Cauchy matrix approach to three non-isospectral nonlinear Schr&ouml;dinger equations

Alemu Yilma Tefera; Shangshuai Li; Da-jun Zhang

doi:10.1088/1572-9494/ad35b1

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Communications in Theoretical Physics ›› 2024, Vol. 76 ›› Issue (5) : 55001. DOI: 10.1088/1572-9494/ad35b1

Mathematical Physics

Cauchy matrix approach to three non-isospectral nonlinear Schrödinger equations

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Abstract

This paper aims to develop a direct approach, namely, the Cauchy matrix approach, to non-isospectral integrable systems. In the Cauchy matrix approach, the Sylvester equation plays a central role, which defines a dressed Cauchy matrix to provide τ functions for the investigated equations. In this paper, using the Cauchy matrix approach, we derive three non-isospectral nonlinear Schrödinger equations and their explicit solutions. These equations are generically related to the time-dependent spectral parameter in the Zakharov–Shabat–Ablowitz–Kaup–Newell–Segur spectral problem. Their solutions are obtained from the solutions of unreduced non-isospectral nonlinear Schrödinger equations through complex reduction. These solutions are analyzed and illustrated to show the non-isospectral effects in dynamics of solitons.

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Cauchy matrix approach / Sylvester equation / nonlinear Schrödinger equation / non-isospectral integrable system / explicit solution

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Alemu Yilma Tefera, Shangshuai Li, Da-jun Zhang. Cauchy matrix approach to three non-isospectral nonlinear Schrödinger equations[J]. Communications in Theoretical Physics, 2024, 76(5): 55001 https://doi.org/10.1088/1572-9494/ad35b1

1. Introduction

The nonlinear Schrödinger (NLS) equation,

\begin{array}{rcl} i q_{t} - q_{x x} - 2 ϵ | q |^{2} q = 0, \end{array}

(1.1a)

as one of the most famous integrable systems, is known as a ‘universal’ model [1], which means it appears as a governing model in various physical phenomena. Here ‘i’ is the imaginary unit, ε = ± 1, ∣q∣² = qq^* and * stands for complex conjugate. It emerges in describing wave packages in deep water [2, 3], plasma physics [4], optical fiber [5, 6], etc. In addition, the NLS equation with various external potentials (known also as the Gross–Pitaevskii (GP) equation [7–9]) is also the governing model in nonlinear optics and Bose–Einstein condensates (BEC) [10]. One can refer to [11] for more references and applications of the NLS equation and its extensions.

The NLS equation with x-coefficient can describe nonlinear waves in non-uniformity media [12–15]. Such equations have been shown to be integrable in the sense of having Lax pairs, with the spectral parameter η satisfying η_t ≠ 0, which are referred to non-isospectral nonlinear Schrödinger equations (NNLSEs). In this paper, we will investigate the following three NNLSEs:

\begin{array}{rcl} i q_{1, t} - q_{1, x x} - 2 | q_{1} |^{2} q_{1} + 2 α {x q}_{1} = 0, \end{array}

(1.2a)

\begin{array}{rcl} i q_{2, t} - q_{2, x x} - 2 | q_{2} |^{2} q_{2} + i β {({x q}_{2})}_{x} = 0, \end{array}

(1.2b)

\begin{array}{rcl} i q_{3, t} - x (q_{3, x x} + 2 | q_{3} |^{2} q_{3}) - 2 q_{3, x} - 2 q_{3} \partial^{- 1} | q_{3} |^{2} = 0, \end{array}

(1.2c)

where α and β are real constants, and ∂⁻¹ stands for the integration operator with respect to x. We denote these equations NNLSE-I, NNLSE-II, and NNLSE-III for short, respectively. They correspond to time-dependant spectral parameter η with time evolutions η_t = α, η_t = − iβη and η_t = −2η², respectively, where

α, β \in R

, e.g. [16]. Although the NNLSE-I and NNLSE-II can be converted to the NLS equation (1.1a) via gauge transformations, e.g. [16], they are physically useful in BEC: the NNLSE-I is the GP equation with a linear potential [17], while the NNLSE-II can provide solutions to the GP equation with a parabolic potential and a gain term e.g. [18]. The NNLSE-III can provide space-time localized soliton waves on zero background [16, 19]. So far, integrable methods, such as the inverse scattering transform [12–14] and bilinear method [16], have been applied to obtain explicit solutions of the above equations. Yet in this paper, we construct their solutions by means of a completely direct method, namely, the Cauchy matrix approach.

The Cauchy matrix approach is a method to construct and study integrable equations by means of the Sylvester-type equations. It is first systematically introduced in [20] to investigate integrable quadrilateral equations and later developed in [21, 22] to more general cases. It has also been applied to the Zakharov–Shabat–Ablowitz–Kaup–Newell–Segur (ZS-AKNS) system [23], equations with self-consistent sources [24], and the self-dual Yang-Mills equation [25, 26], etc. The purpose of this paper is not only to construct solutions to the three NNLSEs in (1.2), but also to extend the Cauchy matrix approach to the non-isospectral case, as the Sylvester-type equation in the Cauchy matrix scheme of the ZS-AKNS system is a typical type (see [22, 27] for the Korteweg–de Vries (KdV) and Kadomtsev–Petviashvili (KP) type equations). One will see that the non-isospectral extension of the Cauchy matrix scheme is quite non-trivial compared with the isospectral case [23].

This paper is organized as follows. Our plan is in the first step to derive three unreduced non-isospectral Schrödinger systems using the Cauchy matrix approach. This will be described in section 2. Then in section 3, we present solution formulae of these unreduced systems. These formulae guide us to implement reduction so that solutions of the NNLSEs are obtained, which will be done in section 4. Dynamics of these solutions are illustrated also in this section. Finally, conclusions are given in section 5. There is an appendix section where solutions of the Sylvester equation with lower triangular Toeplitz matrices are presented.

2. Cauchy matrix approach to unreduced NNLSEs

In this section, we describe the Cauchy matrix approach for unreduced NNLSEs.

2.1. Sylvester equation and master functions

We start from the Sylvester equation of the following type (see [23, 25]):

\begin{array}{rcl} K M - M K = r s^{T}, \end{array}

(2.1)

in which the involved elements are block matrices in the form of

\begin{array}{rcl} \begin{array}{l} K = (\begin{array}{cc} K_{1} & 0 \\ 0 & K_{2} \end{array}), M = (\begin{array}{cc} 0 & M_{1} \\ M_{2} & 0 \end{array}), \\ r = (\begin{array}{cc} r_{1} & 0 \\ 0 & r_{2} \end{array}), s = (\begin{array}{cc} 0 & s_{1} \\ s_{2} & 0 \end{array}), \end{array} \end{array}

(2.2)

where

K_{i} \in C_{N \times N} [t], M_{i} \in C_{N \times N} [x, t], r_{i}, s_{i} \in C_{N \times 1} [x, t]

, for i = 1, 2. An equivalent form of (2.1) is given by

\begin{array}{rcl} K_{1} M_{1} - M_{1} K_{2} = r_{1} s_{2}^{T}, \end{array}

(2.3a)

\begin{array}{rcl} K_{2} M_{2} - M_{2} K_{1} = r_{2} s_{1}^{T} . \end{array}

(2.3b)

We assume matrices K₁ and K₂ do not share any eigenvalues, so that the Sylvester equation (2.1) has a unique solution M for given K, r, s [28]. By these elements we define master functions

\begin{array}{rcl} S^{(i, j)} ≐ s^{T} K^{j} {(, I_{2 N} + M)}^{- 1} K^{i} r = (\begin{array}{cc} s_{1}^{(i, j)} & s_{2}^{(i, j)} \\ s_{3}^{(i, j)} & s_{4}^{(i, j)} \end{array}), (i, j \in Z), \end{array}

(2.4)

where I_2N is the 2N th-order unit matrix and of which the more explicit versions are

\begin{array}{rcl} s_{1}^{(i, j)} = - s_{2}^{T} K_{2}^{j} {(, I_{N} - M_{2} M_{1})}^{- 1} M_{2} K_{1}^{i} r_{1}, \end{array}

(2.5a)

\begin{array}{rcl} s_{2}^{(i, j)} = s_{2}^{T} K_{2}^{j} {(, I_{N} - M_{2} M_{1})}^{- 1} K_{2}^{i} r_{2}, \end{array}

(2.5b)

\begin{array}{rcl} s_{3}^{(i, j)} = s_{1}^{T} K_{1}^{j} {(, I_{N} - M_{1} M_{2})}^{- 1} K_{1}^{i} r_{1}, \end{array}

(2.5c)

\begin{array}{rcl} s_{4}^{(i, j)} = - s_{1}^{T} K_{1}^{j} {(, I_{N} - M_{1} M_{2})}^{- 1} M_{1} K_{2}^{i} r_{2} . \end{array}

(2.5d)

In addition, a difference-product formula can be formulated from (2.1) as

\begin{array}{rcl} S^{(i + 1, j)} - S^{(i, j + 1)} = S^{(0, j)} S^{(i, 0)} . \end{array}

(2.6)

The proof can be found in [23, 25].

