Elastic and nonelastic interactional solutions for the (2 + 1)-dimensional Ito equation

Abstract

In this paper, based on the bilinear form and two new test functions, for the (2 + 1)-dimensional Ito equation, we obtain non-elastic interactional solutions composed of three different types of waves including the solitary wave, the periodic wave and the lump wave, and elastic interactional solutions with the solitary wave and the lump wave. We also obtain the periodic wave solutions and two-lump wave solutions.

Keywords:

• (2 + 1)-dimension Ito equation
• solitary wave
• periodic wave
• lump wave
• interactional solution

1. Introduction

The exact solutions of non-linear evolution equations have attracted much attention due to their wide applications in fluid mechanics, optics and plasma. There are many methods to obtain exact solutions, such as the inverse scattering method (Ablowitz & Segur, 1981), the Bäklund transformation method (Rogers & Schief, 2002), the Darboux transformation method (Gu et al., 2005) and the Hirota bilinear method (Hirota, 2004), etc. The Hirota bilinear method is a practical method in finding multiple soliton solutions, rational solutions and periodic solutions. In this paper, we will study the exact solutions of the (2 + 1)-dimensional Ito equation by using of the Hirota bilinear method with two new test functions.

In Ito (1980), with the transformation $u=2{(\hspace{0.17em}\text{ln}\hspace{0.17em}f)}_{\mathit{xx}}$, from the bilinear form (1) $$(D_t^2+D_t{D}_x^3)f·f=0,$$(1) where $$D_x^m{D}_t^{n}f·g=({\partial }_x-{\partial }_{x\prime })({\partial }_t-{\partial }_{t\prime })f(x,t)g(x\prime ,t\prime ){|}_{x\prime =x,t\prime =t},$$

Ito obtained the (1 + 1)-dimensional Ito equation (2) $$u_{\mathit{tt}}+u_{\mathit{xxxt}}+3(2u_x{u}_t+u{u}_{\mathit{xt}})+3u_{\mathit{xx}}{\int }_{-\infty }^x{u}_{t}dx\prime =0,$$(2) which has the same 1-soliton solution as that of the KdV equation. However, the multi-soliton solutions differ in the phase shift. Some explicit solutions of the (1 + 1)-dimensional Ito Equation (2)(2) $$u_{\mathit{tt}}+u_{\mathit{xxxt}}+3(2u_x{u}_t+u{u}_{\mathit{xt}})+3u_{\mathit{xx}}{\int }_{-\infty }^x{u}_{t}dx\prime =0,$$(2) have been obtained in Hu and Li (1991); Zhang and Chen (1991); Li and Zeng (2007). In WazWaz (2008), with the generalized bilinear form of (1) (3) $$(D_t^2+D_t{D}_x^3+\alpha D_t{D}_y+\beta D_t{D}_x)f·f=0,$$(3) and through the transformation (4) $$u(x,y,t)=2{(\hspace{0.17em}\text{ln}\hspace{0.17em}f(x,y,t\left)\right)}_{\mathit{xx}},$$(4)

Wazwaz presented the (2 + 1)-dimensional Ito equation (5) $$u_{\mathit{tt}}+u_{\mathit{xxxt}}+3(2u_x{u}_t+u{u}_{\mathit{xt}})+3u_{\mathit{xx}}{\int }_{-\infty }^x{u}_{t}dx\prime +\alpha u_{\mathit{yt}}+\beta u_{\mathit{xt}}=0,$$(5) where α and β are constants.

For the (2 + 1)-dimensional Ito Equation (5)(5) $$u_{\mathit{tt}}+u_{\mathit{xxxt}}+3(2u_x{u}_t+u{u}_{\mathit{xt}})+3u_{\mathit{xx}}{\int }_{-\infty }^x{u}_{t}dx\prime +\alpha u_{\mathit{yt}}+\beta u_{\mathit{xt}}=0,$$(5) , based on the bilinear form and different test functions, different types of solutions have been obtained. In WazWaz (2008), Wazwaz obtained multiple-soliton solutions by using exp-type test function. In Li and Zhao (2009), Li and Zhao obtained periodic solutions and soliton solutions with exp + cos + exp-type test function. The three-wave solutions were obtained in Zhao, Dai, and Wang (2010) with exp + cos + sinh-type test function. The breather-wave and the rational rogue wave solutions were provided in Wang, Tian, Qin, and Zhang (2017) with exp + cos-type and exp-type test function. In Tang, Tao, and Guan (2016); Yang, Ma, and Qin (2017); Ma, Yong, and Zhang (2017); Zou, Yu, Tian, Feng, and Li (2018), the lump and the lump-soliton solutions were obtained with rational-type, rational + exp-type and rational + cosh-type test functions.

