Unconventional characteristic line for the nonautonomous KP equation

Applied Mathematics Letters 100 (2020) 106047

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Applied Mathematics Letters www.elsevier.com/locate/aml

Unconventional characteristic line for the nonautonomous KP equation Xin Yu a , Zhi-Yuan Sun a,b ,∗ a

Ministry-of-Education Key Laboratory of Fluid Mechanics and National Laboratory for Computational Fluid Dynamics, Beijing University of Aeronautics and Astronautics, Beijing 100191, China b International Research Institute for Multidisciplinary Science, Beijing University of Aeronautics and Astronautics, Beijing 100191, China

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Article history: Received 29 July 2019 Received in revised form 4 September 2019 Accepted 5 September 2019 Available online 12 September 2019 Keywords: Unconventional characteristic line Nonautonomous KP equation Bilinear method

abstract The unconventional characteristic line for a nonautonomous KP equation is constructed, which reveals that the propagation of solitons can be different from the traditional ones. Based on the Painlevé analysis and bilinear method, we obtain the soliton solution, and further study the polarity of characteristic line, the effect of initial phase, as well as the inelastic interaction and soliton resonance. We expect these results to be helpful in understanding relevant phenomena in water waves and dust acoustic waves. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction The Kadomtsev–Petviashvili (KP) equation has been extensively investigated mathematically and physically in fluids, plasmas and other fields [1–4]. Because of the diversity of boundaries, media and control equations, the KP equation also has different forms, including the nonautonomous ones [1,5–15]. The characteristic line, on the other hand, is important for solving partial differential equations and can provide an intuitive understanding of the propagation of shallow water waves modeled by the KP equation [1–4]. The conventional characteristic line (face) for KP equation is usually linear, as the following [2,4] k x + l y + ω t = const .

(1)

In addition, the KP equation also has the parabolic characteristic line [3,15] k(t) x + l1 (t) y + l2 (t) y 2 + ω(t) = const , 1

(2)

where the solitons might decay as (x2 + y 2 ) 2 → ∞ except for the direction along the characteristic line. ∗ Corresponding author. E-mail address: [email protected] (Z.-Y. Sun).

https://doi.org/10.1016/j.aml.2019.106047 0893-9659/© 2019 Elsevier Ltd. All rights reserved.

X. Yu and Z.-Y. Sun / Applied Mathematics Letters 100 (2020) 106047

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In this paper, we will consider such a nonautonomous KP equation [1,5–15], [ut + a(t) uux + b(t) uxxx ]x + c(y, t) uyy + d(t) uxy + [e1 (y, t) + e2 (t)x] uxx + f (t)ux + g(t)uy = 0 ,

(3)

where x, y and t are scaled space and time coordinates respectively, u is the function of x, y and t, representing the amplitude of the shallow water wave in fluids and representing the electrostatic wave potential in plasmas [5–15], and a(t), b(t), c(y, t), d(t), e1 (y, t), e2 (t), f (t) and g(t) are inhomogeneous coefficients. When g(t) = 0 and c(y, t) = c(t), Eq. (3) reduces to the case in Ref. [15]. It is worth noting that here we suppose g(t) > 0 or g(t) < 0. In other words, the model in this paper is totally different from that in Ref. [15]. For the existence of term uy in Eq. (3), we will construct a new kind of characteristic line for the nonautonomous KP equation. In Section 2, the Painlev´e analysis [16] and bilinear method [17] will be applied to solve Eq. (3). In Section 3, we will show how the characteristic line polarity, initial phase and wave number influence the soliton propagation and interaction. Finally, Section 4 will be the conclusion. 2. Soliton solutions The integrable conditions by the Painlev´e analysis for Eq. (3) can be derived as ∫ 3 1 a(t) = 6 ρ b(t) 4 g(t) 4 e [f (t)−2e2 (t)]dt , ∫ [ ] c(y, t) = g(t) 2y − 2 d(t)dt + λ ,

e1 (y, t) = −

(4) (5)

e2 (t)2 y 3b′ (t)2 y 3g ′ (t)2 y 3e2 (t)b′ (t)y − + + g(t) 8b(t)2 g(t) 8g(t)3 4b(t)g(t)

b′′ (t)y g ′′ (t)y e2 (t)g ′ (t)y e′2 (t)y + − − 4g(t)2 g(t) 4b(t)g(t) 4g(t)2 √ ∫ + α2 (t) 2y − 2 d(t)dt + λ + α1 (t) ,