2.2. Unreduced NNLSE-I

To derive an unreduced form of the NNLSE-I equation, let us introduce dispersion relations of r and s as follows,

\begin{array}{rcl} r_{x} = \frac{1}{2} K r a, s_{x} = - \frac{1}{2} K^{T} s a, \end{array}

(2.7a)

\begin{array}{rcl} r_{t} = (- \frac{1}{2} K^{2} + α x) r a, s_{t} = (\frac{1}{2} {(, K^{T})}^{2} - α x) s a, \end{array}

(2.7b)

where

\begin{array}{rcl} a = (\begin{array}{cc} 1 & 0 \\ 0 & - 1 \end{array}) \end{array}

(2.8)

and matrix K obeys the evolution

\begin{array}{rcl} K_{t} = \frac{d}{d t} K (t) = 2 α, α \in R . \end{array}

(2.9)

In addition, we assume K_t and K commute. Then one can derive the evolution of M by taking a derivative of the Sylvester equation (2.1), which gives rise to

\begin{array}{rcl} K_{t} M + K M_{t} - M_{t} K - M K_{t} = r_{t} s^{T} + r s_{t}^{T} . \end{array}

Substituting (2.7b) and (2.9) into it yields

\begin{array}{rcl} K M_{t} - M_{t} K = \frac{1}{2} (- K^{2} r a s^{T} + r a s^{T} K^{2}), \end{array}

which leads us to a Sylvester equation,

\begin{array}{rcl} K (M_{t} + \frac{1}{2} (K r a s^{T} + r a s^{T} K)) - (M_{t} + \frac{1}{2} (K r a s^{T} + r a s^{T} K)) K = 0. \end{array}

It has a unique zero solution in the light of assumption that K₁ and K₂ do not share any eigenvalues. Thus we have

\begin{array}{rcl} M_{t} = - \frac{1}{2} (K r a s^{T} + r a s^{T} K) . \end{array}

(2.10)

One can also find [23]

\begin{array}{rcl} M_{x} = \frac{1}{2} r a s^{T} . \end{array}

(2.11)

Next we are going to derive the derivative of S^(i,j). Let us define the auxiliary vector,

\begin{array}{rcl} u^{(i)} = {(I_{2 N} + M)}^{- 1} K^{i} r, \end{array}

(2.12)

which is connected with S^(i,j) by

\begin{array}{rcl} S^{(i, j)} = s^{T} K^{j} u^{(i)} . \end{array}

(2.13)

The derivative of u⁽ⁱ⁾ with respect to x reads [23]

\begin{array}{rcl} u_{x}^{(i)} = \frac{1}{2} (, u^{(i + 1)} a - u^{(0)} a S^{(i, 0)}) . \end{array}

(2.14)

To derive the derivative of u⁽ⁱ⁾ with respect to t, taking t-derivative in (2.12) yields

\begin{array}{rcl} M_{t} u^{(i)} + (I_{2 N} + M) u_{t}^{(i)} = i K^{i - 1} K_{t} r + K^{i} r_{t}, \end{array}

(2.15)

and we substitute(2.7b), (2.9) and (2.10) into (2.15), and then left-multiplied by

{(I_{2 N} + M)}^{- 1}

, we get

\begin{array}{rcl} u_{t}^{(i)} = \frac{1}{2} (, 4 i α u^{(i - 1)} - u^{(i + 2)} a + u^{(1)} a S^{(i, 0)} + u^{(0)} a S^{(i, 1)}) + α x u^{(i)} a . \end{array}

(2.16)

Using the relation (2.13), it is easy to get x-derivative of S^(i,j), which reads [23]

\begin{array}{rcl} S_{x}^{(i, j)} = \frac{1}{2} (, S^{(i + 1, j)} a - a S^{(i, j + 1)} - S^{(0, j)} a S^{(i, 0)}) . \end{array}

(2.17)

For the t-derivative of S^(i,j), from (2.13) we have

\begin{array}{rcl} S_{t}^{(i, j)} = s_{t}^{T} K^{j} u^{(i)} + j s^{T} K^{j - 1} K_{t} u^{(i)} + s^{T} K^{j} u_{t}^{(i)} . \end{array}

(2.18)

Then, substituting (2.7b), (2.9) and (2.16) into (2.18), we have

\begin{array}{rcl} \begin{array}{l} S_{t}^{(i, j)} = \frac{1}{2} (, a S^{(i, j + 2)} - S^{(i + 2, j)} a + S^{(1, j)} a S^{(i, 0)} + S^{(0, j)} a S^{(i, 1)}) \\ + α (, x (S^{(i, j)} a - a S^{(i, j)}) + 2 (j S^{(i, j - 1)} + i S^{(i - 1, j)})) . \end{array} \end{array}

(2.19a)

One can repeatedly use (2.17) and get second-order derivatives of S^(i,j) with respect to x, which which reads

\begin{array}{rcl} \begin{array}{l} S_{x x}^{(i, j)} = \frac{1}{4} (S^{(i, j + 2)} + S^{(i + 2, j)} - 2 a S^{(i + 1, j + 1)} a \\ + 2 a S^{(0, j + 1)} a S^{(i, 0)} - 2 S^{(0, j)} a S^{(i + 1, 0)} a \\ - S^{(1, j)} S^{(i, 0)} + S^{(0, j)} S^{(i, 1)} \\ + 2 S^{(0, j)} a S^{(0, 0)} a S^{(i, 0)}) . \end{array} \end{array}

(2.19b)

Next, let us define

\begin{array}{rcl} U ≐ S^{(0, 0)}, u_{i} = s_{i}^{(0, 0)}, (i = 1, 2, 3, 4), \end{array}

(2.20)

one has the following by taking i = j = 0 in (2.17) and (2.19):

\begin{array}{rcl} \begin{array}{l} U_{x} = \frac{1}{2} (S^{(1, 0)} a - a S^{(0, 1)} - U a U), \\ U_{x x} = \frac{1}{4} (S^{(2, 0)} + S^{(0, 2)} - 2 a S^{(1, 1)} a + 2 a S^{(0, 1)} a U - 2 U a S^{(1, 0)} a \\ - S^{(1, 0)} U + U S^{(0, 1)} + 2 U a U a U), \\ U_{t} = \frac{1}{2} (a S^{(0, 2)} - S^{(2, 0)} a \\ + S^{(1, 0)} a U + U a S^{(0, 1)}) + α x [U, a], \end{array} \end{array}

where [A, B] = AB − BA. Besides, by the difference-product formula (2.6), we obtain the following relations by choosing (i, j) = (0, 1), (1, 0), (0, 0), respectively:

\begin{array}{rcl} \begin{array}{l} S^{(0, 2)} = S^{(1, 1)} - S^{(0, 1)} U, \\ , S^{(2, 0)} = S^{(1, 1)} + U S^{(1, 0)}, \\ S^{(0, 1)} = S^{(1, 0)} - U^{2} . \end{array} \end{array}

Then by direct calculation we find

\begin{array}{rcl} a U_{t} - U_{x x} = - \frac{1}{2} [U, a] ([U, a] U - [S^{(1, 0)}, a]) + α x a [U, a] . \end{array}

(2.21)

Unfolding (2.21) we obtain a closed system of u₂ and u₃ as the unreduced form of the NNLSE-I:

\begin{array}{rcl} u_{2, t} - u_{2, x x} - 2 u_{2}^{2} u_{3} + 2 α {x u}_{2} = 0, \end{array}

(2.22a)

\begin{array}{rcl} u_{3, t} + u_{3, x x} + 2 u_{2} u_{3}^{2} - 2 α {x u}_{3} = 0. \end{array}

(2.22b)

2.3. Unreduced NNLSE-II

To obtain an unreduced form of the NNLSE-II, we introduce the following dispersion relations:

\begin{array}{rcl} r_{x} = \frac{1}{2} K r a, s_{x} = - \frac{1}{2} K^{T} s a, \end{array}

(2.23a)

\begin{array}{rcl} \begin{array}{l} r_{t} = \frac{1}{2} (, - K^{2} + i β x K,) r a + \frac{1}{2} i β r, \\ s_{t} = - \frac{1}{2} (, - {(K^{T})}^{2} + i β x K^{T},) s a + \frac{1}{2} i β s, \end{array} \end{array}

(2.23b)

where a is defined as (2.8) and

\begin{array}{rcl} K_{t} = i β K (t), β \in R . \end{array}

(2.24)

Again, we assume K_t and K commute. The evolutions of M and S^(i,j) with respect to x are the same as (2.11) and (2.17), so we only consider the t-derivatives. Using (2.23b) and (2.24) and employing a similar procedure as we derived (2.10), we have

\begin{array}{rcl} M_{t} = \frac{1}{2} (i β x r a s^{T} - K r a s^{T} - r a s^{T} K) . \end{array}

(2.25)

Note that (2.15) and (2.18) are generic, and can use them for this case as well. We substitute (2.25), (2.23b) and (2.24) into (2.15), similar to the treatment in section 2.2, we have

\begin{array}{rcl} \begin{array}{l} u_{t}^{(i)} = \frac{1}{2} (, 2 i β i u^{(i)} - u^{(i + 2)} a + i β u^{(i)} \\ + u^{(1)} a S^{(i, 0)} + u^{(0)} a S^{(i, 1)}) \\ + \frac{1}{2} i β x (, u^{(i + 1)} a - u^{(0)} a S^{(i, 0)}) . \end{array} \end{array}

(2.26)

Then, substituting (2.23b), (2.24) and (2.26) into (2.18), we get the derivative of S^(i,j) with respect to t:

\begin{array}{rcl} \begin{array}{l} S_{t}^{(i, j)} = \frac{1}{2} (, a S^{(i, j + 2)} - S^{(i + 2, j)} a + 2 i β (1 + i + j) S^{(i, j)} \\ + S^{(1, j)} a S^{(i, 0)} + S^{(0, j)} {a S}^{(i, 1)}) \\ + \frac{1}{2} i β x (, S^{(i + 1, j)} a - a S^{(i, j + 1)} - S^{(0, j)} a S^{(i, 0)}) . \end{array} \end{array}

(2.27)

Thus we have

\begin{array}{rcl} \begin{array}{l} U_{t} = \frac{1}{2} (, a S^{(0, 2)} - S^{(2, 0)} a + 2 i β U + S^{(1, 0)} a U \\ + U a S^{(0, 1)} + i β x (S^{(1, 0)} a - a S^{(0, 1)} - U a U)) . \end{array} \end{array}

Through a direct calculation, we find

\begin{array}{rcl} \begin{array}{l} a U_{t} - U_{x x} - i β a {(x U)}_{x} = - \frac{1}{2} [U, a] ([U, a] U - [S^{(1, 0)}, a]) . \end{array} \end{array}

(2.28)

which reveals an equation set of u₂ and u₃ as the unreduced NNLSE-II:

\begin{array}{rcl} u_{2, t} - u_{2, x x} - 2 u_{3} u_{2}^{2} - i β {({x u}_{2})}_{x} = 0, \end{array}

(2.29a)

\begin{array}{rcl} u_{3, t} + u_{3, x x} + 2 u_{2} u_{3}^{2} - i β {({x u}_{3})}_{x} = 0. \end{array}

(2.29b)

2.4. Unreduced NNLSE-III

In this case, the dispersion relations are:

\begin{array}{rcl} r_{x} = \frac{1}{2} K r a, s_{x} = - \frac{1}{2} K^{T} s a, \end{array}

(2.30a)

\begin{array}{rcl} r_{t} = - \frac{1}{2} K^{2} r a x - K r, s_{t} = \frac{1}{2} {(, K^{T})}^{2} s a x - K^{T} s, \end{array}

(2.30b)

where

\begin{array}{rcl} K_{t} = - K^{2} \end{array}

(2.31)

and we assume KK_t = K_tK. The evolution relations of M,

u_{t}^{(i)}

and S^(i,j) with respect to t are presented as below:

\begin{array}{rcl} M_{t} = - \frac{1}{2} x (K r a s^{T} + r a s^{T} K), \end{array}

