Rational breather waves in a nonlinear vibration system

Abstract

A nonlinear vibration system behaviors always periodically; however, sometimes a vibrating structure, e.g. an empty oil tank, will be collapsed suddenly. This can be explained as a kind of rogue waves. This paper gives a general approach to search for rational breather waves to nonlinear equations including nonlinear vibration systems and dispersive wave systems. A breather type of rogue wave solution which contains two peaks whose amplitudes are two to three times higher than its surrounding waves and generally forms in a short time is constructed. Moreover, by using three-wave method, new breather-type multisolitary coherent structures are presented, the fusion interactions of localized structures are discussed and graphically investigated, showing some novel features and interesting behaviors.

Introduction

Nonlinear vibration systems generally are characterized by instable response, chaos, bifurcation phenomena, and other relatively complex behaviors of vibration, and it occurs everywhere, almost all research focuses on the periodic property; however, there are some interesting phenomena which cannot be explained by periodic solutions. For example, a thin nanofiber membrane^1–5 under some perturbation will change its shape to a strange form, Yu et al. ¹ explained that the surface motion will be blocked. An empty oil tank vibrates in a small amplitude, but suddenly it will be collapsed, crimped fibers are obtained by vibration of axially moving jets. ⁶ All these phenomena can be explained by the rogue wave.^7,8 Dan et al. ⁹ applied Hall–Petch effect or the geometrical potential theory^10–12 to stabilize bubble walls so that bubbles can be enlarged during the spinning process. To investigate wave propagation processes for nonlinear problems, some effective methods have been proposed.^13–20

Recently, Zh et al. ²¹ introduced a homoclinic breather limit method (HBLM) to seek wave solutions to nonlinear evolution equations

$$P (u, u_t, u_x, u_y, \dots )=0$$

By using Painlevé analysis, and a transformation

$$u=T\left(f\right)$$

made for some new and unknown function f, the mentioned nonlinear evolution equation can be converted into Hirota’s bilinear form

$$H(Dt,\hspace{0.17em}Dx;\hspace{0.17em}f)=0$$

Then, by using extended homoclinic test approach, one can get homoclinic (heteroclinic) breather wave solution to the above equation. A rational homoclinic (heteroclinic) wave can be obtained by letting the period of periodic wave go to infinite in the homoclinic (heteroclinic) breather wave solution.

In this letter, we pay our attention to construct new localized structures for the (2 + 1)D modified dispersive water-wave system given as

$$u_{\mathrm{yt}}+u_{\mathrm{xxy}}-2v_{\mathrm{xx}}-{(u^2)}_{\mathrm{xy}}=0$$

(1)

$$v_t-v_{\mathrm{xx}}-2{(uv)}_x=0$$

(2)

These equations modeled nonlinear and dispersive long gravity waves traveling in two horizontal directions on shallow waters of uniform depth derived by Boiti et al. ²² The system describes a lot of vibration processes in mathematical physics, e.g. the motion of the linearized vibrating string or membrane, and the vibration of a moving belt has similar equations like those given in equations (1) and (2). ²³ Some proper results including periodic folded wave, nonpropagating and propagating solutions, abundant propagating localized excitations, and so on were obtained^24–29 by a variety of approaches, and the system can also be effectively solved by the exp-function method, and the homotopy perturbation method and the variational iteration method.^30–32

Present work deals with new localized coherent structures of system (1) and (2) by using HBLM and three-wave method.

Rational breather wave

In this section, we first consider system (1) and (2) with the dependent variable transformation with the logarithm of function f(x, y, t)

$$u={\left(\ln f\right)}_x+u_0,\hspace{1em}v={\left(\ln f\right)}_{xy}+v_0$$

(3)

where $u_0\left(x,y,t\right),v_0\left(x,y,t\right)$ are known solutions of system expressed by equations (1) and (2).