+

(6)

where α1 (t) and α2 (t) are introduced arbitrary functions of t, ρ is a nonzero constant, λ is a constant, ′ denotes the derivative with respect to t, and b(t) > 0, g(t) > 0. Using the conditions (4)–(6) and the following dependent variable transformation, ∫ 1 1 2 u = b(t) 4 g(t)− 4 e− [f (t)−2e2 (t)]dt (logΦ)xx + Ψ (x, y, t) , (7) ρ Ψ (x, y, t) = Ψ1 (t)x + Ψ2 (y, t) , ] 1 [ Ψ1 (t) = −β3 (t) − e2 (t) , a(t) √ ∫ a(t)Ψ1 (t)2 y f (t)Ψ1 (t)y Ψ1′ (t)y Ψ2 (y, t) = − − − + β2 (t) 2y − 2 d(t)dt + λ + β1 (t) , g(t) g(t) g(t) we can transform Eq. (3) into its bilinear form as below, [ ∂ ∂ ] Dx Dt + b(t) Dx4 + c(y, t) Dy2 + d(t) Dx Dy + φ1 (x, y, t) Dx2 + φ2 (t) + g(t) Φ · Φ = 0, ∂x ∂y where 3 4

1 4

φ1 (x, y, t) = 6 ρ b(t) g(t) e

∫

[f (t)−2e2 (t)]dt

√ ∫ [ ] β2 (t) 2y − 2 d(t)dt + λ + β1 (t)

(8) (9) (10)

(11)

X. Yu and Z.-Y. Sun / Applied Mathematics Letters 100 (2020) 106047

yb′′ (t) yg ′′ (t) β3 (t)2 y 3b′ (t)2 y 3g ′ (t)2 y β3′ (t)y − + − − + 4b(t)g(t) 4g(t)2 g(t) 8b(t)2 g(t) 8g(t)3 g(t) √ ∫ 3b′ (t)β3 (t)y g ′ (t)β3 (t)y + α2 (t) 2y − 2 d(t)dt + λ + α1 (t) − xβ3 (t) , − − 4b(t)g(t) 4g(t)2 +

φ2 (t) =

b′ (t) g ′ (t) − , 4b(t) 4g(t)

∂ ∂ Φ · Φ = 2ΦΦx , Φ · Φ = 2ΦΦy . ∂x ∂y

3

(12)

(13) (14)

Thereinto Φ is a function of x, y and t, β1 (t), β2 (t) and β3 (t) are arbitrary functions of t, and Dxm Dtn is the Hirota bilinear derivative operator [2,17]. Based on Eq. (11), the multi-soliton solution of Eq. (3) can be obtained by the standard bilinear method. For the convenience of discussing the problem, we only list its two-soliton solution here. The expression Φ in two soliton solution can be Φ = 1 + exp(ξ1 ) + exp(ξ2 ) + exp(ξ1 + ξ2 + A12 ) , where

√

∫ 2y − 2

ξj = kj (t) x + lj1 (t) y + lj2 (t)

(15)

kj (t) = kj0 e

∫

d(t)dt + λ + ωj (t) + ξj0 (j = 1, 2),

(16)

β3 (t) dt

(17)

, [ ] ′ b(t)kj (t)g (t) − g(t) kj (t)b (t) + 4b(t)kj′ (t) lj1 (t) = , 4b(t)g(t)2 ∫ g(t)lj1 (t) [ dt −2 kj (t) lj2 (t) = e lj0 ∫ ∫ 2g(t)lj1 (t) ∫ ) ] ( [ k (t) −2e2 (t)+f (t)] dt 1 3 j − e kj (t) 6ρb(t) 4 g(t) 4 β2 (t) + e [2e2 (t)−f (t)] dt α2 (t) dt , ∫ [ ∫ 3 1 1 ωj (t) = − 4b(t)kj (t)3 + 24e− (2e2 (t)−f (t)) dt ρb(t) 4 g(t) 4 β1 (t)kj (t) 4 [ ] ∫ 4 (λ − 2 d(t) dt)lj1 (t)2 + lj1 (t) + lj2 (t)2 g(t) + kj (t) [ ] 2 4 α1 (t)kj (t) + d(t)lj1 (t)kj (t) + kj′ (t) b′ (t) g ′ (t) ] + + − dt , kj (t) b(t) g(t) ′

eAjl =

2 4 3 3 4 2 2 2 2 2 −3k20 k10 + 6k20 k10 − 3k20 k10 + l20 k10 − 2k20 l10 l20 k10 + k20 l10 2 2 l2 , 4 2 2 2 4 3 3 −3k20 k10 − 6k20 k10 − 3k20 k10 + l20 k10 − 2k20 l10 l20 k10 + k20 10

(18)

(19)

(20)

(21)

where kj0 , lj0 and ξj0 are all real constants. 3. Unconventional characteristic line From expression (16), we can record the characteristic line in the x–y plane as √ ∫ C = kj (t) x + lj1 (t) y + lj2 (t) 2y − 2 d(t)dt + λ + ωj (t) + ξj0 ,

(22)

which is not straight line or parabola. Without loss of generality, we take b(t) = f (t) = g(t) = 1 and 1 d(t) = e2 (t) = 0 for an example. In order to make soliton decay as (x2 + y 2 ) 2 → ∞, i.e. Ψ (x, y, t) = 0, we need to add an additional condition β3 (t) = −e2 (t). The introduced arbitrary functions α1 (t), α2 (t) β1 (t),

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Fig. 1. Two solitons at t = 0 with unconventional characteristic lines given by Expressions (7)–(10) and (15)–(21): (a) l10 = 1, l20 = 2, ξ10 = 3, ξ20 = −3; (b) k10 = 1, k20 = −1, l10 = 1, l20 = 5, ξ10 = −1, ξ20 = −28.