(2.32a)

\begin{array}{rcl} u_{t}^{(i)} = \frac{1}{2} x (, u^{(1)} a S^{(i, 0)} + u^{(0)} a S^{(i, 1)} - u^{(i + 2)} a) - (i + 1) u^{(i + 1)}, \end{array}

(2.32b)

\begin{array}{rcl} \begin{array}{l} S_{t}^{(i, j)} = \frac{1}{2} x (a S^{(i, j + 2)} - S^{(i + 2, j)} a + S^{(1, j)} a S^{(i, 0)} + S^{(0, j)} a S^{(i, 1)}) \\ - (j + 1) S^{(i, j + 1)} - (i + 1) S^{(i + 1, j)} . \end{array} \end{array}

(2.32c)

Let i = j = 0 in (2.32c) we find

\begin{array}{rcl} \begin{array}{l} U_{t} = \frac{1}{2} x (a S^{(0, 2)} - S^{(2, 0)} a + S^{(1, 0)} a U + U a S^{(0, 1)}) - S^{(0, 1)} - S^{(1, 0)} . \end{array} \end{array}

By a direct calculation, we have

\begin{array}{rcl} \begin{array}{l} a U_{t} - x U_{x x} - 2 U_{x} = - \frac{1}{2} x [U, a] ([U, a] U - [S^{(1, 0)}, a]) \\ - a S^{(1, 0)} - S^{(1, 0)} a + U a U, \end{array} \end{array}

(2.33)

which leads to

\begin{array}{rcl} u_{2, t} - x (u_{2, x x} + 2 u_{3} u_{2}^{2}) - 2 u_{2, x} - 2 u_{2} \partial^{- 1} (u_{2} u_{3}) = 0, \end{array}

(2.34a)

\begin{array}{rcl} u_{3, t} + x (u_{3, x x} + 2 u_{2} u_{3}^{2}) + 2 u_{3, x} + 2 u_{3} \partial^{- 1} (u_{2} u_{3}) = 0, \end{array}

(2.34b)

where the relation

\begin{array}{rcl} u_{4} - u_{1} = - 2 \partial^{- 1} (u_{2} u_{3}) \end{array}

(2.35)

has been utilized, which has been proved in [23].

3. Explicit solutions for the unreduced NNLSEs

We have derived the unreduced NNLSEs. Their solutions are given by S^(0,0) which is determined by K, M, r and s through the formula (2.4). In isospectral case (see [23]) K is a constant matrix and one can equivalently consider its canonical form, i.e. diagonal or Jordan forms or their combinations, and the resulted solutions can be classified by the canonical forms of K. However, it is much different in non-isospectral case as K is no longer a constant matrix and must obey evolutions such as (2.9), (2.24) and (2.31). That means, in principle, we can not classify solutions by considering the canonical forms of K.

In this section, for convenience, we only consider the case of K being diagonal. There will be a case of K composed by lower Toeplitz matrices to be presented in appendix A.

In the following, let us take the unreduced NNLSE-I (2.22) as an example. Consider K = diag(K₁, K₂) with K₁, K₂ being the following diagonal forms

\begin{array}{rcl} K_{1} = d i a g (k_{1}, k_{2}, \dots, k_{N}), K_{2} = d i a g (l_{1}, l_{2}, \dots, l_{N}), \end{array}

(3.1)

where

\begin{array}{rcl} k_{j} (t) = 2 α t + c_{j}, l_{j} (t) = 2 α t + d_{j}, c_{j}, d_{j} \in C, \end{array}

(3.2)

such that K satisfies (2.9). The dispersion relation (2.7) yields

\begin{array}{rcl} \begin{array}{l} r_{1, x} = \frac{1}{2} K_{1} r_{1}, r_{2, x} = - \frac{1}{2} K_{2} r_{2}, r_{1, t} \\ = (- \frac{1}{2} K_{1}^{2} + α x) r_{1}, r_{2, t} = (\frac{1}{2} K_{2}^{2} - α x) r_{2}, \\ s_{1, x} = \frac{1}{2} K_{1}^{T} s_{1}, s_{2, x} = - \frac{1}{2} K_{2}^{T} s_{2}, s_{1, t} = (- \frac{1}{2} {(, K_{1}^{T})}^{2} + α x) s_{1}, s_{2, t} \\ = (\frac{1}{2} {(, K_{2}^{T})}^{2} - α x) s_{2} . \end{array} \end{array}

(3.3a)

(3.3a) has the following solutions:

\begin{array}{rcl} r_{1} = {(ρ_{1}, ρ_{2}, \dots, ρ_{N})}^{T}, s_{1} = {(σ_{1}, σ_{2}, \dots, σ_{N})}^{T}, \end{array}

(3.4a)

\begin{array}{rcl} r_{2} = {(ϱ_{1}, ϱ_{2}, \dots, ϱ_{N})}^{T}, s_{2} = {(ϖ_{1}, ϖ_{2}, \dots, ϖ_{N})}^{T}, \end{array}

(3.4b)

where

\begin{array}{rcl} ρ_{j} = e^{ξ_{j}}, ξ_{j} = \frac{k_{j} (t)}{2} x - \frac{k_{j}^{3} (t)}{12 α} + ξ_{j}^{(0)}, \end{array}

(3.5a)

\begin{array}{rcl} ϱ_{j} = e^{η_{j}}, η_{j} = - \frac{l_{j} (t)}{2} x + \frac{l_{j}^{3} (t)}{12 α} + η_{j}^{(0)}, \end{array}

(3.5b)

\begin{array}{rcl} σ_{j} = e^{ζ_{j}}, ζ_{j} = \frac{k_{j} (t)}{2} x - \frac{k_{j}^{3} (t)}{12 α} + ζ_{j}^{(0)}, \end{array}

(3.5c)

\begin{array}{rcl} ϖ_{j} = e^{ς_{j}}, ς_{j} = - \frac{l_{j} (t)}{2} x + \frac{l_{j}^{3} (t)}{12 α} + ς_{j}^{(0)}, \end{array}

(3.5d)

and

ξ_{j}^{(0)}, η_{j}^{(0)}, ζ_{j}^{(0)}, ς_{j}^{(0)}

are constants. Then, the set of Sylvester equations (2.3) allow solutions

M_{1} = {({(M_{1})}_{i, j})}_{N \times N}

and

M_{2} = {({(M_{2})}_{i, j})}_{N \times N}

where

\begin{array}{rcl} {(M_{1})}_{i, j} = \frac{ρ_{i} ϖ_{j}}{k_{i} - l_{j}}, {(M_{2})}_{i, j} = \frac{ϱ_{i} σ_{j}}{l_{i} - k_{j}} . \end{array}

(3.6)

Finally, we reach to the explicit expressions of u₂ and u₃:

\begin{array}{rcl} u_{2} = s_{2}^{T} {(, I_{N} - M_{2} M_{1})}^{- 1} r_{2}, \end{array}

(3.7a)

\begin{array}{rcl} u_{3} = s_{1}^{T} {(, I_{N} - M_{1} M_{2})}^{- 1} r_{1} . \end{array}

(3.7b)

which satisfy the unreduced NNLSE-I (2.22).

For the unreduced NNLSE-II (2.29) and the unreduced NNLSE-III (2.34), there solutions can be expressed through the formulae (3.7) with (3.4) and (3.6) but where ρ_j, ϱ_j, σ_j, ϖ_j and k_j, l_j are defined differently. For the unreduced NNLSE-II (2.29), we have

\begin{array}{rcl} ρ_{j} = e^{ξ_{j}}, ξ_{j} = \frac{k_{j} (t)}{2} x + \frac{i k_{j}^{2} (t)}{4 β} + \frac{1}{2} i β t + ξ_{j}^{(0)}, \end{array}

(3.8a)

\begin{array}{rcl} ϱ_{j} = e^{η_{j}}, η_{j} = - \frac{l_{j} (t)}{2} x - \frac{i l_{j}^{2} (t)}{4 β} + \frac{1}{2} i β t + η_{j}^{(0)}, \end{array}

(3.8b)

\begin{array}{rcl} σ_{j} = e^{ζ_{j}}, ζ_{j} = \frac{k_{j} (t)}{2} x + \frac{i k_{j}^{2} (t)}{4 β} + \frac{1}{2} i β t + ζ_{j}^{(0)}, \end{array}

(3.8c)

\begin{array}{rcl} ϖ_{j} = e^{ς_{j}}, ς_{j} = - \frac{l_{j} (t)}{2} x - \frac{i l_{j}^{2} (t)}{4 β} + \frac{1}{2} i β t + ς_{j}^{(0)}, \end{array}

(3.8d)

where

\begin{array}{rcl} k_{j} (t) = c_{j} e^{i β t}, l_{j} (t) = d_{j} e^{i β t} . \end{array}

(3.9)

For the unreduced NNLSE-III (2.34), we have

\begin{array}{rcl} ρ_{j} = e^{ξ_{j}}, ξ_{j} = \frac{k_{j} (t)}{2} x + l n (k_{j} (t)) + ξ_{j}^{(0)}, \end{array}

(3.10a)

\begin{array}{rcl} ϱ_{j} = e^{η_{j}}, η_{j} = - \frac{l_{j} (t)}{2} x + l n (l_{j} (t)) + η_{j}^{(0)}, \end{array}

(3.10b)

\begin{array}{rcl} σ_{j} = e^{ζ_{j}}, ζ_{j} = \frac{k_{j} (t)}{2} x + l n (k_{j} (t)) + ζ_{j}^{(0)}, \end{array}

(3.10c)

\begin{array}{rcl} ϖ_{j} = e^{ς_{j}}, ς_{j} = - \frac{l_{j} (t)}{2} x + l n (l_{j} (t)) + ς_{j}^{(0)}, \end{array}

(3.10d)

where

\begin{array}{rcl} k_{j} (t) = \frac{1}{t - c_{j}}, l_{j} (t) = \frac{1}{t - d_{j}} . \end{array}

(3.11)

4. Reduction to three NNLSEs and their solutions

4.1. General case

The reduction from the unreduced NNLSEs (i.e. (2.22), (2.29), (2.34)) to the three NNLSEs in (1.2) is

\begin{array}{rcl} u_{2} = u_{3}^{*}, \end{array}

(4.1)

together with replacing t → it. To achieve the above relation, we introduce constraint on K₁ and K₂ such that

\begin{array}{rcl} K_{2} = - K_{1}^{*} . \end{array}

(4.2)

Then, from the dispersion relations (2.7), (2.23) and (2.30), one can always get⁴(⁴In the diagonal case one should suitably take constants

ξ_{j}^{(0)}, η_{j}^{(0)}, ζ_{j}^{(0)}, ς_{j}^{(0)}

such that

η_{j} = ξ_{j}^{*}, ς_{j} = ζ_{j}^{*}

\begin{array}{rcl} r_{2} = r_{1}^{*}, s_{2} = s_{1}^{*} . \end{array}

(4.3)

Next, the original Sylvester equations (2.3) yield

\begin{array}{rcl} \begin{array}{l} K_{1} M_{1} + M_{1} K_{1}^{*} = r_{1} s_{1}^{†}, - K_{1}^{*} M_{2} - M_{2} K_{1} = r_{1}^{*} s_{1}^{T}, \end{array} \end{array}

and hence we have

\begin{array}{rcl} M_{2} = - M_{1}^{*} \end{array}

(4.4)

thanks to the uniqueness of the solutions of the Sylvester equation. Here

s_{1}^{†} = {(s_{1}^{T})}^{*}

. It then follows that

\begin{array}{rcl} \begin{array}{l} u_{2} = s_{2}^{T} {(, I_{N} - M_{2} M_{1})}^{- 1} r_{2} = s_{1}^{†} (I_{N} - M_{1}^{*} M_{2}^{*}) r_{1}^{*} = u_{3}^{*} . \end{array} \end{array}

In conclusion, for the three NNLSEs in (1.2), their solutions can be expressed in the form

\begin{array}{rcl} \begin{array}{l} q_{j} = u_{3} = s_{1}^{T} {(, I_{N} + M_{1} M_{1}^{*})}^{- 1} r_{1}, (j = 1, 2, 3), \end{array} \end{array}

(4.5)

with r₁, s₁ and M₁ accordingly.