Substituting equation (3) into equations (1) and (2), by choosing $u_0=0, v_0=0$ , we can simplify equations (1) and (2) as follows

$${\left(\frac{f{f}_{xx}-f_x{f}_t+f_x{f}_{xx}-f{f}_{xxx}}{f^2}\right)}_y=0\mathrm{ }$$

(4)

Therefore, we only need to solve the following equation

$${ff}_{xt}-f_x{f}_t +f_x{f}_{\mathrm{xx}}-{ff}_{\mathit{xxx}} =0$$

(5)

and from which we can obtain the solutions of system expressed by equations (1) and (2).

So, along with the homoclinic (heteroclinic) breather limit method, we suppose

$$f\left(x,y,t\right)=a_1\left(y\right)e^{{\zeta }_1}+a_2\left(y\right)\cos \left[p\left(y\right)\left(x+q\left(y\right)\right)\right]+a_3\left(y\right)e^{{\zeta }_2}$$

(6)

with ${\zeta }_i=k_{i}x+m_i\left(y\right)+{\omega }_i\left(y\right)t,\left(i=1,2\right).$ Substituting equation (6) into equation (5), with the aid of symbolic computation Mathematica, we find that equation (5) is automatically satisfied with the following relations

$${\omega }_1=k_1^2+p^2,\hspace{1em}{\omega }_2=k_2^2+p^2$$

(7)

with k_i(i = 1, 2) the arbitrary constants. Where $a_i(y),m_i(y)(i=1,2)$ and p(y), q(y) are arbitrary functions of y.

Inserting equation (6) with equation (7) into equation (3), we obtain a new type of two-wave solution of system expressed by equations (1) and (2) in the form

$$u_1=\frac{k_1{a}_1{e}^{{\zeta }_1}-p{a}_2\sin \left(\zeta \right)+k_2{a}_3{e}^{{\zeta }_2}}{a_1{e}^{{\zeta }_1}+a_2\cos \left(\zeta \right)+a_3{e}^{{\zeta }_2}}$$

$$\begin{array}{l}{v}_1=\frac{k_1{e}^{{\zeta }_1}\left[a_1^{\prime }+k_1{a}_1{m}_1^{\prime }\right]+k_2{e}^{{\zeta }_2}\left[a_3^{\prime }+k_2{a}_3{m}_2^{\prime }\right]-p\left[a_2{\mathrm{pq}}^{\prime }\mathrm{cos\zeta }+a_2^{\prime }\mathrm{sin\zeta }\right]}{a_1{e}^{{\zeta }_1}+a_2\cos \left(\zeta \right)+a_3{e}^{{\zeta }_2}}\\ \hspace{1em}-\frac{(k_1{a}_1{e}^{{\zeta }_1}-p{a}_2\sin \left(\zeta \right)+k_2{a}_3{e}^{{\zeta }_2})[a_2^{\prime }\mathrm{cos\zeta }-a_2{\mathrm{pq}}^{\prime }\mathrm{sin\zeta }+e^{{\zeta }_1}(a_1^{\prime }+k_1{a}_1{m}_1^{\prime })+e^{{\zeta }_2}(a_3^{\prime }+k_2{a}_3{m}_2^{\prime })]}{{[a_1{e}^{{\zeta }_1}+a_2\cos \left(\zeta \right)+a_3{e}^{{\zeta }_2}]}^2}\end{array}$$

where ζ = p(y)[x + q(y)], $a_i=a_i\left(y\right),m_j=m_j\left(y\right)\left(i=1,2,3;\hspace{0.5em}j=1,2\right),$ p = p(y) and q(y) are arbitrary functions of y, $k_i(i=1,2)$ are arbitrary constants.