Solitary wave solutions, periodic wave solutions and lump wave solutions of non-linear evolution equations are of great significance in theory and practice (see Chen, Yan, & Zhang, 2003; Wazwaz, 2016; Zhang & Chen, 2016; Zhang, Liu, & Ren, 2008; Luo, Duan, Liu, & Liu, 2010; Ma, 2015; Zhang, Dong, Zhang, & Yang, 2017). The interactions of solitary waves, periodic waves and lump waves for non-linear evolution equations have attracted much attention in Kaup (1981); Fokas, Pelinovsky, and Sulem (2001); Lu, Tian, and Grimshaw (2004); Baronio, Degasperis, Conforti, and Wabnitz (2012); Rao, Cheng, and He (2017); Huang and Chen (2017); Wang, Wang, Zhang, and Temuer (2017). For the above mentioned exact solutions of the (2 + 1)-dimensional Ito Equation (5)(5) $$u_{\mathit{tt}}+u_{\mathit{xxxt}}+3(2u_x{u}_t+u{u}_{\mathit{xt}})+3u_{\mathit{xx}}{\int }_{-\infty }^x{u}_{t}dx\prime +\alpha u_{\mathit{yt}}+\beta u_{\mathit{xt}}=0,$$(5) , the interactions of solitary waves are elastic and the interactions of different types of waves are inelastic. The interactions consist of only two types of waves in solitary wave, periodic wave and lump wave.

There are two aims in our work. One aim is to obtain periodic solutions and non-elastic interactional solutions composed of three different types of waves including the solitary wave, the periodic wave and the lump wave by constructing the rational + cosh + cos-type test function. This is described in Section 2. The other aim is to obtain an elastic interactional solution with the solitary wave and the lump wave by constructing the rational + rational*exp type test function and this is studied in detail in Section 3. Finally, we give our conclusions in Section 4.

2. Periodic wave solutions and nonelastic interactional solutions

In this section, based on the bilinear form (3)(3) $$(D_t^2+D_t{D}_x^3+\alpha D_t{D}_y+\beta D_t{D}_x)f·f=0,$$(3) , we construct rational + cosh + cos-type test function and discuss periodic wave solutions and interactional solutions with the solitary wave, the periodic wave and the lump wave for the (2 + 1)-dimensional Ito Equation (5)(5) $$u_{\mathit{tt}}+u_{\mathit{xxxt}}+3(2u_x{u}_t+u{u}_{\mathit{xt}})+3u_{\mathit{xx}}{\int }_{-\infty }^x{u}_{t}dx\prime +\alpha u_{\mathit{yt}}+\beta u_{\mathit{xt}}=0,$$(5) .

We assume the test function has the following form (6) $$\begin{array}{c}f={(a_{1}x+a_{2}y+a_{3}t+a_4)}^2+{(a_{5}x+a_{6}y+a_{7}t+a_8)}^2+a_9\\ +p\cosh (m_{1}x+m_{2}y+m_{3}t)+q\hspace{0.17em}\text{cos}\hspace{0.17em}(k_{1}x+k_{2}y+k_{3}t),\end{array}$$(6) where a_i $(i=1,2,\dots ,9)$, k_j (j = 1, 2, 3), p and q are real constants, m_j (j = 1, 2, 3) are real or pure imaginary constants and are real without specification. If a_i = 0 $(i=1,2,\dots ,9)$, the test function f is similar to that in Li (2009), Zhao (2010), Wang (2017), and Zou et al. (2018). If p = 0 or q = 0, the test function f is similar to that in Tang et al. (2016), Yang et al. (2017), and Ma et al. (2017).