In the following, for the three NNLSEs, we will look at their explicit solutions and illustrate their dynamics.

4.2. Explicit solutions of the NNLSE-I and dynamics

4.2.1. Formulation of solitons

With the above results, we rewrite the Sylvester equation and dispersion relations as

\begin{array}{rcl} K_{1} M_{1} + M_{1} K_{1}^{*} = r_{1} s_{1}^{†}, \end{array}

(4.6a)

\begin{array}{rcl} r_{1, x} = \frac{1}{2} K_{1} r_{1}, s_{1, x} = \frac{1}{2} K_{1}^{T} s_{1}, \end{array}

(4.6b)

\begin{array}{rcl} r_{1, t} = i (- \frac{1}{2} K_{1}^{2} + α x) r_{1}, s_{1, t} = i (- \frac{1}{2} {(, K_{1}^{T})}^{2} + α x) s_{1}, \end{array}

(4.6c)

where

\begin{array}{rcl} K_{1, t} = 2 i α I_{N}, α \in R . \end{array}

(4.7)

Then, solutions of the NNLSE-1 (1.2a) are given by (4.5) where M₁, r₁, s₁ satisfy the above settings. In particular, when

\begin{array}{rcl} K_{1} = d i a g (k_{1}, k_{2}, \dots, k_{N}) \end{array}

(4.8)

we have

\begin{array}{rcl} r_{1} = {(ρ_{1}, ρ_{2}, \dots, ρ_{N})}^{T}, s_{1} = {(σ_{1}, σ_{2}, \dots, σ_{N})}^{T}, \end{array}

(4.9a)

\begin{array}{rcl} {(M_{1})}_{i, j} = \frac{ρ_{i} σ_{j}^{*}}{k_{i} + k_{j}^{*}}, \end{array}

(4.9b)

where

\begin{array}{rcl} k_{j} (t) = 2 i α t + c_{j}, \end{array}

(4.10a)

\begin{array}{rcl} ρ_{j} = e^{ξ_{j}}, ξ_{j} = \frac{k_{j} (t)}{2} x - \frac{k_{j}^{3} (t)}{12 α} + ξ_{j}^{(0)}, \end{array}

(4.10b)

\begin{array}{rcl} σ_{j} = e^{ζ_{j}}, ζ_{j} = \frac{k_{j} (t)}{2} x - \frac{k_{j}^{3} (t)}{12 α} + ζ_{j}^{(0)} . \end{array}

(4.10c)

Note that

c_{j}, ξ_{j}^{(0)}, ζ_{j}^{(0)} \in C

, and throughout this section we write

\begin{array}{rcl} \begin{array}{l} c_{j} = a_{j} + i b_{j}, ξ_{j}^{(0)} = d_{j} + i e_{j}, \\ ζ_{j}^{(0)} = f_{j} + i h_{j}, \\ a_{j}, b_{j}, d_{j}, e_{j}, f_{j}, h_{j} \in R . \end{array} \end{array}

4.2.2. One-soliton solution

For N = 1 case, we have the following:

\begin{array}{rcl} K_{1} = k_{1}, r_{1} = ρ_{1} = e^{ξ_{1}}, s_{1} = σ_{1} = e^{ζ_{1}}, M_{1} = \frac{ρ_{1} σ_{1}^{*}}{k_{1} + k_{1}^{*}}, \end{array}

(4.11)

where ρ₁, σ₁ are defined as in (4.10). Substitute them into (4.5) one obtains

\begin{array}{rcl} q_{1} = \frac{4 a_{1}^{2} e^{ζ_{1} + ξ_{1}}}{4 a_{1}^{2} + e^{R e [ζ_{1} + ξ_{1}]}}, \end{array}

(4.12)

where for complex number

z = x + i y, x, y \in R

R e [z] = x

. Then the square module of one-soliton solution, namely, the envelope of the soliton, becomes:

\begin{array}{rcl} \begin{array}{l} | q_{1} |^{2} = a_{1}^{2} {s e c h}^{2} \\ \times (d_{1} + f_{1} + a_{1} x - \frac{a_{1}^{3} - 3 a_{1} {(, b_{1} + 2 t α)}^{2}}{6 α} - l n (2 a_{1})) . \end{array} \end{array}

(4.13)

For the shape and motion of ∣q₁∣² given by (4.13), we illustrate them in figure 1.

Figure 1. The shape and motion of the envelope of one-soliton solution given by (4.13) for $c_{1} = 0.4, α = 0.2, ξ_{1}^{(0)} = ζ_{1}^{(0)} = 0$ . (a) 3D plot. (b) 2D plot of (a) at t = 5 (red dot-dashed curve), t = −3 (blue dashed curve) and t = −4 (green solid curve).

Full size|PPT slide

The soliton ∣q₁∣² travels with a fixed amplitude

A = a_{1}^{2}

. The top trace of ∣q₁∣² is a parabola, which can be derived as

\begin{array}{rcl} x (t) = \frac{a_{1}^{3} - 3 a_{1} {(, b_{1} + 2 α t)}^{2} - 6 α (d_{1} + f_{1}) + 6 α l n (2 a_{1})}{6 α a_{1}}, \end{array}

(4.14)

of which the vertex point is

\begin{array}{rcl} (x, t) = (\frac{a_{1}^{3} - 6 α (d_{1} + f_{1}) + 6 α l n (2 a_{1}))}{6 α a_{1}}, - \frac{b_{1}}{2 α}), \end{array}

and the velocity of the soliton is

\begin{array}{rcl} x^{'} (t) = - 2 (2 α t + b_{1}) . \end{array}

4.2.3. Two-soliton solution

For N = 2 case, we have

\begin{array}{rcl} K_{1} = (\begin{array}{cc} k_{1} & 0 \\ 0 & k_{2} \end{array}), r_{1} = {(ρ_{1}, ρ_{2})}^{T}, s_{1} = {(σ_{1}, σ_{2})}^{T}, \end{array}

(4.15)

and the dressed Cauchy matrix comes to be

\begin{array}{rcl} M_{1} = (\begin{array}{cc} \frac{ρ_{1} σ_{1}^{*}}{k_{1} + k_{1}^{*}} & \frac{ρ_{1} σ_{2}^{*}}{k_{1} + k_{2}^{*}} \\ \frac{ρ_{2} σ_{1}^{*}}{k_{2} + k_{1}^{*}} & \frac{ρ_{2} σ_{2}^{*}}{k_{2} + k_{2}^{*}} \end{array}) . \end{array}

Then the explicit formula of the two-soliton solution (4.5) becomes:

\begin{array}{rcl} q_{1} = \frac{(1 + M_{11}) e^{ζ_{2} + ξ_{2}} - M_{12} e^{ζ_{1} + ξ_{2}} - M_{21} e^{ζ_{2} + ξ_{1}} + (1 + M_{22}) e^{ζ_{1} + ξ_{1}}}{1 + M_{11} - M_{12} M_{21} + M_{22} (1 + M_{11})}, \end{array}

(4.16)

where

\begin{array}{rcl} \begin{array}{l} (\begin{array}{cc} M_{11} & M_{12} \\ M_{21} & M_{22} \end{array}) = (\begin{array}{cc} \frac{e^{2 R e [ξ_{1} + ζ_{1}]}}{4 a_{1}^{2}} + \frac{e^{ξ_{1} + ζ_{1} + ξ_{2}^{*} + ζ_{2}^{*}}}{B^{2}} & \frac{1}{2} e^{ζ_{2} + ξ_{1}} (\frac{e^{ξ_{1}^{*} + ζ_{1}^{*}}}{a_{1} B^{*}} + \frac{e^{ξ_{2}^{*} + ζ_{2}^{*}}}{a_{2} B}) \\ \frac{1}{2} e^{ζ_{1} + ξ_{2}} (\frac{e^{ξ_{1}^{*} + ζ_{1}^{*}}}{a_{1} B^{*}} + \frac{e^{ξ_{2}^{*} + ζ_{2}^{*}}}{a_{2} B}) & \frac{e^{2 R e [ξ_{2} + ζ_{2}]}}{4 a_{2}^{2}} + \frac{e^{ξ_{2} + ζ_{2} + ξ_{1}^{*} + ζ_{1}^{*}}}{{(B^{*})}^{2}} \end{array}), \\ B = c_{1} + c_{2}^{*} = a_{1} + a_{2} + i (b_{1} - b_{2}) . \end{array} \end{array}

For the shape and motion of ∣q₁∣², we illustrate them in figure 2.

Figure 2. The shape and motion of the two-soliton solution given by ∣q₁∣² with (4.16) for $c_{1} = 0.2 + 0.2 i, c_{2} = 0.2, α = 0.2, ξ_{1}^{(0)} = ξ_{2}^{(0)} = ζ_{1}^{(0)} = ζ_{2}^{(0)} = 0$ . (a) 3D plot. (b) 2D plot of (a) at t = 2 (red dot-dashed curve), t = 0 (blue dashed curve) and t = −2 (brown solid curve).