Solution $(u_1,v_1)$ is a type of dromion solutions with oscillating tails in x and y directions. For instance, by choosing

$$k_1=-k_2=p,\hspace{1em}{m}_1=-m_2=m\left(y\right),\hspace{1em}{a}_1=a_3=a(y),\hspace{1em}{a}_2=-2a(y)$$

in solution $(u_1,v_1)$ , it can be rewritten in the form

$$u_2=\frac{p\sin \left[p\left(x+q\left(y\right)\right)\right]+e^{2p^{2}t}p\sinh [p\left(x+m\left(y\right)\right)]}{e^{2p^{2}t}p\cosh \left[p\left(x+m\left(y\right)\right)\right]-\cos [p\left(x+q\left(y\right)\right)]}$$

$$\begin{array}{l}{v}_2=p^2[\frac{q^{\prime }\cos \left[p\left(x+q\left(y\right)\right)\right]+m^{\prime }{e}^{2p^{2}t}\cosh \left[p\left(x+m\left(y\right)\right)\right]}{e^{2p^{2}t}\cosh \left[p\left(x+m\left(y\right)\right)\right]-\cos \left[p\left(x+q\left(y\right)\right)\right]}\\ \hspace{1em}-\frac{\left[\sin \left(p\left(x+q\left(y\right)\right)\right)+e^{2p^{2}t}\sinh \left(p\left(x+q\left(y\right)\right)\right)\right][q^{\prime }\sin \left(p\left(x+q\left(y\right)\right)\right]+m^{\prime }{e}^{2p^{2}t}\sinh [p\left(x+m\left(y\right)\right)]}{{\left(e^{2p^{2}t}\cosh \left[p\left(x+m\left(y\right)\right)\right]-\cos \left[p\left(x+q\left(y\right)\right)\right]\right)}^2}]\end{array}$$

Solution (u₂, v₂) shows elastic interaction between a backward-direction periodic wave and homoclinic wave of different direction (see Figure 1).

Figure 1.

The spatial structure of v₂ at m(y) = y, q(y) = −y, p = 1, t = 0.1.

Considering a limit behavior of $(u_2,v_2)$ as the period $\frac{2\pi }{p}$ of periodic wave cos[p(x + q(y))] goes to infinite, we obtain a rational solution of systems (1) and (2) which gives as

$$u_3=\frac{2[\left(x+m\left(y\right)\right)+\left(x+q\left(y\right)\right)]}{{[x+m\left(y\right)]}^2+{[x+q\left(y\right)]}^2+4t}$$

$$v_3=\frac{2\left[q^{\prime }\left[4t-2{\left(x+q\left(y\right)\right)}^2+{\left(m\left(y\right)-q\left(y\right)\right)}^2\right]+m^{\prime }\left[4t-2{\left(x+q\left(y\right)\right)}^2+{\left(m\left(y\right)-q\left(y\right)\right)}^2\right]\right]}{{{[(x+m\left(y\right))]}^2+{[(x+q\left(y\right))}^2+4t]}^2}$$

where m( y) and q( y) are arbitrary functions of y.

Solution (u₃, v₃) contains two waves with different velocities and directions. For fixed x and t, (u₃, v₃) → (0, 0) as q(y)(or m(y)) → ±∞. It is a breather type of rogue wave solution which contains two peaks whose amplitudes are two to three times higher than its surrounding waves and generally forms in a short time (see Figure 2(a)). It indicates that 2D rogue wave which is localized in both x- and y-directions may exist in the framework of MDWWS.

Figure 2.

(a) The spatial structure of rogue wave of v₃ at m(y) = 2y, q(y) = −2y, t = 1 and (b) the spatial structure of higher rogue wave of v₃ with equation (8) at t = 0.05.

In another, owing to the arbitrary functions m(y), q(y) involved in (u₃, v₃), we can also search for higher order rational solution for system expressed by equations (1) and (2). In fact, if m(y) and q(y) are choosing as

$$q\left(y\right)={\mathrm{2y}}^3+1,\hspace{1em}m\left(y\right)=-y^3$$

(8)

respectively, we can obtain the higher order rational solution of systems (1) and (2). Figure 2(b) describes the figure of a higher order rogue wave for the rational solution.

