1) was first derived in work [2] and now this equation is called as the Nizhnik-Novikov-Veselov equation [3, 4]. Author [1] have used the traveling wave transformation u(x, y, t) = v(ξ) in Eq. (1), where ξ = kx + ly + wt. From Eq. (1) he obtained the nonlinear ordinary differential equations kl(k + l3)v′′ + 3(k + l)v + klwv = 0. (2) In fact, the general solution of Eq. (2) is well known. Let us present this solution. Multiplying Eq. (2) on v′ and integrating with respect to ξ we obtain the equation in the form (v′)2 = − 2 kl v − ω k3 + l3 v + c1, (3) where c1 is an arbitrary constant. Substituting v(ξ) = −2 k l ℘(ξ)− k l w 6 (k3 + l3) . (4) into Eq. (3) we have the equation for the Weierstrass elliptic function in the form (℘′)2 = 4℘ − g2℘− g3, (5)