Full size|PPT slide

4.2.4. Double-pole solution

The matrix K₁ can also be a triangle Toeplitz matrix (see appendix A), which will lead to the so-called multiple-pole solutions. We will present solution formula in appendix A for the Sylvester equations in (2.3) where K₁ and K₂ are triangular Toeplitz matrices. For the double-pole solution, it can be obtained by setting

\begin{array}{rcl} \begin{array}{l} K_{1} = (\begin{array}{cc} k_{1} & 0 \\ \partial_{c_{1}} k_{1} & k_{1} \end{array}), r_{1} = {(ρ_{1}, \partial_{c_{1}} ρ_{1})}^{T}, s_{1} \\ = {(σ_{1}, \partial_{c_{1}} σ_{1})}^{T}, \end{array} \end{array}

(4.17)

where k₁, ρ₁, σ₁ are defined as in (4.10). Then the dressed Cauchy matrix can be constructed via appendix A as

\begin{array}{rcl} M_{1} = (\begin{array}{cc} ρ_{1} & 0 \\ \partial_{c_{1}} ρ_{1} & ρ_{1} \end{array}) (\begin{array}{cc} \frac{1}{k_{1} + k_{1}^{*}} & - \frac{{(\partial_{c_{1}} k_{1})}^{*}}{{(k_{1} + k_{1}^{*})}^{2}} \\ - \frac{\partial_{c_{1}} k_{1}}{{(k_{1} + k_{1}^{*})}^{2}} & \frac{2 (\partial_{c_{1}} k_{1}) {(\partial_{c_{1}} k_{1})}^{*}}{{(k_{1} + k_{1}^{*})}^{3}} \end{array}) (\begin{array}{cc} σ_{1}^{*} & {(\partial_{c_{1}} σ_{1})}^{*} \\ {(\partial_{c_{1}} σ_{1})}^{*} & 0 \end{array}) . \end{array}

(4.18)

The explicit formula of the double-pole solution (4.5) reads

\begin{array}{rcl} q_{1} = \frac{16 a_{1}^{3} e^{ξ_{1} + ζ_{1}} (, 16 a_{1}^{5} (1 + D^{2}) - (a_{1} - 2 D^{*} + a_{1} {(, D^{2})}^{*}) D^{2} e^{2 R e [ξ_{1} + ζ_{1}]})}{256 a_{1}^{8} + 32 A_{1} a_{1}^{4} e^{2 R e [ξ_{1} + ζ_{1}]} + | D |^{4} e^{4 R e [ξ_{1} + ζ_{1}]}}, \end{array}

(4.19)

where

\begin{array}{rcl} \begin{array}{l} A_{1} = (3 | D |^{2} - 4 a_{1} R e [D] (1 + | D |^{2}) \\ + 2 a_{1}^{2} (1 + D^{2}) (1 + {(D^{2})}^{*})), \\ D = \frac{x}{2} - \frac{k_{1}^{2}}{4 α} . \end{array} \end{array}

The shape and motion of ∣q₁∣² are illustrated in figure 3.

Figure 3. The shape and motion of the double-pole soliton ∣q₁∣² with (4.19) for $c_{1} = - 0.4, α = 0.2, ξ_{1}^{(0)} = ζ_{1}^{(0)} = 0$ . (a) 3D plot. (b) 2D plot (a) at t = 4 (red dashed curve), t = 2 (green solid curve) and t = −2 (blue dot-dashed curve).

Full size|PPT slide

4.3. Explicit solutions of the NNLSE-II and dynamics

4.3.1. Formulation of solitons

Solution q₂ of the NNLSE-II equation (1.2b) can be expressed through (4.5) where M₁, r₁, s₁ are determined by

\begin{array}{rcl} K_{1} M_{1} + M_{1} K_{1}^{*} = r_{1} s_{1}^{†}, \end{array}

(4.20a)

\begin{array}{rcl} r_{1, x} = \frac{1}{2} K_{1} r_{1}, s_{1, x} = \frac{1}{2} K_{1}^{T} s_{1}, \end{array}

(4.20b)

\begin{array}{rcl} \begin{array}{l} r_{1, t} = - \frac{1}{2} (, i K_{1}^{2} + β x K_{1} + β,) r_{1}, \\ s_{1, t} = - \frac{1}{2} (, i {(, K_{1}^{T})}^{2} + β x K_{1}^{T} + β,) s_{1}, \end{array} \end{array}

(4.20c)

and

\begin{array}{rcl} K_{1, t} = - β K_{1} (t), β \in R . \end{array}

(4.21)

In the case of K₁ being a diagonal matrix (4.8), r₁, s₁, M₁ are given as (4.9) where

\begin{array}{rcl} k_{j} (t) = c_{j} e^{- β t}, \end{array}

(4.21a)

\begin{array}{rcl} ρ_{j} = e^{ξ_{j}}, ξ_{j} = \frac{1}{2} {x k}_{j} (t) + \frac{i k_{j}^{2} (t)}{4 β} - \frac{β}{2} t + ξ_{j}^{(0)}, \end{array}

(4.21b)

\begin{array}{rcl} σ_{j} = e^{ζ_{j}}, ζ_{j} = \frac{1}{2} {x k}_{j} (t) + \frac{i k_{j}^{2} (t)}{4 β} - \frac{β}{2} t + ζ_{j}^{(0)} . \end{array}

(4.21c)

4.3.2. One-soliton solution

The one-soliton solution corresponds to the N = 1 case, in which we have

\begin{array}{rcl} K_{1} = k_{1}, r_{1} = ρ_{1} = e^{ξ_{1}}, s_{1} = σ_{1} = e^{ζ_{1}}, M_{1} = \frac{ρ_{1} σ_{1}^{*}}{k_{1} + k_{1}^{*}}, \end{array}

(4.23)

where k₁, ρ₁, σ₁ are defined as in (4.21). Substituting them into (4.5) one obtains

\begin{array}{rcl} q_{2} = \frac{4 a_{1}^{2} e^{ξ_{1} + ζ_{1}}}{4 a_{1}^{2} + e^{2 (, β t + R e [ξ_{1} + ζ_{1}],)}}, \end{array}

(4.24)

and then the envelope reads

\begin{array}{rcl} | q_{2} |^{2} = a_{1}^{2} e^{- 2 t β} {s e c h}^{2} [a_{1} e^{- β t} (x - \frac{b_{1}}{β}) - l n (2 a_{1}) + d_{1} + f_{1}] . \end{array}

(4.25)

equation (4.25) describes a solitary wave traveling with an amplitude

a_{1}^{2} e^{- 2 β t}

, top trace

\begin{array}{rcl} x (t) = \frac{(l n (2 a_{1}) - (d_{1} + f_{1}),) e^{β t}}{a_{1}} + \frac{b_{1} e^{- β t}}{β} \end{array}

(4.26)

and velocity

\begin{array}{rcl} \begin{array}{l} x^{'} (t) = \frac{β (, l n (2 a_{1}) - (d_{1} + f_{1}),) e^{β t}}{a_{1}} - b_{1} e^{- β t} . \end{array} \end{array}

When

\frac{(l n (2 a_{1}) - (d_{1} + f_{1}))}{a_{1}} \frac{b_{1}}{β} > 0

, the top trace has a similar shape to

s g n [\frac{b_{1}}{β}] \cosh β t;

while when

\frac{(l n (2 a_{1}) - (d_{1} + f_{1}))}{a_{1}} \frac{b_{1}}{β} < 0

, the top trace has a similar shape to

- s g n [\frac{b_{1}}{β}] \sinh β t

. From (4.26) we can also find that when

(l n (2 a_{1}) - (d_{1} + f_{1})) = b_{1} = 0

, there is x(t) = 0, which corresponds to a stationary soliton as shown in figure 4. Note that the non-isospectral effects affect amplitude, velocity and shape of (4.25).

Figure 4. The shape and motion of stationary one-soliton solution given by (4.25) for $c_{1} = 0.5, β = 0.04, ξ_{1}^{(0)} = ζ_{1}^{(0)} = 0$ . (a) 3D plot. (b) 2D plot of (a) at t = −6 (blue dashed curve), t = 0 (green solid curve) and t = 5 (red dot-dashed curve).

Full size|PPT slide

4.3.3. Two-soliton solution

When N = 2 we have

\begin{array}{rcl} \begin{array}{l} K_{1} = (\begin{array}{cc} k_{1} & 0 \\ 0 & k_{2} \end{array}), r_{1} = {(ρ_{1}, ρ_{2})}^{T}, \\ s_{1} = {(σ_{1}, σ_{2})}^{T}, M_{1} = (\begin{array}{cc} \frac{ρ_{1} σ_{1}^{*}}{k_{1} + k_{1}^{*}} & \frac{ρ_{1} σ_{2}^{*}}{k_{1} + k_{2}^{*}} \\ \frac{ρ_{2} σ_{1}^{*}}{k_{2} + k_{1}^{*}} & \frac{ρ_{2} σ_{2}^{*}}{k_{2} + k_{2}^{*}} \end{array}), \end{array} \end{array}

(4.27)

where k_j, ρ_j, σ_j are defined as in (4.21). Then, the explicit formula the two-soliton solution (4.5) becomes:

\begin{array}{rcl} q_{2} = \frac{(1 + H_{11}) e^{ζ_{2} + ξ_{2}} - H_{12} e^{ζ_{1} + ξ_{2}} - H_{21} e^{ζ_{2} + ξ_{1}} + (1 + H_{22}) e^{ζ_{1} + ξ_{1}}}{1 + H_{11} - H_{12} H_{21} + H_{22} (1 + H_{11})}, \end{array}

(4.28)

where

\begin{array}{rcl} \begin{array}{l} (\begin{array}{cc} H_{11} & H_{12} \\ H_{21} & H_{22} \end{array}) \\ = (\begin{array}{cc} \frac{1}{4} e^{2 t β + ζ_{1} + ξ_{1}} (\frac{e^{ζ_{1}^{*} + ξ_{1}^{*}}}{a_{1}^{2}} + \frac{4 e^{ξ_{2}^{*} + ζ_{2}^{*}}}{B^{2}}) & \frac{1}{2} e^{2 t β + ζ_{2} + ξ_{1}} (\frac{e^{ζ_{1}^{*} + ξ_{1}^{*}}}{a_{1} B^{*}} + \frac{e^{ξ_{2}^{*} + ζ_{2}^{*}}}{a_{2} B}) \\ \frac{1}{2} e^{2 t β + ζ_{1} + ξ_{2}} (\frac{e^{ζ_{1}^{*} + ξ_{1}^{*}}}{a_{1} B^{*}} + \frac{e^{ξ_{2}^{*} + ζ_{2}^{*}}}{a_{2} B}) & \frac{1}{4} e^{2 t β + ζ_{2} + ξ_{2}} (\frac{4 e^{ζ_{1}^{*} + ξ_{1}^{*}}}{{(B^{*})}^{2}} + \frac{e^{ξ_{2}^{*} + ζ_{2}^{*}}}{a_{2}^{2}}) \end{array}), \\ B = c_{1} + c_{2}^{*} = a_{1} + a_{2} + i (b_{1} - b_{2}) . \end{array} \end{array}

The shape and motion of ∣q₂∣² are illustrated in figure 5.