Fig. 2. Two solitons at t = 0 with unconventional characteristic lines given by Expressions (7)–(10) and (15)–(21): l10 = 1, l20 = −1, (a) ξ10 = 4, ξ20 = 3; (b) ξ10 = −5, ξ20 = 3.

β2 (t) are further assumed to be zero, and the characteristic line at a given time is finally transformed into a simple and clear form √ C = kj0 x + lj0 2y + λ , (23) whose nature is essentially equivalent to expression (22). Here we study how the unconventional characteristic lines affect soliton propagation and interaction. We define the polarity of the characteristic line as Pj = kj0 lj0 ,

(24)

which describes the opening direction of the unconventional characteristic line. Figs. 1–3 are illustrated with parameters as k10 = 1, k20 = 2, ρ = 1 and λ = 40 except for k20 = −1 in Fig. 1(b). As shown in Fig. 1(a), the opening directions of two parallel solitons are the same for P1 P2 > 0 and k10 /k20 = l10 /l20 . However, although the amplitudes of two solitons in Fig. 1(b) are equal, the polarities of the characteristic lines are opposite for P1 P2 < 0. Fig. 2 shows the effect of initial phase on soliton interaction. The value |ξ10 − ξ20 | describes the effective √ interaction distance between solitons. We can find two solitons with 2y + λ > 0 move from separation state in Fig. 2(a) to interaction state in Fig. 2(b) when the initial phase is changed. Note that although the soliton characteristic line is unconventional, its features of amplitude interaction, caused by the wave numbers, are the same as those for the solitons with conventional characteristic line. Fig. 3(a) gives the inelastic interaction for unconventional characteristic line solitons with eA12 = 0, in which two separate solitons merge into a single one. Fig. 3(b) gives the soliton resonance for unconventional characteristic line solitons with eA12 → ∞, where two solitons merge into one at first and then separate. If necessary, the change of amplitude with time or variable coefficients can also be discussed similar to Refs. [15,18].

X. Yu and Z.-Y. Sun / Applied Mathematics Letters 100 (2020) 106047

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Fig. 3. Two solitons at t = 0 with unconventional characteristic lines given by Expressions (7)–(10) and (15)–(21): (a) l10 = 0, √ l20 = 2 3, ξ10 = 8, ξ20 = −3; (b) l10 = −3.19617, l20 = 4, ξ10 = 15, ξ20 = −15.

4. Conclusions A nonautonomous KP equation, Eq. (3), has been investigated in this paper, which is a generalized model describing the shallow water waves and dust acoustic waves [5–15]. The Painlev´e analysis has been applied to Eq. (3) to test its integrability, during which three conditions are derived indicating five independent variable coefficients. Eq. (3) is also changed into its bilinear form and solved by using the bilinear method. The two soliton solutions have been listed and illustrated in their explicit forms. Different from the conventional and parabola characteristic lines, a new kind of characteristic line, √ ∫ C = kj (t) x + lj1 (t) y + lj2 (t)

2y − 2

d(t)dt + λ + ωj (t) + ξj0 ,

has been constructed. We have defined the characteristic line polarity and shown how solitons propagate with the different opening directions in Fig. 1. The initial phase and wave number have been used to explain their influence on soliton modes, including the interaction state, inelastic interaction and soliton resonance in Figs. 2 and 3. The above results in this paper are all based on the unconventional line in the x–y plane and could be extended to the multi solitons. Acknowledgments We express our sincere thanks to the editor and referees for their valuable suggestions. This work has been supported by the National Natural Science Foundation of China under Grant Nos. 11902016 and 11302014, and by the Fundamental Research Funds for the Central Universities. References [1] M.J. Ablowitz, P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge Univ. Press, New York, 1991. [2] R. Hirota, The Direct Method in Soliton Theory, Cambridge Univ. Press, Cambridge, 2004. [3] D. David, N. Kamran, D. Levi, P. Winternitz, Subalgebras of loop algebras and symmetries of the Kadomtsev–Petviashvili equation, Phys. Rev. Lett. 55 (1985) 2111. [4] J. Satsuma, N-soliton solution of the two-dimensional Korteweg-deVries equation, J. Phys. Soc. Japan 40 (1976) 286. [5] L.L. Li, B. Tian, C.Y. Zhang, T. Xu, On a generalized Kadomtsev–Petviashvili equation with variable coefficients via symbolic computation, Phys. Scr. 76 (2007) 411. [6] Y.T. Gao, B. Tian, Cylindrical Kadomtsev–Petviashvili model, nebulons and symbolic computation for cosmic dust ion-acoustic waves, Phys. Lett. A 349 (2006) 314. [7] Y.T. Gao, B. Tian, On the non-planar dust-ion-acoustic waves in cosmic dusty plasmas with transverse perturbations, Europhys. Lett. 77 (2007) 15001.

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