New coherent structures

Now, let us consider equation (5) by using the three wave type of Ansatz function³³ in the form

$$f\left(x,y,t\right)=a_1\left(y\right)e^{{\zeta }_1}+a_2\left(y\right)\cos \left({\zeta }_2\right)+a_3(y)e^{\mathrm{\omega t}}\cosh ({\zeta }_3)+a_4(y)e^{{\zeta }_4}$$

(9)

where ${\zeta }_i=k_{i}x+m_i(y)+{\omega }_{i}t\left(i=1,2,3,4\right)$ and $a_i(y),m_i(y)(i=1,2,3,4)$ are only the arbitrary functions of y, $k_i(i=1,2,3,4),{\mathrm{ \omega }}_i(i=1,2,3)$ and ω are arbitrary constants, respectively.

Substituting Ansatz (9) into equation (5) with Mathematica, we find that equation (5) can be automatically satisfied with the following relations

$$k_1=-k_4,\hspace{1em}{k}_2=k_2,\hspace{1em}{k}_3=k_3$$

(10)

$$\omega =k_2^2+k_3^2,\hspace{1em}{\omega }_1={\omega }_4=k_1^2+k_2^2,\hspace{1em}{\omega }_2={\omega }_3=0$$

(11)

$$m_1\left(y\right)=-m_4\left(y\right),\hspace{1em}{m}_2\left(y\right)=m_2\left(y\right),\hspace{1em}{m}_3\left(y\right)=m_3\left(y\right)$$

(12)

$$a_1\left(y\right)=a_1\left(y\right),\hspace{1em}{a}_2\left(y\right)=a_2\left(y\right),\hspace{1em}{a}_3\left(y\right)=a_3\left(y\right),\hspace{1em}{a}_4\left(y\right)=a_4\left(y\right)$$

(13)

Choosing a = a₁(y) = a₄(y) as arbitrary constants, we can explicitly derive the three-wave solutions of system expressed by equations (1) and (2) as follows

$$u_4=\frac{2{ak}_1{e}^{{\omega }_{1}t}\sinh \left(\zeta \right)-a_2{k}_2\sin \left({\zeta }_2\right)+a_3{k}_3{e}^{\mathrm{\omega t}}\sinh ({\zeta }_3)}{2{ae}^{{\omega }_{1}t}\cosh \left(\zeta \right)+a_2\cos \left({\zeta }_2\right)+a_3{e}^{\mathrm{\omega t}}\cosh ({\zeta }_3)}$$

$$\begin{array}{l}{v}_4=\frac{2{ak}_1{m}^{\prime }{e}^{{\omega }_{1}t}\cosh \left(\zeta \right)-k_2[a_2{m}_2^{\prime }\cos \left({\zeta }_2\right)+a_2^{\prime }\sin ({\zeta }_2)]+k_3{e}^{\mathrm{\omega t}}[a_3{m}_3^{\prime }\cosh \left({\zeta }_3\right)+a_3^{\prime }\sinh ({\zeta }_3)]}{2{ae}^{{\omega }_{1}t}\cosh \left(\zeta \right)+a_2\cos \left({\zeta }_2\right)+a_3{e}^{\mathrm{\omega t}}\cosh ({\zeta }_3)}\\ \hspace{1em}-\frac{e^{{\omega }_{2}t}\left[2{ak}_1{e}^{{\omega }_{1}t}\sinh \left(\zeta \right)-a_2{k}_2\sin \left({\zeta }_2\right)+a_3{k}_3{e}^{{\omega }_{2}t}\sinh \left({\zeta }_3\right)\right][a_3^{\prime }\cosh \left({\zeta }_3\right)+a_3{m}_2^{\prime }\sinh \left({\zeta }_3\right)]}{{(2{\mathrm{ae}}^{{\omega }_{1}t}\cosh \left(\zeta \right)+a_2\cos \left({\zeta }_2\right)+a_3{e}^{\mathrm{\omega t}}\cosh ({\zeta }_3))}^2}\\ \hspace{1em}+\frac{\left[2{ak}_1{e}^{{\omega }_{1}t}\sinh \left(\zeta \right)-a_2{k}_2\sin \left({\zeta }_2\right)+a_3{k}_3{e}^{\mathrm{\omega t}}\sinh \left({\zeta }_3\right)\right][a_2{m}_2^{\prime }\sin \left({\zeta }_2\right)-a_2^{\prime }\cos \left({\zeta }_2\right)-2{\mathrm{am}}_1^{\prime }{e}^{{\omega }_{1}t}\sinh \left(\zeta \right)]}{{(2{ae}^{{\omega }_{1}t}\cosh \left(\zeta \right)+a_2\cos \left({\zeta }_2\right)+a_3{e}^{\mathrm{\omega t}}\cosh ({\zeta }_3))}^2}\end{array}$$

