Soliton solution for a highly nonlinear difference-differential equation

Generalizing homotopy analysis method to solve Lotka–Volterra equation

“…Note that the solution series (22) contain the auxiliary parameter h, which we can choose properly by plotting the so-called h-curves to ensure solution series converge, as suggested by Liao [5]. In this way, we choose a valid value of h. Here we choose the auxiliary parameter h = −1 by h-curve.…”

“…It is important to ensure that the solution series (22) are convergent. Note that the solution series (22) contain the auxiliary parameter h, which we can choose properly by plotting the so-called h-curves to ensure solution series converge, as suggested by Liao [5].…”

“…We get all the solutions as follows: u n,0 (t) = n + t u n,1 (t) = h(n + 1)t + 3ht 2 u n,2 (t) = (1 + 6n)h(1 + h)t + (36h 2 n + 6h 2 + 3h)t 2 + 18h 2 t 3 · · · · · · · · · · · · · · · · · · · · · (22) Then the solution expression (14) can be written in an accurate form as follows u n (t) = +∞ m=1 a n,m (h)t m . To increase the accuracy and convergence of the solution, Liao [5] has applied the homotopy analysis method to provide an analytic solution for the classical problem of nonlinear progressive waves in deep water and developed a new technique, namely the Homotopy-Padé technique.…”

“…It should be emphasized that u n,m (t), (m ≥ 1) is governed by the linear equation (21) with the linear initial conditions (22). We get all the solutions as follows: u n,0 (t) = n + t u n,1 (t) = h(n + 1)t + 3ht 2 u n,2 (t) = (1 + 6n)h(1 + h)t + (36h 2 n + 6h 2 + 3h)t 2 + 18h 2 t 3 · · · · · · · · · · · · · · · · · · · · · (22) Then the solution expression (14) can be written in an accurate form as follows u n (t) = +∞ m=1 a n,m (h)t m .…”

“…Since the work of Fermi, Pasta and Ulam in the 1950s [14], there were quite a number of research works developed during the last decades on DDEs. For instance, Levi and his co-workers analyzed the condition for existence of higher symmetries for a class of DDEs [15,16], Yamilov and his coworkers [17,18] made outstanding contribution to the classification of DDEs, integrability tests and connections between integrable PDEs and DDEs [19,20] and a lot of works were developed to analyze the properties of solutions of DDEs [21][22][23][24][25][26].…”