Substituting (6)(6) $$\begin{array}{c}f={(a_{1}x+a_{2}y+a_{3}t+a_4)}^2+{(a_{5}x+a_{6}y+a_{7}t+a_8)}^2+a_9\\ +p\cosh (m_{1}x+m_{2}y+m_{3}t)+q\hspace{0.17em}\text{cos}\hspace{0.17em}(k_{1}x+k_{2}y+k_{3}t),\end{array}$$(6) into the bilinear form (3)(3) $$(D_t^2+D_t{D}_x^3+\alpha D_t{D}_y+\beta D_t{D}_x)f·f=0,$$(3) and equating all the coefficients of independent variables to zeros, we get a set of algebraic equations: (7) $$\beta a_1+\alpha a_2+a_3=0,\beta a_5+\alpha a_6+a_7=0,a_1{a}_3+a_5{a}_7=0,$$(7) (8) $$m_1{m}_3=0,k_1{k}_3=0,\beta m_1+m_1^3+\alpha m_2+m_3=0,\beta k_1-k_1^3+\alpha k_2+k_3=0.$$(8)

We note that the relations in Equation (7)(7) $$\beta a_1+\alpha a_2+a_3=0,\beta a_5+\alpha a_6+a_7=0,a_1{a}_3+a_5{a}_7=0,$$(7) are the same as those in Yang et al. (2017), Ma et al. (2017), and Zou et al. (2018). The Equation (8)(8) $$m_1{m}_3=0,k_1{k}_3=0,\beta m_1+m_1^3+\alpha m_2+m_3=0,\beta k_1-k_1^3+\alpha k_2+k_3=0.$$(8) indicates four cases (9) $$\begin{array}{l}\left(1\right)m_1=0,k_1=0,m_3=-\alpha m_2,k_3=-\alpha k_2;\\ \left(2\right)m_3=0,k_3=0,\beta m_1+m_1^3+\alpha m_2=0,\beta k_1-k_1^3+\alpha k_2=0;\\ \left(3\right)m_1=0,k_3=0,m_3=-\alpha m_2,\beta k_1-k_1^3+\alpha k_2=0;\\ \left(4\right)m_3=0,k_1=0,\beta m_1+m_1^3+\alpha m_2=0,k_3=-\alpha k_2.\end{array}$$(9)

The case of p = 0 or q = 0 corresponds to that in Yang et al. (2017) and Ma et al. (2017). If $p·q\ne 0$, $\text{cos}\hspace{0.17em}(k_{1}x+k_{2}y+k_{3}t)$ and $\cosh (m_{1}x+m_{2}y+m_{3}t)$ satisfy the principle of superposition of f.

From Equations (7)(7) $$\beta a_1+\alpha a_2+a_3=0,\beta a_5+\alpha a_6+a_7=0,a_1{a}_3+a_5{a}_7=0,$$(7) and (8)(8) $$m_1{m}_3=0,k_1{k}_3=0,\beta m_1+m_1^3+\alpha m_2+m_3=0,\beta k_1-k_1^3+\alpha k_2+k_3=0.$$(8) , we can obtain many different types of exact solutions of the Ito Equation (5)(5) $$u_{\mathit{tt}}+u_{\mathit{xxxt}}+3(2u_x{u}_t+u{u}_{\mathit{xt}})+3u_{\mathit{xx}}{\int }_{-\infty }^x{u}_{t}dx\prime +\alpha u_{\mathit{yt}}+\beta u_{\mathit{xt}}=0,$$(5) . In the following, we analyse two cases in detail.

Case 1. Periodic wave solutions

In this case, we take (10) $$f_1=a_9+p\cosh \left(m_{1}x-\frac{m_1^3+\beta m_1}{\alpha }y\right)+q\hspace{0.17em}\text{cos}\hspace{0.17em}\left(\frac{k_3}{\alpha }y-k_{3}t\right).$$(10)

