Complexitons of the modified KdV equation by Darboux transformation

doi:10.1016/j.amc.2007.06.011

Applied Mathematics and Computation

Volume 196, Issue 2, 1 March 2008, Pages 501-510

https://doi.org/10.1016/j.amc.2007.06.011 Get rights and content

Abstract

Darboux transformation (DT), a comprehensive approach to construct the explicit solutions of the nonlinear evolutionary equation, is applied to construct the new complexiton solution of the negative mKdV equation. We find that the complexiton solution is related to the single complex spectral parameter and that it is completely different from the breather solution for the positive mKdV equation. Consequently, we generalize the concept of complexiton of the KdV equation to the mKdV equation. In addition, the relationship of the spectral parameter and the known solutions of the mKdV equation is clarified and the multi-complexiton solution, multi-complexiton–positon, multi-complexiton–negaton and multi-complexiton–soliton solutions are obtained in the uniform manner by DT. The interaction of complexiton and soliton is also discussed in detail. It is shown that the new complexiton and soliton remain unchanged except for phase shifts after their interaction. At the same time, the superreflectionless property of one-positon potential for the mKdV equation is shown in detail.

Introduction

The concepts of positons and negatons were introduced for the Korteweg-de Vries (KdV) equation [1], [2], [3] $q_{t} - 6 {qq}_{x} + q_{xxx} = 0$

It was pointed out that if the spectral parameter of the associated Schrödinger eigenvalue problem is double and positive, the related solution of Eq. (1.1) is called positon which is usually expressed by means of the trigonometric functions. The positon solution is a long-range analogue of soliton and slowly decreasing, oscillating solution and possesses superreflectionless property. If the spectral parameter is double and negative, the related solution of (1.1) is called negaton which is usually expressed by the hyperbolic functions. Negaton solution is a singular reduced two-soliton solution corresponding to merging two negative eigenvalues of the Schrödinger operator. Later, the concepts of the positons and negatons were extended to other evolution equations [4], [5], [6], [7], [8]. The so-called complexiton solution of Eq.(1.1) was introduced in [9]. Complexiton solution is related to the single complex parameter λ_i of the associated Schrödinger eigenvalue problem and expressed by combinations of the trigonometric functions and hyperbolic functions. It was shown that complexiton solution possess singularities of combination of trigonometric function waves and hyperbolic function waves which have different traveling speeds.

The mKdV equation $q_{t} \pm 6 q^{2} q_{x} + q_{xxx} = 0$ are two important equations in physical contexts [10], [11], [12], [13], [14]. For convenience, we will classify the equation as ‘positive mKdV equation’and ‘negative mKdV equation’ depending on the prefactor ±1 of 6q²q_x. We can easily find that the relationship of the spectral parameter and the known solutions for the mKdV equation: For the positive mKdV equation, the N-soliton and N-pole solutions obtained by the Hirota method in [15], [16] are related to the single and double real spectral parameter, respectively. The breather and degeneration of breather solutions introduced in [17] are corresponding to pairs of complex conjugated spectral parameters. While for the negative mKdV equation, singular soliton and the negaton solutions introduced by Miura transformation in [3] are corresponding to the single and double real spectral parameters, respectively. Positon solutions given in [4], [18] are corresponding to the double pure imaginary spectral parameters. However, the solutions corresponding to pairs of complex conjugated spectral parameters for the negative mKdV equation has not been investigated. Though the superreflectionless property of positons for the mKdV equation was mentioned in [4], [18], the details of the calculation were not given.

In this paper, Darboux transformation (DT), a comprehensive approach to construct the explicit solutions of the nonlinear evolutionary equation, is applied to construct the new complexiton solution of the negative mKdV equation. We find that the complexiton solution is related to the single complex spectral parameter and that it is completely different from the breather solution for the positive mKdV equation. Consequently, we generalize the concept of complexiton of the KdV equation to the mKdV equation. In addition, the multi-complexiton solution, multi-complexiton–positon, multi-complexiton–negaton and multi-complexiton–soliton solutions are obtained in the uniform manner by means of DT. The interaction of complexiton and soliton is also discussed in detail. It is shown that the new complexiton and soliton remain unchanged except for phase shifts after their interaction. At the same time, the superreflectionless property of one-positon potential for the mKdV equation is shown in detail.

