*Problem 5.5:*

Now we consider the following equation

$$\mathbf{u}^{\varepsilon} D\_t^{\delta} w(\mathbf{x}, y, t) = a\_1 \Delta w(\mathbf{x}, y, t) - b\_1 \sin(w(\mathbf{x}, y, t)), \quad (\mathbf{x}, y) \in \Phi\_\prime \quad t \in [0, T], \quad 1 < \delta \le 2,\tag{48}$$

where *a*1 and *b*1 are constants and the initial and boundary conditions are

$$\begin{cases} w(\mathbf{x}, y, 0) = \arctan\left( \exp(\frac{1}{2} - \sqrt{15\mathbf{x}^2 + 15y^2}) \right), \quad w\_l(\mathbf{x}, y, 0) = 0, \quad (\mathbf{x}, y) \in \breve{\Phi} = \Phi \cup \partial\Phi, \\ w(\mathbf{x}, y, t) = 0, \quad (\mathbf{x}, y) \in \partial\Phi, \quad t \in [0, T]. \end{cases} \tag{49}$$

Here, we examine the behaviour of circular ring soliton numerically. Due to pulsating behaviour, such waves are also known as pulsons. We choose different values of parameters *a*1, *b*1 to present surface plots to study the time evolution of the circular ring soliton. We observe the effect of *a*1 and *b*1 on solutions. In Figure 5, numerical solutions for different values of *a*1 and *b*1 have been plotted. Figure 6 shows the numerical solution for *a*1 = 0.05 while varying *b*1. In Figure 7 the results are plotted for *b*1 = 10, in which the wave peak value at the centre becomes lower as *a*1 increases. This reveals that the solitary wave moves in a stable way up to a large time under finite initial condition.

**Figure 5.** Graphical behaviour of problem 5.5 at *δ* = 1.9, *a*1 = 0.1, *b*1 = 10.

**Figure 6.** Graphical behaviour of problem 5.5 at *δ* = 1.9, *a*1 = 0.05.

**Figure 7.** Graphical behaviour of problem 5.5 at *δ* = 1.9, *b*1 = 10.