If m₁ is a real constant, we get a new t-periodic wave solution of the Ito Equation (5)(5) $$u_{\mathit{tt}}+u_{\mathit{xxxt}}+3(2u_x{u}_t+u{u}_{\mathit{xt}})+3u_{\mathit{xx}}{\int }_{-\infty }^x{u}_{t}dx\prime +\alpha u_{\mathit{yt}}+\beta u_{\mathit{xt}}=0,$$(5) (11) $$u_1=\frac{2p{m}_1^2\left(p+q\cosh \left(m_{1}x-\frac{m_1^3+\beta m_1}{\alpha }y\right)\hspace{0.17em}\text{cos}\hspace{0.17em}\left(\frac{k_3}{\alpha }y-k_{3}t\right)+a_9\cosh \left(m_{1}x-\frac{m_1^3+\beta m_1}{\alpha }y\right)\right)}{{\left(a_9+p\cosh \left(m_{1}x-\frac{m_1^3+\beta m_1}{\alpha }y\right)+q\hspace{0.17em}\text{cos}\hspace{0.17em}\left(\frac{k_3}{\alpha }y-k_{3}t\right)\right)}^2}.$$(11)

when $a_9>0$, p > 0, $-a_9<q<a_9$, this solution is non-singular. For given y = 0, taking α = 2, β = 5, $a_9=6,m_1=2,k_3=1$, p = 1, q = 5, the period of the t-periodic wave solution (11)(11) $$u_1=\frac{2p{m}_1^2\left(p+q\cosh \left(m_{1}x-\frac{m_1^3+\beta m_1}{\alpha }y\right)\hspace{0.17em}\text{cos}\hspace{0.17em}\left(\frac{k_3}{\alpha }y-k_{3}t\right)+a_9\cosh \left(m_{1}x-\frac{m_1^3+\beta m_1}{\alpha }y\right)\right)}{{\left(a_9+p\cosh \left(m_{1}x-\frac{m_1^3+\beta m_1}{\alpha }y\right)+q\hspace{0.17em}\text{cos}\hspace{0.17em}\left(\frac{k_3}{\alpha }y-k_{3}t\right)\right)}^2}.$$(11) is $T=2\pi /k_3$ (see Figure 1(a)).

Figure 1. Plots of the t-periodic solution (11) and the double periodic solution (12) for the Ito equation with α = 2, β = 5, $a_9=6,k_3=1$, p = 1, y = 0: (a) t-periodic solution; (b) x − t double periodic solution. (a) m₁ = 2, q = 5. (b) ${\tilde{m}}_1=2$, q = 3.

If $m_1=\mathrm{i}{\tilde{m}}_1$ and ${\tilde{m}}_1$ is a real constant, we find a double periodic wave solution of the Ito Equation (5)(5) $$u_{\mathit{tt}}+u_{\mathit{xxxt}}+3(2u_x{u}_t+u{u}_{\mathit{xt}})+3u_{\mathit{xx}}{\int }_{-\infty }^x{u}_{t}dx\prime +\alpha u_{\mathit{yt}}+\beta u_{\mathit{xt}}=0,$$(5) (12) $${\tilde{u}}_1=\frac{-2p{\tilde{m}}_1^2\left(p+q\hspace{0.17em}\text{cos}\hspace{0.17em}\left({\tilde{m}}_{1}x+\frac{{\tilde{m}}_1^3-\beta {\tilde{m}}_1}{\alpha }y\right)\hspace{0.17em}\text{cos}\hspace{0.17em}\left(\frac{k_3}{\alpha }y-k_{3}t\right)+a_9\hspace{0.17em}\text{cos}\hspace{0.17em}\left({\tilde{m}}_{1}x+\frac{{\tilde{m}}_1^3-\beta {\tilde{m}}_1}{\alpha }y\right)\right)}{{\left(a_9+p\hspace{0.17em}\text{cos}\hspace{0.17em}\left({\tilde{m}}_{1}x+\frac{{\tilde{m}}_1^3-\beta {\tilde{m}}_1}{\alpha }y\right)+q\hspace{0.17em}\text{cos}\hspace{0.17em}\left(\frac{k_3}{\alpha }y-k_{3}t\right)\right)}^2}.$$(12)