Section snippets

Darboux transformation (DT) of the modified KdV equation

At first, we investigate the Darboux transformation of the negative mKdV equation $q_{t} - 6 q^{2} q_{x} + q_{xxx} = 0 .$ The auxiliary linear system associated with Eq. (2.1) (the Lax pair) reads $Φ_{x} = (\begin{matrix} - λ & q \\ r & λ \end{matrix}) Φ,$ $Φ_{t} = (\begin{matrix} 4 λ^{3} - 2 λ qr - {qr}_{x} + {rq}_{x} & - 4 λ^{2} q + 2 λ q_{x} - q_{xx} + 2 q^{2} r \\ - 4 λ^{2} r - 2 λ r_{x} - r_{xx} + 2 {qr}^{2} & - (4 λ^{3} - 2 λ qr - {qr}_{x} + {rq}_{x}) \end{matrix}) Φ,$ with the reduction requirement $q = r,$ where $Φ = (\begin{matrix} φ^{(1)} \\ φ^{(2)} \end{matrix})$ is the solution of (2.2a), (2.2b) corresponding to the spectral parameter λ. Suppose that $Φ_{i} = (\begin{matrix} φ_{i}^{(1)} \\ φ_{i}^{(2)} \end{matrix}) (i = 1, 2, \dots, 2 n)$ are 2n different solutions of the system (2.2a), (2.2b) with the

Complexiton solutions

Solving linear system (2.2a), (2.2b) and (2.3) with λ_j = α_j + iβ_j, α_k, β_k≠0, yields $Φ_{j} = (\begin{matrix} φ_{j}^{(1)} \\ φ_{j}^{(2)} \end{matrix}) = (\begin{matrix} \exp (ϕ_{j}) [\cos (Λ_{j}) + i \sin (Λ_{j})] \\ \exp (- ϕ_{j}) [\cos (Λ_{j}) - i \sin (Λ_{j})] \end{matrix}),$ where $ϕ_{j} = α_{j} x - 4 (α_{j}^{3} - 3 α_{j} β_{j}^{2}) t + α_{j} χ_{j}, Λ_{j} = β_{j} x + 4 (β_{j}^{3} - 3 α_{j}^{2} β_{j}) t + β_{j} χ_{j},$ χ_j is an analytic function of λ_j.

By taking 4n conjugated complex parameters λ_4j = − λ_4j−1 = α_j − iβ_j, λ_4j−2 = − λ_4j−3 = α_j + iβ_j, j = 1, … , n, the solution (2.8a) with n replaced by 2n becomes in the following form: $q [2 n] = 2 \frac{\det (η_{1}^{(1)}, η_{1}^{(2)}, \bar{η_{1}^{(1)}}, \bar{η_{1}^{(2)}}, \dots, η_{n}^{(1)}, η_{n}^{(2)}, \bar{η_{n}^{(1)}}, \bar{η_{n}^{(2)}})}{\det (δ_{1}^{(1)}, δ_{1}^{(2)}, \bar{δ_{1}^{(1)}}, \bar{δ_{1}^{(2)}}, \dots, δ_{n}^{(1)})}$

Complexiton-1–soliton solution

In order to obtain 1-complexiton–1-soliton solution, we take six spectral parameters λ₁ = − λ₂ = α₁ + iβ₁, λ₃ = − λ₄ = α₁ − iβ₁ and λ₅ = λ₆ = λ, Im(λ) = 0. By the property of the determinant, formula (2.8a) with q[0] = 0 gives $q [3] = 2 \frac{G_{3}}{Δ_{3}},$ where $\begin{matrix} G_{3} = - 8 α_{1}^{2} β_{1} [(α_{1}^{2} + β_{1}^{2})^{2} - λ^{4} - 4 λ^{2} β_{1}^{2}] \cosh (2 ϕ_{1}) \sin (2 Λ_{1}) \sinh (2 θ) + 8 α_{1} β_{1}^{2} [(α_{1}^{2} + β_{1}^{2})^{2} - λ^{4} + 4 α_{1}^{2} λ^{2}] \\ \times \sinh (2 ϕ_{1}) \cos (2 Λ_{1}) \sinh (2 θ) + 4 λ β_{1}^{2} [(α_{1}^{2} + β_{1}^{2})^{2} + λ^{4} + 2 λ^{2} β_{1}^{2} - 2 α_{1}^{2} λ^{2}] \sinh^{2} (2 ϕ_{1}) \\ - 4 α_{1}^{2} λ [(α_{1}^{2} + β_{1}^{2})^{2} + λ^{4} + 2 λ^{2} β_{1}^{2} - 2 α_{1}^{2} λ^{2}] \sin^{2} (2 Λ_{1}) + 16 α_{1} β_{1} λ (α_{1}^{2} + β_{1}^{2}) (α_{1}^{2} - β_{1}^{2} - λ^{2}) \sinh (2 ϕ_{1}) \\ \times \sin (2 Λ_{1}) \cosh (2 θ) - 32 α_{1}^{2} β_{1}^{2} λ (α_{1}^{2} + β_{1}^{2}) \cosh ( \end{matrix}$