Figure 5. The shape and motion of the envelope of two-soliton solution (4.28) for $c_{1} = - 0.2 - 0.2 i, c_{2} = 0.2 + 0.1 i, β = 0.1, ξ_{1}^{(0)} = ζ_{1}^{(0)} = ξ_{2}^{(0)} = ζ_{2}^{(0)} = 0$ . (a) 3D plot. (b) 2D plot of (a) at t = −15 (green solid curve), t = −12 (red dashed curve) and t = −6 (blue dashed curve).

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4.3.4. Double-pole solution

Double-pole solution of the NNLSE-II can be given by the formula (4.5) with the setting (4.17) and M₁ takes the form (4.18), where k₁, ρ₁, σ₁ are defined as in (4.21). The explicit formula of such a solution reads

\begin{array}{rcl} q_{2} = \frac{16 a_{1}^{3} e^{ξ_{1} + ζ_{1}} (, 16 a_{1}^{5} (1 + D^{2}) - D^{2} (a_{1} - 2 D^{*} + a_{1} {(, D^{2})}^{*}) e^{2 (β t + R e [ξ_{1} + ζ_{1}])})}{256 a_{1}^{8} + 32 A_{1} a_{1}^{4} e^{2 (β t + R e [ξ_{1} + ζ_{1}])} + | D |^{4} e^{4 (t β + R e [ξ_{1} + ζ_{1}])}}, \end{array}

(4.29)

where

\begin{array}{rcl} \begin{array}{l} A_{1} = 3 | D |^{2} - 4 a_{1} R e [D] (1 + | D |^{2}) + 2 a_{1}^{2} (1 + D^{2}) (1 + {(D^{*})}^{2}), \\ D = \frac{1}{2} x e^{- β t} + \frac{i c_{1} e^{- 2 β t}}{2 β} . \end{array} \end{array}

Shape and motion of ∣q₂∣² are illustrated in figure 6.

Figure 6. The shape and motion of the envelope of the double-pole soliton (4.29) for $c_{1} = 4, β = 0.07, ξ_{1}^{(0)} = 3, ζ_{1}^{(0)} = 0$ . (a) 3D plot. (b) 2D plot (a) at t = 24 (red dot-dashed curve), t = 20 (green solid curve) and t = 15 (blue dashed curve).

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4.4. Explicit solution of the NNLSE-III and dynamics

4.4.1. Formulation of solitons

For the NNLSE-III equation (1.2c), its solutions are formulated by (4.5) where M₁, r₁, s₁ are determined by

\begin{array}{rcl} K_{1} M_{1} + M_{1} K_{1}^{*} = r_{1} s_{1}^{†}, \end{array}

(4.30a)

\begin{array}{rcl} r_{1, x} = \frac{1}{2} K_{1} r_{1}, s_{1, x} = \frac{1}{2} K_{1}^{T} s_{1}, \end{array}

(4.30b)

\begin{array}{rcl} r_{1, t} = - i (\frac{1}{2} x K_{1}^{2} + K_{1}) r_{1}, s_{1, t} = - i (\frac{1}{2} x {(, K_{1}^{T})}^{2} + K_{1}) s_{1}, \end{array}

(4.30c)

and

\begin{array}{rcl} K_{1, t} = - i K_{1}^{2} . \end{array}

(4.31)

In the case of K₁ being a diagonal matrix (4.8), r₁, s₁, M₁ are given as (4.9) where

\begin{array}{rcl} k_{j} (t) = \frac{1}{i t - c_{j}}, \end{array}

(4.32a)

\begin{array}{rcl} ρ_{j} = e^{ξ_{j}}, ξ_{j} = \frac{1}{2} {x k}_{j} (t) + l n (k_{j} (t)) + ξ_{j}^{(0)}, \end{array}

(4.32b)

\begin{array}{rcl} σ_{j} = e^{ζ_{j}}, ζ_{j} = \frac{1}{2} {x k}_{j} (t) + l n (k_{j} (t)) + ζ_{j}^{(0)} . \end{array}

(4.32c)

4.4.2. One-soliton solution

For one-soliton, we have

\begin{array}{rcl} K_{1} = k_{1}, r_{1} = ρ_{1} = e^{ξ_{1}}, s_{1} = σ_{1} = e^{ζ_{1}}, M_{1} = \frac{ρ_{1} σ_{1}^{*}}{k_{1} + k_{1}^{*}}, \end{array}

(4.33)

where k₁, ρ₁, σ₁ are defined as in (4.32). From (4.5) we have

\begin{array}{rcl} q_{3} = \frac{4 a_{1}^{2} e^{ξ_{1} + ζ_{1}}}{4 a_{1}^{2} + {(a_{1}^{2} + {(b_{1} - t)}^{2})}^{2} e^{2 (, R e [ξ_{1} + ζ_{1}],)}}, \end{array}

(4.34)

which yields

\begin{array}{rcl} \begin{array}{l} | q_{3} |^{2} = \frac{a_{1}^{2}}{{(a_{1}^{2} + {(b_{1} - t)}^{2})}^{2}} {s e c h}^{2} \\ [d_{1} + f_{1} - \frac{a_{1} x}{a_{1}^{2} + {(b_{1} - t)}^{2}} - l n (2 a_{1})] . \end{array} \end{array}

(4.35)

The envelope is depicted in figure 7. It is interesting that ∣q₃∣² has a time-dependent amplitude

\frac{a_{1}^{2}}{{(a_{1}^{2} + {(b_{1} - t)}^{2})}^{2}}

, which indicates that the soliton is a localized wave with respect to both space and time. In addition, the top trace for (4.35) reads

\begin{array}{rcl} x (t) = \frac{A (a_{1}^{2} + {(b_{1} - t)}^{2})}{a_{1}}, w h e r e A = d_{1} + f_{1} - l n (2 a_{1}), \end{array}

which, in general, is a parabola curve, and along which the soliton travels and changes its direction at the vertex (t, x) = (b₁, Aa₁) where ∣q₃∣² takes maximum value

\frac{1}{a_{1}^{2}}

, see, e.g. Figure 7(c). A special case takes place when A = 0, which yields a stationary soliton, as depicted in figure 7(a).

Figure 7. The shape and motion the envelope of one-soliton solution given by (4.35) for (a) $c_{1} = 0.5, ξ_{1}^{(0)} = ζ_{1}^{(0)} = 0$ . (b) 2D plot of (a) at t = 0 (blue dashed curve), t = 0.3 (red dot-dashed curve) and t = 0.5 (green solid curve). (c) $c_{1} = 1 - 0.5 i, ξ_{1}^{(0)} = ζ_{1}^{(0)} = 2$ , (d) 2D plot of (c) at t = −0.5 (black dashed curve), t = 0 (red solid curve) and t = 0.5 (blue dashed curve).

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4.4.3. Two-soliton solution

The two-soliton solution is given by (4.5) where K₁, r₁, s₁, M₁ are given as in (4.27) but k_j, ρ_j, σ_j are defined as in (4.32). The solution is written as

\begin{array}{rcl} q_{3} = \frac{(1 + F_{11}) e^{ζ_{2} + ξ_{2}} - F_{12} e^{ζ_{1} + ξ_{2}} - F_{21} e^{ζ_{2} + ξ_{1}} + (1 + F_{22}) e^{ζ_{1} + ξ_{1}}}{1 + F_{11} - F_{12} F_{21} + F_{22} (1 + F_{11})}, \end{array}

(4.36)

where

\begin{array}{rcl} \begin{array}{l} (\begin{array}{cc} F_{11} & F_{12} \\ F_{21} & F_{22} \end{array}) \\ = (\begin{array}{cc} \frac{e^{2 R e [ξ_{1} + ζ_{1}]}}{{(k_{1} + k_{1}^{*})}^{2}} + \frac{e^{ξ_{1} + ζ_{1} + ζ_{2}^{*} + ξ_{2}^{*}}}{{(k_{1} + k_{2}^{*})}^{2}} & \frac{e^{2 R e [ξ_{1}] + ζ_{2} + ζ_{1}^{*}}}{(k_{1} + k_{1}^{*}) (k_{1}^{*} + k_{2})} + \frac{e^{2 R e [ζ_{2}] + ξ_{1} + ξ_{2}^{*}}}{(k_{1} + k_{2}^{*}) (k_{2} + k_{2}^{*})} \\ \frac{e^{2 R e [ζ_{1}] + ξ_{2} + ξ_{1}^{*}}}{(k_{1} + k_{1}^{*}) (k_{1}^{*} + k_{2})} + \frac{e^{2 R e [ξ_{2}] + ζ_{1} + ζ_{2}^{*}}}{(k_{1} + k_{2}^{*}) (k_{2} + k_{2}^{*})} & \frac{e^{2 R e [ξ_{2} + ζ_{2}]}}{{(k_{2} + k_{2}^{*})}^{2}} + \frac{e^{ξ_{1}^{*} + ζ_{1}^{*} + ζ_{2} + ξ_{2}}}{{(k_{1}^{*} + k_{2})}^{2}} \end{array}) . \end{array} \end{array}

For the shape and motion of the envelope ∣q₃∣², we illustrate them in figure 8, where (a) shows the scattering of two solitons and (b) describes interactions of two different stationary solitons.

Figure 8. The shape and motion of the envelope of two-soliton solution (4.36) for (a) $c_{1} = - 1.7 + 1.5 i, c_{2} = - 1.7 - 2.5 i, ξ_{1}^{(0)} = ζ_{1}^{(0)} = ξ_{2}^{(0)} = ζ_{2}^{(0)} = - 1$ . (b) $c_{1} = - 4 - 0.5 i, c_{2} = 0.5 - i, ξ_{1}^{(0)} = ζ_{1}^{(0)} = ξ_{2}^{(0)} = ζ_{2}^{(0)} = 0$ .

