The evolutionary standard NPDEs involve a first-order partial derivative in respect with time and describe the unidirectional motion of a single wave. The dual/two-mode equations are NPDEs of second order in time, and govern the propagation of two wave modes, in the same direction simultaneously, with the same dispersion relation but with different phase velocity, linear and nonlinear parameters. At present, investigations on the two-mode problems are mainly based on the Korsunsky proposed method [
2]. It shows that for deriving the two-mode PDEs, it is necessary to collect, as two distinct components, the nonlinear terms
and the linear terms
, other than
. For more information and developments that have been achieved for two-mode PDEs, we recommend the following articles and the references therein [
3,
4,
5,
6]. The dynamics of the two-mode KdV equation associated to the standard mode third order KdV equation, was studied by various analytical methods: the reductive perturbation [
7], the Hamiltonian system [
8], or the Bell polynomials [
9]. In [
10], it was found that the two modes are solitons that separate without any change of their initial shapes and velocities except for the phase shifts after each collision. Also in [
11] the bright, dark, periodic, and singular-periodic dual-wave solutions are constructed for the two-mode Sawada-Kotera equation arising in fluids by the modified Kudryashov and new auxiliary equation methods. A finite series in terms of tanh-sech functions is proposed as a suggested solution for dual-mode version of the nonlinear Schrödinger equation [
12]. More exactly dual-mode dark and singular soliton solutions have been obtained. The innovative tanh-expansion method and Kudryashov technique are used in [
13] to the dual-mode Kadomtsev-Petviashvili equation to find the necessary constraint conditions that guarantee the existence of soliton solutions. Multiple kink solutions are pointed out in [
14] for the two-mode Sharma-Tasso-Olver equation, and for the two-mode fourth-order Burgers equation by using the Cole-Hopf transformation combined with the simplified Hirota’s method. Three different techniques including the tanh-expansion method, the rational sine–cosine method and the Kudryashov-expansion method have been applied in [
15] in order to study the dynamic behaviours for a dual-mode generalized Hirota–Satsuma coupled KdV system.
The contributions of this work are two-fold. First, we find explicit dual-waves solutions for the dual/two-mode Caudrey-Dodd-Gibbon (TMCDG) equation for arbitrary nonlinearity and dispersion parameters,
and
. Previosly, only the case
was considered in [
16], using the Hirota method. Second, we study the influence of the mentioned parameters as well as of
which stands for phase velocity, on the wave propagations, showing how the dual-wave propagation depends on them.
The paper is organized as it follows: After the Introduction, in section 2, an overview on the general form of the TMCDG equation is provided. In section 3 we present basic facts on the Kudryashov method [
17,
18] and the exponential expansion method [
19]. The findings of our investigation, when the previous methods were applied to TMCDG equation, are pointed out in section 4. A general discussion and some graphical representations of the acquired solutions are presented in section 5.
Section 6 is dedicated to some conclusions and final remarks.