Complexiton solutions of high order

By taking 4n conjugated complex parameters λ_4j = − λ_4j−1 = α_j − iβ_j, λ_4j−2 = − λ_4j−3 = α_j + iβ_j, j = 1, … , n and calculating the limit α_j + iβ_j → α₁ + iβ₁, α_j − iβ_j → α₁ − iβ₁

j = 2, … , n in (3.3), we obtain the complexiton solutions of (n − 1)-order $\begin{matrix} q [2 n] = 2 \frac{\det (Re η_{1}^{(1)}, Re η_{1}^{(2)}, Im η_{1}^{(1)}, Im η_{1}^{(2)}, \dots, \partial_{α_{1}}^{n - 1} Re η_{1}^{(1)}, \partial_{α_{1}}^{n - 1} Re η_{1}^{(2)}, \partial_{α_{1}}^{n - 1} Im η_{1}^{(1)}, \partial_{α_{1}}^{n - 1} Im η_{1}^{(2)}}{\det (Re δ_{1}^{(1)}, Re δ_{1}^{(2)}, Im δ_{1}^{(1)}, Im δ_{1}^{(2)}, \dots, \partial_{α_{1}}^{n - 1} Re δ_{1}^{(1)}, \partial_{α_{1}}^{n - 1} Re δ_{1}^{(2)}, \partial_{α_{1}}^{n - 1} Im δ_{1}^{(1)}, \partial_{α_{1}}^{n - 1} Im δ_{1}^{(2)})} \end{matrix} .$ Similarly, we can obtain more general complexiton solutions of high order $q [2 n] = 2 \frac{H_{1}}{Q_{1}}$ where $\begin{matrix} H_{1} = \end{matrix}$

Multi-complexiton solution, multi-complexiton–positon, multi-complexiton–negaton and multi-complexiton–soliton

In order to obtain multi-complexiton solution, multi-complexiton–positon, multi-complexiton–negaton and multi-complexiton–soliton, we take $2 (k + \sum_{j = 1}^{k} σ_{j})$ the real or pure imaginary spectral parameters $λ_{2 m - 1} = - λ_{2 m} = ξ_{m}, m = 1, \dots, (k + \sum_{j = 1}^{k} σ_{j})$ and $4 (p + \sum_{j = 1}^{p} n_{j})$ complex conjugated spectral parameters $λ_{4 l} = - λ_{4 l - 1} = α_{l} - i β_{l}, λ_{4 l - 2} = - λ_{4 l - 3} = α_{l} + i β_{l}, l = 1, \dots, (p + \sum_{j = 1}^{p} n_{j})$ , and calculate the limit $ξ_{k + \sum_{j = 1}^{m - 1} σ_{j} + 1}, \dots, ξ_{k + \sum_{j = 1}^{m} σ_{j}} \to ξ_{m}, m = 1, \dots, k$ and $α_{p + \sum_{j = 1}^{l - 1} n_{j} + 1} + i β_{p + \sum_{j = 1}^{l - 1} n_{j} + 1}, \dots, α_{p + \sum_{j = 1}^{l} n_{j}} + i β_{p + \sum_{j = 1}^{l} n_{j}} \to α_{l} + i β_{l}, α_{p + \sum_{j = 1}^{l - 1} n_{j} + 1} - i β_{p + \sum_{j = 1}^{l - 1}}$

