The features of resonance phenomena are an important problem, both experimentally and theoretically. For the integrable systems, resonance in solitons may lead to various types of new excitations such as the breathers [1], the soliton fissions, the soliton fusions [2], the rational-exponential waves [3] and so on. The soliton molecule, which can be treated as the soliton bound state, has attracted considerable attention. The soliton molecules were first predicted theoretically in the framework of the nonlinear Schrödinger-Ginzburg-Landau equation [4] and the coupled nonlinear Schrödinger equations [5]. Recently, a new velocity resonance mechanism is introduced to form soliton molecules [6, 7]. For velocity resonance, high-order dispersive terms play a key role in the nonlinear integrable systems [6]. Based on the velocity resonance, the soliton molecules of the (2+1)-dimensional fifth-order Korteweg–de Vries (KdV) equation [8], the complex modified KdV equation [9] and the (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation [10] are constructed by using the Darboux transformation and the variable separation method. Meanwhile, some novel interactions between soliton molecules and breather solutions, and between soliton molecules and dromions, are presented by the velocity resonance mechanism [8–11]. The interaction between solitons and other nonlinear excitations plays an important topic [12–20]. The consistent Riccati expansion (CRE) or the consistent tanh expansion (CTE) method can be applied to find these types of solutions [21]. The method has been applied to various nonlinear systems, including the modified Kadomtsev-Petviashvili (KP) equation [22], the modified KdV-Calogero-Bogoyavlenkskii-Schiff equation [23], the supersymmetric integrable systems [24] and the non-integrable cubic generalised KP equation [25]. In this paper, the main purpose of our work is to construct soliton molecules, interaction between a soliton molecule and one-soliton, and interaction between solitons and cnoidal periodic wave solutions for an extended modified KdV (mKdV) equation.

In the sense of Marchant et al [26, 27], one can get the extended mKdV equation. The extended mKdV equation readswhere α and β are the third-order and fifth-order dispersion coefficients matching with the relevant nonlinear terms. The extended mKdV equation (1) describes the evolution of steeper waves with shorter wavelengths than in the mKdV equation [28]. The extended mKdV equation (1) becomes the focusing form and defocusing form with σ = +1 and σ = −1. The equation (1) gives rise to multiple soliton solutions and multiple singular soliton solutions for σ = +1 and σ = −1, respectively [28]. The extended mKdV equation (1) for the focusing form, i.e. σ = +1 readsThe extended mKdV equation of the focusing form (2) possesses the Lax pair, the infinitely conservation laws, the Darboux transformation, the Painlevé property and the multi-soliton solution [29]. The long-time asymptotic behavior for the extended mKdV equation (2) has been analyzed recently [30].

This paper is organized as follows. In section 2, the soliton molecules and interaction between a soliton molecule and one-soliton of the extended mKdV equation are obtained by a new resonance condition. In section 3, the nonlocal symmetry and its application of the extended mKdV equation can be constructed by the truncated Painlevé method. In section 4, the extended mKdV equation is used in the CRE method. The interacted one-soliton with periodic waves can be derived by using a nonauto-Bäcklund transformation theorem. Some concluding remarks will be made in the last section.

To determine the multi-soliton solution of the extended mKdV equation (2), the dependent variable transformation reads [29]The auxiliary functions of F(x, t) and G(x, t) for the three-soliton solution are selected aswith θ_i = k_ix + ω_it + c_i. By substituting (4) and (3) into (2), the phases shift a_ij and b₁₂₃ readand the dispersion relationIf the parameters a₁₃, a₂₃, b₁₂₃ and $\exp ({\theta }_3)$ in (4) select zero, the three-soliton solution will reduce to the two-soliton solution.