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4.4.4. Double-pole solution

Double-pole solution of the NNLSE-III is given by the formula (4.5) with the setting (4.17) and M₁ takes the form (4.18), where k₁, ρ₁, σ₁ are defined as in (4.32). Its explicit formula is

\begin{array}{rcl} q_{3} = \frac{(1 + R_{22} - (R_{12} + R_{21}) V + (1 + R_{11}) V^{*},) e^{ξ_{1} + ζ_{1}}}{1 + R_{11} - R_{12} R_{21} + R_{22} + R_{11} R_{22}}, \end{array}

(4.37)

where

\begin{array}{rcl} V = \frac{1}{2} k_{1}^{2} x + k_{1}, (\begin{array}{cc} R_{11} & R_{12} \\ R_{21} & R_{22} \end{array}) = M_{1} M_{1}^{*} . \end{array}

The shape and motion of ∣q₃∣² are illustrated in figure 9. When

ξ_{1}^{(0)} \neq ζ_{1}^{(0)}

we get two solitons moving along the same parabolic top trace, as shown in figure 9(a). When

ξ_{1}^{(0)} = ζ_{1}^{(0)} = 0

we get two overlapped solitons as shown in figure 9(c).

Figure 9. The shape and motion of the envelope of the double-pole solution (4.37) for (a) $c_{1} = - 2 + 2 i, ξ_{1}^{(0)} = - 2, ζ_{1}^{(0)} = - 3$ . (b) 2D plot (a) at t = 3 (blue dot-dashed curve), t = 2 (red dashed curve) and t = 1.5 (purple solid curve). (c) $c_{1} = 0.4, ξ_{1}^{(0)} = ζ_{1}^{(0)} = 0$ . (d) 2D plot (c) at t = 0.2 (green solid curve), t = 0.1 (red dot-dashed curve) and t = −0.4 (blue dashed curve).

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5. Conclusions

In this paper we have developed the Cauchy matrix approach to the NNLSEs, which serve as example models in the ZS-AKNS hierarchy. We believe that solutions of other order equations in the non-isospectral ZS-AKNS hierarchy, such as the non-isospectral sine-Gordon equation, the non-isospectral modified KdV (mKdV) equation and the non-isospectral Hirota equation (combination of the NLS and the mKdV) can be obtained along with this line.

In the Cauchy matrix approach, the Sylvester equation (e.g. (2.1)) plays an central role, which defines a dressed Cauchy matrix to provide τ functions (i.e. ∣I_2N + M∣) for the investigated equations. In non-isospectral case, one needs to suitably select dispersion relations of the time part (e.g. (2.7b), (2.23b) and (2.30b)) according to the time-evolution of the spectral parameters. One needs also to formulate special relations (e.g. (2.35)) to figure out the integration term (e.g. in (2.34)). Apparently, compared with the isospectral case [23], the non-isospectral extension of the Cauchy matrix scheme is quite non-trivial. In addition, in the isospectral case (see [23]) K is a constant matrix and it can be proved that K and its any similar form lead to same S^(i,j) therefore one only needs to consider its canonical form and the resulted solutions can be classified by the canonical forms of K. However, as we have seen that in non-isospectral case K is no longer a constant matrix, usually K and its similar form do not obey the same evolutions (e.g. (2.9), (2.24) and (2.31)). Therefore in the non-isospectral case, we can not classify solutions by considering the canonical forms of K. In appendix A we will formulate solutions of the Sylvester equations in (2.3) where K₁ and K₂ are triangular Toeplitz matrices, which are used to get multiple-pole solutions. Note that the Sylvester equation (2.1) to formulate the ZS-AKNS system is different from the one for the KdV type and KP type equations (see [22, 27]). Extension of the Cauchy matrix approach to the non-isospectral KdV and KP type equations (as well as the non-isospectral equations with sources, e.g. [30]) will be considered elsewhere.

Appendix A. Solutions to (2.3) with triangular Toeplitz matrices

We sketch a procedure to construct solutions to the Sylvester equations in (2.3) when K₁ and K₂ are lower triangular Toeplitz matrices. A lower triangular Toeplitz matrix is a square matrix in the following form and can be considered to be generated by a certain function:

\begin{array}{rcl} F_{c}^{[N]} [, f (c)] = {(\begin{array}{ccccc} f & 0 & 0 & \dots & 0 \\ \frac{\partial_{c} f}{1!} & f & 0 & \dots & 0 \\ \frac{\partial_{c}^{2} f}{2!} & \frac{\partial_{c} f}{1!} & f & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ \frac{\partial_{c}^{N - 1} f}{(N - 1)!} & \frac{\partial_{c}^{N - 2} f}{(N - 2)!} & \frac{\partial_{c}^{N - 3} f}{(N - 3)!} & \dots & f \end{array})}_{N \times N}, \end{array}

(A.1)

where f(c) is the generating function. Note that the subindex ‘c’ indicates that the lower triangular Toeplitz matrix is generated by taking derivatives with respect to c. We also introduce a symmetric matrix generated by f(c), denoted as

\begin{array}{rcl} H_{c}^{[N]} [f (c)] = {(\begin{array}{ccccc} f & \frac{\partial_{c} f}{1!} & \frac{\partial_{c}^{2} f}{2!} & . . . & \frac{\partial_{c}^{N - 1} f}{(N - 1)!} \\ \frac{\partial_{c} f}{1!} & \frac{\partial_{c}^{2} f}{2!} & \frac{\partial_{c}^{3} f}{3!} & . . . & 0 \\ \frac{\partial_{c}^{2} f}{2!} & \frac{\partial_{c}^{3} f}{3!} & \frac{\partial_{c}^{4} f}{4!} & . . . & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ \frac{\partial_{c}^{N - 1} f}{(N - 1)!} & 0 & 0 & . . . & 0 \end{array})}_{N \times N} . \end{array}

(A.2)

As a special property of such two types of matrices, we mention that [26]

\begin{array}{rcl} \begin{array}{l} F_{c}^{[N]} [f (c) g (c)] = F_{c}^{[N]} [f (c)] F_{c}^{[N]} [g (c)], \\ H_{c}^{[N]} [f (c) g (c)] = H_{c}^{[N]} [f (c)] F_{c}^{[N]} [g (c)] . \end{array} \end{array}

(A.3)

For more properties, one can refer to [29] and proposition 3 in [26].

In the following we will use the notations k₁, l₁, ρ₁, ϱ₁, σ₁ and ϖ₁ that we introduced in section 3 but we do not specify them since the following description is generic and true for all the three NNLSEs. We consider k₁ and l₁ to be functions of c₁ and d₁, respectively, e.g. (3.2), (3.9) and (3.11). Let

\begin{array}{rcl} K_{1} = F_{c_{1}}^{[N]} [, k_{1}], K_{2} = F_{d_{1}}^{[N]} [, l_{1}] . \end{array}

(A.4a)

Then, it can be verified that K = diag(K₁, K₂) satisfies the evolutions (2.9), (2.24) and (2.31) when k₁, l₁ are defined as (3.2), (3.9) and (3.11), respectively. Next, define

\begin{array}{rcl} r_{1} = F_{1} e_{N}, s_{2} = H_{2} e_{N}, F_{1} = F_{c_{1}}^{[N]} [, ρ_{1}], H_{2} = H_{d_{1}}^{[N]} [, ϖ_{1}], \end{array}

(A.4b)

\begin{array}{rcl} r_{2} = F_{2} e_{N}, s_{1} = H_{1} e_{N}, F_{2} = F_{d_{1}}^{[N]} [, ϱ_{1}], H_{1} = H_{c_{1}}^{[N]} [, σ_{1}], \end{array}

(A.4c)

where

\begin{array}{rcl} e_{N} = {(\underset{N - d i m e n s i o n a l}{\underset{⏟}{1, 0, 0, \dots, 0}})}^{T} . \end{array}

Then, the above defined elements satisfy the dispersion relations (2.7), (2.23) and (2.30) when k₁, l₁ are defined as (3.2), (3.9) and (3.11), respectively.

Next, we look for solution M₁ and M₂ of the Sylvester equations in (2.3) in the form

\begin{array}{rcl} M_{1} = F_{1} G_{1} H_{2}, M_{2} = F_{2} G_{2} H_{1}, \end{array}

(A.5)

where G₁ and G₂ are unknowns. In these settings, equation (2.3a) can be rewritten as

\begin{array}{rcl} K_{1} F_{1} G_{1} H_{2} - F_{1} G_{1} H_{2} K_{2} = F_{1} e_{N} e_{N}^{T} H_{2} . \end{array}

(A.6)

Using the relations [26]

\begin{array}{rcl} K_{1} F_{1} = F_{1} K_{1}, H_{2} K_{2} = K_{2}^{T} H_{2} . \end{array}

(A.6) reduces to

\begin{array}{rcl} K_{1} G_{1} - G_{1} K_{2}^{T} = e_{N} e_{N}^{T} . \end{array}

(A.7)

For convenience, we write G₁ = (g₁, g₂, …, g_N) where

g_{j} = {(g_{1, j}, g_{2, j}, \dots, g_{N, j})}^{T}

. Then, the first column of (A.7) reads

\begin{array}{rcl} (\begin{array}{c} k_{1} g_{1, 1} \\ (\partial_{c_{1}} k_{1}) g_{1, 1} + k_{1} g_{2, 1} \\ \frac{1}{2!} (\partial_{c_{1}}^{2} k_{1}) g_{1, 1} + (\partial_{c_{1}} k_{1}) g_{2, 1} + k_{1} g_{3, 1} \\ ⋮ \\ \frac{1}{(N - 1)!} (\partial_{c_{1}}^{N - 1} k_{1}) g_{1, 1} + \dots + k_{1} g_{N, 1} \end{array}) - l_{1} (\begin{array}{c} g_{1, 1} \\ g_{2, 1} \\ g_{3, 1} \\ ⋮ \\ g_{N, 1} \end{array}) = (\begin{array}{c} 1 \\ 0 \\ 0 \\ ⋮ \\ 0 \end{array}), \end{array}

which gives rise to

\begin{array}{rcl} g_{1, 1} = \frac{1}{k_{1} - l_{1}}, g_{m, 1} = - \frac{1}{k_{1} - l_{1}} (\sum_{j = 1}^{m - 1} \frac{\partial_{c_{1}}^{j} k_{1}}{j!} g_{m - j, 1}), m = 2, 3, \dots, N, \end{array}

(A.8)

from which one can successively determine g_2,1, g_3,1, ..., g_N,1. For example, we have

\begin{array}{rcl} \begin{array}{l} g_{2, 1} = - \frac{\partial_{c_{1}} k_{1}}{{(k_{1} - l_{1})}^{2}}, g_{3, 1} \\ = \frac{{(\partial_{c_{1}} k_{1})}^{2}}{{(k_{1} - l_{1})}^{3}} \\ - \frac{\partial_{c_{1}}^{2} k_{1}}{2! {(k_{1} - l_{1})}^{2}} . \end{array} \end{array}