One-positon solution and the superreflectionless phenomenon

The one-positon solution of the negative mKdV equation (2.1) introduced in [18] is given by $q [2] = \frac{4 μ_{1} [\sin (2 {\bar{θ}}_{1}) - 2 μ_{1} γ_{1} \cos (2 {\bar{θ}}_{1})]}{\sin^{2} (2 {\bar{θ}}_{1}) - 4 μ_{1}^{2} γ_{1}^{2}},$ $where {\bar{θ}}_{j} = μ_{j} (x + 4 μ_{j}^{2} t + χ_{j}), γ_{j} = x + 12 μ_{j}^{2} t + m_{j}, m_{j} = χ_{j} + λ_{j} \frac{\partial χ_{j}}{\partial λ_{j}} .$ The solution of the linear system (2.2a), (2.2b) and (2.3) with λ₁ = iμ₁ and q[2] is $Φ_{1} = (\begin{matrix} φ_{1} \\ φ_{2} \end{matrix}), φ_{1} = \frac{K_{1}}{\sin^{2} (2 {\bar{θ}}_{1}) - 4 μ_{1}^{2} γ_{1}^{2}}, φ_{2} = \frac{K_{2}}{\sin^{2} (2 {\bar{θ}}_{1}) - 4 μ_{1}^{2} γ_{1}^{2}},$ where $\begin{matrix} K_{1} = [(μ^{2} + μ_{1}^{2}) \sin^{2} (2 {\bar{θ}}_{1}) + 4 μ_{1}^{2} γ_{1}^{2} (μ_{1}^{2} - μ^{2}) - i μ_{1} μ \sin (4 {\bar{θ}}_{1}) + 4 i μ_{1}^{2} μ γ_{1}] \exp (- i \bar{θ}) \\ + [- 4 i μ μ_{1}^{2} γ_{1} \cos (2 {\bar{θ}}_{1}) + 2 i μ_{1} μ \sin (2 {\bar{θ}}_{1}) - 4 μ_{1}^{3} γ_{1} \sin (2 {\bar{θ}}_{1})] \exp (i \bar{θ}), \end{matrix}$ $\begin{matrix} K_{2} = [(μ^{2} + μ_{1}^{2}) \sin^{2} (2 {\bar{θ}}_{1}) + 4 μ_{1}^{2} \end{matrix}$

Conclusion

The mKdV equation (1.2) is an important equation (2.2a), (2.2b) in physical contexts. The sign of the nonlinear term of the equation play an important role in the property of the solution. In this paper, Darboux transformation (DT), a comprehensive approach to construct the explicit solutions of the nonlinear evolutionary equation, is applied to construct the new complexiton solution of the negative mKdV equation. We find that the complexiton solution is related to the single complex spectral

Acknowledgement

This work is supported by National Basic Research Program of China (973 Program) (grant No. 2007CB814800) and the National Natural Science of China (grant No. 10601028).

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For different integrable systems, these new positon, negaton and complexiton solutions are analytical or singular. Much more exact solutions of integrable systems with different methods are constructed directly in [13–16]. We will construct new positon, negaton and complexiton solutions for the coupled KdV–mKdV system from its Darboux transformation with zero constant seed solution by selecting the spectral parameters as real or complex number in the wave functions (9) and (10).
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Complexitons of the modified KdV equation by Darboux transformation

Abstract

Introduction

Section snippets

Darboux transformation (DT) of the modified KdV equation

Complexiton solutions

Complexiton-1–soliton solution

Complexiton solutions of high order

Multi-complexiton solution, multi-complexiton–positon, multi-complexiton–negaton and multi-complexiton–soliton

One-positon solution and the superreflectionless phenomenon

Conclusion

Acknowledgement

Phys. Lett. A

Phys. Lett. A

Chaos, Solitons & Fractals

Phys. Lett.

Negaton and positon solutions of the KdV and mKdV hierarchy

J. Phys. A: Math. Gen.

Positons of the modified Korteweg de Vries equation

Ann. Phys.

Negaton and positon solutions of the soliton equation with self-consistent sources

J. Phys. A: Math. Gen.

The solutions of the NLS equations with self-consistent sources

J. Phys. A: Math. Gen.

Positon solutions of the sine-Gordon equation

J. Math. Phys.