when $a_9>0,\left|p\right|+\left|q\right|<a_9$, this solution is non-singular. Taking α = 2, β = 5, $a_9=6,{\tilde{m}}_1=2,k_3=1$, p = 1, q = 3, the periods of the x-t double periodic wave solution (12)(12) $${\tilde{u}}_1=\frac{-2p{\tilde{m}}_1^2\left(p+q\hspace{0.17em}\text{cos}\hspace{0.17em}\left({\tilde{m}}_{1}x+\frac{{\tilde{m}}_1^3-\beta {\tilde{m}}_1}{\alpha }y\right)\hspace{0.17em}\text{cos}\hspace{0.17em}\left(\frac{k_3}{\alpha }y-k_{3}t\right)+a_9\hspace{0.17em}\text{cos}\hspace{0.17em}\left({\tilde{m}}_{1}x+\frac{{\tilde{m}}_1^3-\beta {\tilde{m}}_1}{\alpha }y\right)\right)}{{\left(a_9+p\hspace{0.17em}\text{cos}\hspace{0.17em}\left({\tilde{m}}_{1}x+\frac{{\tilde{m}}_1^3-\beta {\tilde{m}}_1}{\alpha }y\right)+q\hspace{0.17em}\text{cos}\hspace{0.17em}\left(\frac{k_3}{\alpha }y-k_{3}t\right)\right)}^2}.$$(12) of the Ito Equation (5)(5) $$u_{\mathit{tt}}+u_{\mathit{xxxt}}+3(2u_x{u}_t+u{u}_{\mathit{xt}})+3u_{\mathit{xx}}{\int }_{-\infty }^x{u}_{t}dx\prime +\alpha u_{\mathit{yt}}+\beta u_{\mathit{xt}}=0,$$(5) are $T_1=2\pi /k_3$ and $T_2=2\pi /{\tilde{m}}_1$ (see Figure 1(b)).

Case 2. Non-elastic interactional solutions

In this case, we take (13) $$\begin{array}{c}{f}_2={\left(a_{1}x+a_{2}y+a_{3}t+a_4\right)}^2+{\left(a_{5}x+a_{6}y+a_{7}t+a_8\right)}^2+a_9\\ +p\cosh \left(m_{1}x-\frac{m_1^3+\beta m_1}{\alpha }y\right)+q\hspace{0.17em}\text{cos}\hspace{0.17em}\left(k_{1}x+\frac{k_1^3-\beta k_1}{\alpha }y\right),\end{array}$$(13) where a_i $(i=1,2,\dots ,9)$ are defined by (7)(7) $$\beta a_1+\alpha a_2+a_3=0,\beta a_5+\alpha a_6+a_7=0,a_1{a}_3+a_5{a}_7=0,$$(7) .

Through the transformation (4)(4) $$u(x,y,t)=2{(\hspace{0.17em}\text{ln}\hspace{0.17em}f(x,y,t\left)\right)}_{\mathit{xx}},$$(4) , we get a new solution of the Ito Equation (5)(5) $$u_{\mathit{tt}}+u_{\mathit{xxxt}}+3(2u_x{u}_t+u{u}_{\mathit{xt}})+3u_{\mathit{xx}}{\int }_{-\infty }^x{u}_{t}dx\prime +\alpha u_{\mathit{yt}}+\beta u_{\mathit{xt}}=0,$$(5) (14) $$u_2(x,y,t)=2{(\hspace{0.17em}\text{ln}\hspace{0.17em}{f}_2)}_{\mathit{xx}}=\frac{2\left(f_{2xx}{f}_2-f_{2x}^2\right)}{f_2^2}.$$(14)

For the polynomial function with $p=q=0$, as $a_1{a}_6-a_2{a}_5\ne 0$ and $a_9>0$, we find a lump solution and one can refer to Yang et al. (2017); Ma et al. (2017); Zou et al. (2018) for its properties.

If we take (15) $$\begin{array}{l}\alpha =5,\beta =1,a_1=1,a_2=3,a_3=-16,a_4=-1,a_5=3,\\ a_6=-\frac{5}{3},a_7=\frac{16}{3},a_8=2,a_9=25,m_1=1,k_1=\frac{3}{2},\end{array}$$(15) for given t = 0, this solution describes the non-elastic interactions of three different types of waves including a lump wave, a periodic wave and a solitary wave (see Figure 2). When $\left|t\right|$ increases, the semi-lump wave (see Figure 2(b)) which describes non-elastic interactions of a lump wave and a periodic wave tends to be swallowed by a solitary wave (see Figure 3).