where ζ = k₁x + m₁( y) and ${\zeta }_i=k_{1}x+m_i\left(y\right)(i=2,3).$

Solution (u₄, v₄) involves two kinds of transcendental functions: trigonometric and hyperbolic functions. It presents a kind of interaction solutions between trigonometrical waves and exponential waves.

We show its typical spatial structure in Figure 3 with the parameters selected as

$$\left(\begin{array}{lll}{a}_1\hfill & a_2\hfill & a_3\hfill \\ k_1\hfill & k_2\hfill & k_3\hfill \\ m_1\hfill & m_2\hfill & m_3\hfill \end{array}\right)=\left(\begin{array}{ccc}\hfill 0.2\hfill & 0.5\hfill & \hfill 1.5\hfill \\ \hfill 0.1\hfill & \hfill 1\hfill & \hfill 1\hfill \\ 10y\hfill & 3y\hfill & 0.3y\hfill \end{array}\right)$$

Figure 3.

The spatial structure of two solitons fusion into one for v₄ at a = 1 and t = 0.

Figure 3 shows that two line solitons come into interaction, fusion to one new soliton with some oscillations whose amplitude is higher than the two line solitons and later break up again to form two solitons which are actually the original soliton, respectively.

Owing to the arbitrary functions $a_i(y)$ and $m_i(y)$ involved in the solution (u₄, v₄), it is convenient to excite soliton structures. After attempts, we construct a new class of novel localized structures as follows.

If we choose the arbitrary functions $a_i(y)$ , $m_i(y)$ (i = 1, 2, 3) and arbitrary constants $k_i$ (i = 1, 2, 3) as

$$\left(\begin{array}{ccc}{a}_1& a_2& a_3\\ k_1& k_2& k_3\\ m_1& m_2& m_3\end{array}\right)=\left(\begin{array}{ccc}2& 1& 1\\ 1& 5& -0.1\\ y& 3& -y^2\end{array}\right)$$

respectively, the fusion between two peakons phenomena is obtained (see Figure 4).

Figure 4.

The spatial structure of two periodic peakons fusion into one line soliton of v₄ at (a) t = 0, (b) t = 0.05, (c) t = 0.1, (d) t = 0.5, (e) t = 1, and (f) t = 2.

Figure 4 shows that two periodic peakons fuse to one single line soliton with the time increasing. After interaction, the final line soliton preserves its shapes and amplitudes and does not move in any direction.

Conclusion

In this paper, the HBLM is proposed to the (2 + 1)-dimensional modified dispersive water-wave system to seek rogue wave solutions. A new family of two-wave solution, rational breather wave solution is obtained. Based on the rational solution, rogue wave solution which has two to three times amplitude higher than its surrounding waves generally forms in a short time. It is a new discovery that rogue wave solutions can come from breather solitary wave solutions for the 2D framework. Moreover, we also derived three-wave solution for the MDWWS by using the three-wave method. The arbitrary functions in the obtained solutions imply that these solutions have rich spatial structures. And it may be helpful in future studies for the intricate nature world.