Invariant solutions of a nonlinear wave equation with a small dissipation obtained via approximate symmetries

Abstract

In this paper, it is shown how a combination of approximate symmetries of a nonlinear wave equation with small dissipations and singularity analysis provides exact analytic solutions. We perform the analysis using the Lie symmetry algebra of this equation and identify the conjugacy classes of the one-dimensional subalgebras of this Lie algebra. We show that the subalgebra classification of the integro-differential form of the nonlinear wave equation is much larger than the one obtained from the original wave equation. A systematic use of the symmetry reduction method allows us to find new invariant solutions of this wave equation.

Appendix: subalgebra classification for the integro-differential case

The Lie symmetry subalgebra for the integro-differential case given in Sect. 3 can be written as the semi-direct sum

(135)

The algebra $\{X_5,X_6\}$ is Abelian and its subalgebra classification is given by

$$\begin{aligned} \{0\},\qquad \{X_5\},\qquad \{X_6\},\qquad \{X_5+aX_6\}(a\ne 0),\qquad \{X_5,X_6\} \end{aligned}$$

(136)

Using the method of splitting and non-splitting subalgebras as given in [12], we classify the one-dimensional subalgebras of the semi-direct sum (135). A basis element for each one-dimensional invariant subalgebra of ${\mathcal {L}}$ is transformed by the Baker-Campbell-Hausdorff formula in order to determine which other invariant subalgebras it is conjugate to. For instance, if we consider the subalgebra $X=\{X_1\}$ and take an arbitrary element of the group generated by ${\mathcal {L}}$, $e^Y$, where Y is the generator

$$\begin{aligned} Y=\alpha X_1+\beta X_2+\gamma X_3+\delta X_4+\zeta X_5+\eta X_6 \end{aligned}$$

(137)

we obtain

$$\begin{aligned} e^YX_1e^{-Y}=X_1-\zeta X_1+\dfrac{\zeta ^2}{2\!}-\cdots = e^{-\zeta }X_1 \end{aligned}$$

(138)

so the subalgebra $\{X_1\}$ is conjugate only to itself. Applying this procedure to the other one-dimensional invariant subalgebras of ${\mathcal {L}}$, we obtain the following list of 63 one-dimensional subalgebras.

The following list constitutes the classification of the one-dimensional subalgebras of the symmetry Lie algebra for both cases of Eq. (120) (where the symbol $X_6$ represents the symmetry generator (123) or the symmetry generator (130) respectively) into conjugacy classes.