Figure 2. Plots of the non-elastic interactional solution (14) with (13) and (15) for the Ito equation. (a) p = 0, q = 0, t = 0. (b) p = 0, q = 5, t = 0. (c) p = 4, q = 5, t = 0.

Figure 3. Plots of the non-elastic interactional solution (14) with (13) and (15) for the Ito equation. (a) p = 4, q = 5, t = −4. (b) p = 4, q = 5, t = −2. (c) p = 4, q = 5, t = 2. (d) p = 4, q = 5, t = 4.

If we take (16) $$\begin{array}{l}\alpha =1,\beta =3,a_1=a_5=0,a_2=-a_3=4,a_4=1,\\ a_6=-a_7=3,a_8=5,a_9=10,m_1=2,k_1=\frac{3}{2},\end{array}$$(16) for $a_1{a}_6-a_2{a}_5=0$, this solution describes the non-elastic interactions of a solitary wave and a periodic wave (see Figure 4).

Figure 4. Plots of the non-elastic interactional solution (14) with (13) and (16) for the Ito equation. (a) p = 1/100, q = 0, t = 0. (b) p = 0, q = 5, t = 0. (c) p = 1/100, q = 5, t = 0.

3. Elastic interactional solutions and two-lump wave solution

In this section, we propose the following novel test function (17) $$f=g+h{\mathrm{e}}^{\xi },\hspace{1em}\xi =c_{1}x+c_{2}y+c_{3}t,$$(17) with (18) $$g=a_1{(x+a_{2}y+a_3)}^2+a_4{(y+a_{5}t+a_6)}^2+a_7,h=b_1{(x+b_{2}y+b_3)}^2+b_4{(y+b_{5}t+b_6)}^2+b_7,$$(18) where a_i, b_i $(i=1,2,\dots ,7)$ and c_j (j = 1, 2, 3) are real constants. Substituting (17)(17) $$f=g+h{\mathrm{e}}^{\xi },\hspace{1em}\xi =c_{1}x+c_{2}y+c_{3}t,$$(17) into the bilinear form (3)(3) $$(D_t^2+D_t{D}_x^3+\alpha D_t{D}_y+\beta D_t{D}_x)f·f=0,$$(3) and equating all the coefficients of independent variables to zeros, we obtain a set of parameters: (19) $$\xi =c_{1}x+c_{2}y-(c_1^3+\beta c_1+\alpha c_2)t,g=a_1{\left(x-\frac{\beta }{\alpha }y+a_3\right)}^2+a_4{\left(y-\alpha t+a_6\right)}^2+a_7,$$(19) (20) $$h=b_1{\left(x-\frac{\beta }{\alpha }y+a_3-\frac{2}{c_1}\right)}^2+\frac{a_4{b}_1}{a_1}{\left(y-\alpha t+a_6-\frac{2\alpha }{c_1^3+\beta c_1+\alpha c_2}\right)}^2+\frac{a_7{b}_1}{a_1}.$$(20)

In order to get non-singular solution of the Ito Equation (5)(5) $$u_{\mathit{tt}}+u_{\mathit{xxxt}}+3(2u_x{u}_t+u{u}_{\mathit{xt}})+3u_{\mathit{xx}}{\int }_{-\infty }^x{u}_{t}dx\prime +\alpha u_{\mathit{yt}}+\beta u_{\mathit{xt}}=0,$$(5) , we suppose $a_1>0,a_4>0,a_7>0$ and $b_1>0$.