We shall study the soliton molecule with a new resonance condition. The new condition $(k_i\ne k_j)$ of velocity resonance readsThe ith and jth solitons of resonant condition (7) are bounded to form a soliton molecule or an asymmetric soliton by selecting the parameters in (7). We take the two-soliton and the new resonance condition to describe the soliton molecule in figures 1–3. For the figure 1, the parameters are selected asFor the figure 2, the parameters are selected asThe parameters are selected as the following form in figure 3 From the figures 1–3, the soliton molecule can be classified into three cases, i.e., k₁ > 0, k₂ > 0; k₁k₂ < 0 and k₁ < 0, k₂ < 0. It represents that the soliton molecule can consist of the bright and bright soliton, the dark and bright soliton, and the dark and dark soliton, respectively. The velocities of two solitons in the molecule are the same value with different amplitude.

To study the interaction between a soliton molecule and one-soliton, we use the three-soliton. We describe this phenomenon in figure 4. For figure 4, the parameters are selected asThe interaction between a soliton molecule and one-soliton is elastic from figure 4.

The nonlocal symmetry for the extended mKdV equation can be constructed with the truncated Painlevé method [31–36]. The Laurent series of the extended mKdV equation (2) readswhere the function f(x, t) = 0 is the equation of singularity manifold. The functions of u₀ and u₁ are functions of x and t. By balancing the coefficients of each power of f independently, we getand the Schwarzian extended mKdV formwhere K and the Schwarzian derivative C satisfyBy the truncated Painlevé method, the nonlocal symmetry of the extended mKdV equation readsThe nonlocal symmetry (16) is the residual of the singularity manifold f. To solve the initial value problem of the Lie’s first principle of the nonlocal symmetry, we localize the nonlocal symmetry to the Lie point symmetry by introducing a new fieldThe initial value problem related by the nonlocal symmetry (17) becomesBy solving the above initial value problem, the Bäcklund transformation (BT) theorem of the enlarged systems (2), (14) and (17) is obtained.

The results presented in this paper can be also applied to the defocusing form of the extended mKdV equation. The extended mKdV equation (1) becomes the defocusing form, i.e. σ = −1. In this part, we shall use the CRE/CTE method to study the defocusing form of the extended mKdV equation. Based on the CRE/CTE method [21], the truncated solution of the defocusing form of the extended mKdV equation reads as the following form by using leading order analysiswhere u₀, u₁ and f are arbitrary functions and R satisfies the Riccati equationwith a₀, a₁ and a₂ being arbitrary constants. Substituting (20) into (2) and balancing the coefficients of R lead toandwhere K(x, t) and C(x, t) satisfy (15). The CRE will be transformed to the CTE with a₀ = 1, a₁ = 0 and a₂ = −1. The special solution of the Riccati equation readsIt demonstrates that a CRE solvable model would be a CTE solvable model, and vice versa. We can construct the theorem, which is treated as a nonauto-BT theorem between the solution of f and u.

The solution of the extended mKdV equation with the defocusing form can be obtained by using the above theorem. We shall list two cases by means of the theorem.

Case I. A trivial solution of the defocusing form iswhere k is an arbitrary constant and ω is determined by the dispersion relation. The one-soliton solution of the defocusing form reads as the following form by using the nonauto-BT theorem

Case II. In order to obtain the interaction between one-soliton and other nonlinear excitations of the defocusing form, this type solution can be determined by a sum of a trivial solution and an arbitrary functionwhere k, ω, k₁ and ω₁ are arbitrary constants. The elliptic function equation can be obtained by substituting (28) into (23)withBy the nonauto-BT, the interaction between one-soliton and cnoidal waves of the defocusing form reads asIf we get the solution of the elliptic function equation (29), the explicit interaction between one-soliton and cnoidal periodic wave solutions for the defocusing form will be obtained by the expression (30).

In summary, the soliton molecules and some novel interaction solutions of the extended mKdV equation were studied by the velocity resonance mechanism and the CRE method. The soliton molecule and the interaction between a soliton molecule and one-soliton were constructed by the new resonance condition of velocity resonance. The interaction between a soliton molecule and one-soliton is elastic by some detail analysis. The nonlocal symmetry and the BT theorem of the extended mKdV equation were derived by the truncated Painlevé method. Based on the CRE/CTE method, the extended mKdV equation is a CRE/CTE solvable system. A nonauto-BT theorem was constructed by using the CRE/CTE method. The interaction between one-soliton and cnoidal periodic waves was given by the nonauto-BT theorem.