$$\begin{aligned} \begin{aligned}&{\mathcal {L}}_1=\{X_1\}, \quad {\mathcal {L}}_2=\{X_2\},\quad {\mathcal {L}}_3=\{X_1+\varepsilon X_2\}, \quad {\mathcal {L}}_4=\{X_3\},\quad {\mathcal {L}}_5=\{X_3+\varepsilon X_1\}, \\&{\mathcal {L}}_6=\{X_3+\varepsilon X_2\},\quad {\mathcal {L}}_7=\{X_3+\varepsilon X_1+aX_2\},\quad {\mathcal {L}}_{8}=\{X_4\}, {\mathcal {L}}_{9}=\{X_4+\varepsilon X_1\}, \\&{\mathcal {L}}_{10}=\{X_4+\varepsilon X_2\},\quad {\mathcal {L}}_{11}=\{X_4+\varepsilon X_1+aX_2\},\quad {\mathcal {L}}_{12}=\{X_4+\varepsilon X_3\}, \\&{\mathcal {L}}_{13}=\{X_4+\varepsilon X_3+aX_1\},\quad {\mathcal {L}}_{14}=\{X_4+\varepsilon X_3+aX_2\},\quad \\&\quad {\mathcal {L}}_{15}=\{X_4+\varepsilon X_3+aX_1+bX_2\}, \\&{\mathcal {L}}_{16}=\{X_5\},\quad {\mathcal {L}}_{17}=\{X_5+\varepsilon X_1\}, \quad {\mathcal {L}}_{18}=\{X_5+\varepsilon X_2\},\quad {\mathcal {L}}_{19}=\{X_5+\varepsilon X_1+aX_2\}, \\&{\mathcal {L}}_{20}=\{X_5+\varepsilon X_3\},\quad {\mathcal {L}}_{21}=\{X_5+\varepsilon X_3+aX_1\},\quad {\mathcal {L}}_{22}=\{X_5+\varepsilon X_3+aX_2\}, \\&{\mathcal {L}}_{23}=\{X_5+\varepsilon X_3+aX_1+bX_2\},\quad {\mathcal {L}}_{24}=\{X_5+\varepsilon X_4\},\quad {\mathcal {L}}_{25}=\{X_5+\varepsilon X_4+aX_1\}, \\&{\mathcal {L}}_{26}=\{X_5+\varepsilon X_4+aX_2\},\quad {\mathcal {L}}_{27}=\{X_5+\varepsilon X_4+aX_1+bX_2\},\quad \\&\quad {\mathcal {L}}_{28}=\{X_5+\varepsilon X_4+aX_3\}, \\&{\mathcal {L}}_{29}=\{X_5+\varepsilon X_4+aX_3+bX_1\},\quad {\mathcal {L}}_{30}=\{X_5+\varepsilon X_4+aX_3+bX_2\}, \\&{\mathcal {L}}_{31}=\{X_5+\varepsilon X_4+aX_3+bX_2+cX_1\},\quad {\mathcal {L}}_{32}=\{X_6\}, {\mathcal {L}}_{33}=\{X_6+\varepsilon X_1\}, \\&{\mathcal {L}}_{34}=\{X_6+\varepsilon X_2\},\quad {\mathcal {L}}_{35}=\{X_6+\varepsilon X_1+aX_2\},\quad {\mathcal {L}}_{36}=\{X_6+\varepsilon X_3\}, \\&{\mathcal {L}}_{37}=\{X_6+\varepsilon X_3+aX_1\},\quad {\mathcal {L}}_{38}=\{X_6+\varepsilon X_3+aX_2\},\quad \\&\quad {\mathcal {L}}_{39}=\{X_6+\varepsilon X_3+aX_1+bX_2\}, \\&{\mathcal {L}}_{40}=\{X_6+\varepsilon X_4\},\quad {\mathcal {L}}_{41}=\{X_6+\varepsilon X_4+aX_1\},\quad {\mathcal {L}}_{42}=\{X_6+\varepsilon X_4+aX_2\}, \\&{\mathcal {L}}_{43}=\{X_6+\varepsilon X_4+aX_1+bX_2\},\quad {\mathcal {L}}_{44}=\{X_6+\varepsilon X_4+aX_3\}, \\&{\mathcal {L}}_{45}=\{X_6+\varepsilon X_4+aX_3+bX_1\},\quad {\mathcal {L}}_{46}=\{X_6+\varepsilon X_4+aX_3+bX_2\}, \\&{\mathcal {L}}_{47}=\{X_6+\varepsilon X_4+aX_3+bX_1+cX_2\},\quad {\mathcal {L}}_{48}=\{X_5+aX_6\},\quad \\&\quad {\mathcal {L}}_{49}=\{X_5+aX_6+\varepsilon X_1\}, \\&{\mathcal {L}}_{50}=\{X_5+aX_6+\varepsilon X_2\},\quad {\mathcal {L}}_{51}=\{X_5+aX_6+\varepsilon X_1+bX_2\},\quad \\&\quad {\mathcal {L}}_{52}=\{X_5+aX_6+\varepsilon X_3\}, \\&{\mathcal {L}}_{53}=\{X_5+aX_6+\varepsilon X_3+bX_1\},\quad {\mathcal {L}}_{54}=\{X_5+aX_6+\varepsilon X_3+bX_2\}, \\&{\mathcal {L}}_{55}=\{X_5+aX_6+\varepsilon X_3+bX_1+cX_2\},\quad {\mathcal {L}}_{56}=\{X_5+aX_6+\varepsilon X_4\}, \\&{\mathcal {L}}_{57}=\{X_5+aX_6+\varepsilon X_4+bX_1\},\quad {\mathcal {L}}_{58}=\{X_5+aX_6+\varepsilon X_4+bX_2\}, \\&{\mathcal {L}}_{59}=\{X_5+aX_6+\varepsilon X_4+bX_1+cX_2\},\quad {\mathcal {L}}_{60}=\{X_5+aX_6+\varepsilon X_4+bX_3\},\quad \\&{\mathcal {L}}_{61}=\{X_5+aX_6+\varepsilon X_4+bX_3+cX_1\},\quad {\mathcal {L}}_{62}=\{X_5+aX_6+\varepsilon X_4+bX_3+cX_2\}, \\&{\mathcal {L}}_{63}=\{X_5+aX_6+\varepsilon X_4+bX_3+cX_1+dX_2\}, \end{aligned} \end{aligned}$$

The subalgebra structure of the integro-differential case is far more extensive than that of the three cases analyzed in Sect. 2.