Note that g, h, $a_1+b_1{\mathrm{e}}^{\xi }$ satisfy the bilinear form Equation (3)(3) $$(D_t^2+D_t{D}_x^3+\alpha D_t{D}_y+\beta D_t{D}_x)f·f=0,$$(3) . Then we obtain three solutions of the Ito Equation (5)(5) $$u_{\mathit{tt}}+u_{\mathit{xxxt}}+3(2u_x{u}_t+u{u}_{\mathit{xt}})+3u_{\mathit{xx}}{\int }_{-\infty }^x{u}_{t}dx\prime +\alpha u_{\mathit{yt}}+\beta u_{\mathit{xt}}=0,$$(5) (21) $$u_1=2{(\hspace{0.17em}\text{ln}\hspace{0.17em}g)}_{\mathit{xx}}=\frac{4a_1\left(g-2a_1{\left(x-\frac{\beta }{\alpha }y+a_3\right)}^2\right)}{g^2},$$(21) (22) $$u_2=2{(\hspace{0.17em}\text{ln}\hspace{0.17em}h)}_{\mathit{xx}}=\frac{4b_1\left(h-2b_1{\left(x-\frac{\beta }{\alpha }y+a_3-\frac{2}{c_1}\right)}^2\right)}{h^2},$$(22) (23) $$u_3=2{(\hspace{0.17em}\text{ln}\hspace{0.17em}(a_1+b_1{\mathrm{e}}^{\xi }\left)\right)}_{\mathit{xx}}=\frac{c_1^2}{2}{\text{sech}}^2\left(\frac{\xi +\hspace{0.17em}\text{ln}\hspace{0.17em}{b}_1-\hspace{0.17em}\text{ln}\hspace{0.17em}{a}_1}{2}\right)$$(23) where g, h and ξ are defined by (19)(19) $$\xi =c_{1}x+c_{2}y-(c_1^3+\beta c_1+\alpha c_2)t,g=a_1{\left(x-\frac{\beta }{\alpha }y+a_3\right)}^2+a_4{\left(y-\alpha t+a_6\right)}^2+a_7,$$(19) and (20)(20) $$h=b_1{\left(x-\frac{\beta }{\alpha }y+a_3-\frac{2}{c_1}\right)}^2+\frac{a_4{b}_1}{a_1}{\left(y-\alpha t+a_6-\frac{2\alpha }{c_1^3+\beta c_1+\alpha c_2}\right)}^2+\frac{a_7{b}_1}{a_1}.$$(20) . For given t, the stagnation points of u₁ and u₂ are (24) $$P_1\left(\beta t-\frac{\beta }{\alpha }{a}_6-a_3,\alpha t-a_6\right),P_{2,3}\left(\beta t-\frac{\beta }{\alpha }{a}_6-a_3\pm \sqrt{\frac{3a_7}{a_1}},\alpha t-a_6\right),$$(24) and (25) $$\begin{array}{l}{Q}_1\left(\beta t-\frac{\beta }{\alpha }{a}_6-a_3+\frac{2\beta }{c_1^3+\beta c_1+\alpha c_2}+\frac{2}{c_1},\alpha t-a_6+\frac{2\alpha }{c_1^3+\beta c_1+\alpha c_2}\right),\\ Q_{2,3}\left(\beta t-\frac{\beta }{\alpha }{a}_6-a_3+\frac{2\beta }{c_1^3+\beta c_1+\alpha c_2}+\frac{2}{c_1}\pm \sqrt{\frac{3a_7}{a_1}},\alpha t-a_6+\frac{2\alpha }{c_1^3+\beta c_1+\alpha c_2}\right).\end{array}$$(25)

From Equations (24)(24) $$P_1\left(\beta t-\frac{\beta }{\alpha }{a}_6-a_3,\alpha t-a_6\right),P_{2,3}\left(\beta t-\frac{\beta }{\alpha }{a}_6-a_3\pm \sqrt{\frac{3a_7}{a_1}},\alpha t-a_6\right),$$(24) and (25)(25) $$\begin{array}{l}{Q}_1\left(\beta t-\frac{\beta }{\alpha }{a}_6-a_3+\frac{2\beta }{c_1^3+\beta c_1+\alpha c_2}+\frac{2}{c_1},\alpha t-a_6+\frac{2\alpha }{c_1^3+\beta c_1+\alpha c_2}\right),\\ Q_{2,3}\left(\beta t-\frac{\beta }{\alpha }{a}_6-a_3+\frac{2\beta }{c_1^3+\beta c_1+\alpha c_2}+\frac{2}{c_1}\pm \sqrt{\frac{3a_7}{a_1}},\alpha t-a_6+\frac{2\alpha }{c_1^3+\beta c_1+\alpha c_2}\right).\end{array}$$(25) , we find that the velocities in x direction and y direction for u₁ and u₂ are $v_{1x}=v_{2x}=\beta ,v_{1y}=v_{2y}=\alpha$, respectively. Obviously, the solutions u₁ and u₂ are almost the same lump waves except for the difference of stagnations, and both of them move along the straight line $x-\beta y/\alpha =-a_3$. Furthermore, u₃ is a solitary wave, the velocities along the x-axis and the y-axis are $v_{3x}=(c_1^3+\beta c_1+\alpha c_2)/c_1$ and $v_{3y}=(c_1^3+\beta c_1+\alpha c_2)/c_2$, respectively. The wave is parallel with the straight line $x+c_{2}y/c_1=\text{const}$.

In the following, we present asymptotic analysis of the solution (4)(4) $$u(x,y,t)=2{(\hspace{0.17em}\text{ln}\hspace{0.17em}f(x,y,t\left)\right)}_{\mathit{xx}},$$(4) with (17)(17) $$f=g+h{\mathrm{e}}^{\xi },\hspace{1em}\xi =c_{1}x+c_{2}y+c_{3}t,$$(17) and we let (26) $$\alpha =6,\beta =5,a_1=2,a_3=-5,a_4=1,a_6=3,a_7=3,b_1=7,c_1=2,c_2=1.$$(26)

Taking $v_{1x}=5,v_{1y}=6,v_{3x}=12,v_{3y}=24$ and $v_{1x}<v_{3x},v_{1y}<v_{3y}$, we find an interactive phenomenon that the solitary wave catches up with the lump wave. In Figure 5(a), the solitary wave u₃ is behind the lump wave u₂. Then the solitary wave u₃ catches up with the lump wave u₂ as shown in Figure 5(b). Finally, the solitary wave surpasses the lump wave in Figure 5(c). Due to the interaction with the solitary wave, the lump wave u₂ is transformed into u₁. The shapes, the amplitudes and the velocities of the two lump waves u₁ and u₂ are the same except for the phase shift. This phenomenon shows the elastic collision. The asymptotic behaviour is given by $$u\to \{\begin{array}{c}{u}_2+u_3,t\to -\infty ,\hfill \\ u_1+u_3,t\to +\infty ,\hfill \end{array}$$ where u₁, u₂ and u₃ are defined by Equations (21)–(23).

Figure 5. Plots of the elastic interactional solution (4) with (17), (19), (20) and (26) for the Ito equation. (a) t = −3. (b) t = 0. (c) t = 4.

When the exponential part of the test function Equation (17)(17) $$f=g+h{\mathrm{e}}^{\xi },\hspace{1em}\xi =c_{1}x+c_{2}y+c_{3}t,$$(17) is a function of x, y and t, the solution is an interactional solution with a solitary wave and a lump wave. In the following, we assume that the exponential part is a function of y and t only, that is (27) $$\begin{array}{l}f=a_1{\left(x-\frac{\beta }{\alpha }y+a_3\right)}^2+a_4{\left(y-\alpha t+a_6\right)}^2+a_7\\ +\left(b_1{\left(x-\frac{\beta }{\alpha }y+a_3\right)}^2+\frac{a_4{b}_1}{a_1}{\left(y-\alpha t+b_6\right)}^2+b_7\right){\mathrm{e}}^{c_2(y-\alpha t)},\end{array}$$(27)

If we take (28) $$\alpha =7,\beta =3,a_1=1,a_3=-2,a_4=2/7,a_6=3,a_7=1/2,b_1=1,b_6=-7,b_7=1,c_2=1,$$(28) from Figure 6, we find that the solution is a two-lump solution and the two-lump wave is located in a line which is parallel with $x=\beta y/\alpha$.

Figure 6. Plots of the 2-lump solution (4) with (17), (27) and (28) for the Ito equation. (a) t = −5. (b) t = 0. (c) t = 5.

4. Conclusion

In this paper, by using Hirota’s bilinear method and by constructing new test functions, we obtained both non-elastic and elastic interactional solutions for the (2 + 1)-dimensional Ito equation. The non-elastic interactional solution showed the interactions among a lump wave, a periodic wave and a solitary wave. The elastic interactional solution showed the interactions between a lump wave and a solitary wave. The asymptotic behaviour of the elastic interaction was analysed. We also obtained periodic solution and two-lump solution of the (2 + 1)-dimensional Ito equation.

Disclosure statement

No potential conflict of interest was reported by the author(s).