Once with g₁ in hand, we can look at the second column of (A.7), which is

\begin{array}{rcl} \begin{array}{l} (\begin{array}{c} k_{1} g_{1, 2} \\ (\partial_{c_{1}} k_{1}) g_{1, 2} + k_{1} g_{2, 2} \\ \frac{1}{2!} (\partial_{c_{1}}^{2} k_{1}) g_{1, 2} + (\partial_{c_{1}} k_{1}) g_{2, 2} + k_{1} g_{3, 2} \\ ⋮ \\ \frac{1}{(N - 1)!} (\partial_{c_{1}}^{N - 1} k_{1}) g_{1, 2} + \dots + k_{1} g_{N, 2} \end{array}) \\ - (\begin{array}{c} (\partial_{d_{1}} l_{1}) g_{1, 1} + l_{1} g_{1, 2} \\ (\partial_{d_{1}} l_{1}) g_{2, 1} + l_{1} g_{2, 2} \\ (\partial_{d_{1}} l_{1}) g_{3, 1} + l_{1} g_{3, 2} \\ ⋮ \\ (\partial_{d_{1}} l_{1}) g_{N, 1} + l_{1} g_{N, 2} \end{array}) = (\begin{array}{c} 0 \\ 0 \\ 0 \\ ⋮ \\ 0 \end{array}) . \end{array} \end{array}

Element of g₂ can be calculated as:

\begin{array}{rcl} \begin{array}{l} g_{1, 2} = \frac{1}{k_{1} - l_{1}} (\partial_{d_{1}} l_{1}) g_{1, 1}, g_{m, 2} \\ = \frac{1}{k_{1} - l_{1}} (, (\partial_{d_{1}} l_{1}) g_{m, 1} \\ - \sum_{j = 1}^{m - 1} \frac{\partial_{c_{1}}^{j} k_{1}}{j!} g_{m - j, 2}), m = 2, 3, \dots, N . \end{array} \end{array}

(A.9)

The first few elements are

\begin{array}{rcl} \begin{array}{l} g_{1, 2} = \frac{\partial_{d_{1}} l_{1}}{{(k_{1} - l_{1})}^{2}}, g_{2, 2} = - \frac{2 (\partial_{c_{1}} k_{1}) (\partial_{d_{1}} l_{1})}{{(k_{1} - l_{1})}^{3}}, \\ g_{3, 2} = \frac{3 {(, \partial_{c_{1}} k_{1})}^{2} (\partial_{d_{1}} l_{1})}{{(k_{1} - l_{1})}^{4}} \\ - \frac{(\partial_{c_{1}}^{2} k_{1}) (\partial_{d_{1}} l_{1})}{{(k_{1} - l_{1})}^{3}} . \end{array} \end{array}

For the n-th column (n > 1) of (A.7), we have

\begin{array}{rcl} K_{1} g_{n} - \sum_{j = 1}^{n - 1} \frac{\partial_{d_{1}}^{j} l_{1}}{j!} g_{n - j} - l_{1} g_{n} = 0, \end{array}

which indicates that

\begin{array}{rcl} g_{n} = \sum_{j = 1}^{n - 1} \frac{\partial_{d_{1}}^{j} l_{1}}{j!} {(, K_{1} - l_{1} I_{N})}^{- 1} g_{n - l} . \end{array}

Finally, G₁ = (g₁, g₂, …, g_N) can be derived.

G₂ can be solved from equation (2.3b) in a similar way. Here we skip the details.

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Acknowledgments

This project is supported by the National Natural Science Foundation of China (No. 12271334).

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Abstract
Key words
Cite this article
1. Introduction
2. Cauchy matrix approach to unreduced NNLSEs
2.1. Sylvester equation and master functions
2.2. Unreduced NNLSE-I
2.3. Unreduced NNLSE-II
2.4. Unreduced NNLSE-III
3. Explicit solutions for the unreduced NNLSEs
4. Reduction to three NNLSEs and their solutions
4.1. General case
4.2. Explicit solutions of the NNLSE-I and dynamics
4.2.1. Formulation of solitons
4.2.2. One-soliton solution
Figure 1. The shape and motion of the envelope of one-soliton solution given by (4.13) for $c_{1} = 0.4, α = 0.2, ξ_{1}^{(0)} = ζ_{1}^{(0)} = 0$ . (a) 3D plot. (b) 2D plot of (a) at t = 5 (red dot-dashed curve), t = −3 (blue dashed curve) and t = −4 (green solid curve).
4.2.3. Two-soliton solution
Figure 2. The shape and motion of the two-soliton solution given by ∣q1∣2 with (4.16) for $c_{1} = 0.2 + 0.2 i, c_{2} = 0.2, α = 0.2, ξ_{1}^{(0)} = ξ_{2}^{(0)} = ζ_{1}^{(0)} = ζ_{2}^{(0)} = 0$ . (a) 3D plot. (b) 2D plot of (a) at t = 2 (red dot-dashed curve), t = 0 (blue dashed curve) and t = −2 (brown solid curve).
4.2.4. Double-pole solution
Figure 3. The shape and motion of the double-pole soliton ∣q1∣2 with (4.19) for $c_{1} = - 0.4, α = 0.2, ξ_{1}^{(0)} = ζ_{1}^{(0)} = 0$ . (a) 3D plot. (b) 2D plot (a) at t = 4 (red dashed curve), t = 2 (green solid curve) and t = −2 (blue dot-dashed curve).
4.3. Explicit solutions of the NNLSE-II and dynamics
4.3.1. Formulation of solitons
4.3.2. One-soliton solution
Figure 4. The shape and motion of stationary one-soliton solution given by (4.25) for $c_{1} = 0.5, β = 0.04, ξ_{1}^{(0)} = ζ_{1}^{(0)} = 0$ . (a) 3D plot. (b) 2D plot of (a) at t = −6 (blue dashed curve), t = 0 (green solid curve) and t = 5 (red dot-dashed curve).
4.3.3. Two-soliton solution
Figure 5. The shape and motion of the envelope of two-soliton solution (4.28) for $c_{1} = - 0.2 - 0.2 i, c_{2} = 0.2 + 0.1 i, β = 0.1, ξ_{1}^{(0)} = ζ_{1}^{(0)} = ξ_{2}^{(0)} = ζ_{2}^{(0)} = 0$ . (a) 3D plot. (b) 2D plot of (a) at t = −15 (green solid curve), t = −12 (red dashed curve) and t = −6 (blue dashed curve).
4.3.4. Double-pole solution
Figure 6. The shape and motion of the envelope of the double-pole soliton (4.29) for $c_{1} = 4, β = 0.07, ξ_{1}^{(0)} = 3, ζ_{1}^{(0)} = 0$ . (a) 3D plot. (b) 2D plot (a) at t = 24 (red dot-dashed curve), t = 20 (green solid curve) and t = 15 (blue dashed curve).
4.4. Explicit solution of the NNLSE-III and dynamics
4.4.1. Formulation of solitons
4.4.2. One-soliton solution
Figure 7. The shape and motion the envelope of one-soliton solution given by (4.35) for (a) $c_{1} = 0.5, ξ_{1}^{(0)} = ζ_{1}^{(0)} = 0$ . (b) 2D plot of (a) at t = 0 (blue dashed curve), t = 0.3 (red dot-dashed curve) and t = 0.5 (green solid curve). (c) $c_{1} = 1 - 0.5 i, ξ_{1}^{(0)} = ζ_{1}^{(0)} = 2$ , (d) 2D plot of (c) at t = −0.5 (black dashed curve), t = 0 (red solid curve) and t = 0.5 (blue dashed curve).
4.4.3. Two-soliton solution
Figure 8. The shape and motion of the envelope of two-soliton solution (4.36) for (a) $c_{1} = - 1.7 + 1.5 i, c_{2} = - 1.7 - 2.5 i, ξ_{1}^{(0)} = ζ_{1}^{(0)} = ξ_{2}^{(0)} = ζ_{2}^{(0)} = - 1$ . (b) $c_{1} = - 4 - 0.5 i, c_{2} = 0.5 - i, ξ_{1}^{(0)} = ζ_{1}^{(0)} = ξ_{2}^{(0)} = ζ_{2}^{(0)} = 0$ .
4.4.4. Double-pole solution
Figure 9. The shape and motion of the envelope of the double-pole solution (4.37) for (a) $c_{1} = - 2 + 2 i, ξ_{1}^{(0)} = - 2, ζ_{1}^{(0)} = - 3$ . (b) 2D plot (a) at t = 3 (blue dot-dashed curve), t = 2 (red dashed curve) and t = 1.5 (purple solid curve). (c) $c_{1} = 0.4, ξ_{1}^{(0)} = ζ_{1}^{(0)} = 0$ . (d) 2D plot (c) at t = 0.2 (green solid curve), t = 0.1 (red dot-dashed curve) and t = −0.4 (blue dashed curve).
5. Conclusions
Appendix A. Solutions to (2.3) with triangular Toeplitz matrices
References
Acknowledgments
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2024-02-19	2024-03-18	2024-03-20	2024-04-22
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2024-04-22

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Abstract

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Cite this article

1. Introduction

2. Cauchy matrix approach to unreduced NNLSEs

2.1. Sylvester equation and master functions

2.2. Unreduced NNLSE-I

2.3. Unreduced NNLSE-II

2.4. Unreduced NNLSE-III

3. Explicit solutions for the unreduced NNLSEs

4. Reduction to three NNLSEs and their solutions

4.1. General case

4.2. Explicit solutions of the NNLSE-I and dynamics

4.2.1. Formulation of solitons

4.2.2. One-soliton solution

Figure 1. The shape and motion of the envelope of one-soliton solution given by (4.13) for c1=0.4,α=0.2,ξ1(0)=ζ1(0)=0. (a) 3D plot. (b) 2D plot of (a) at t = 5 (red dot-dashed curve), t = −3 (blue dashed curve) and t = −4 (green solid curve).

4.2.3. Two-soliton solution

Figure 2. The shape and motion of the two-soliton solution given by ∣q1∣2 with (4.16) for c1=0.2+0.2i,c2=0.2,α=0.2,ξ1(0)=ξ2(0)=ζ1(0)=ζ2(0)=0. (a) 3D plot. (b) 2D plot of (a) at t = 2 (red dot-dashed curve), t = 0 (blue dashed curve) and t = −2 (brown solid curve).

4.2.4. Double-pole solution

Figure 3. The shape and motion of the double-pole soliton ∣q1∣2 with (4.19) for c1=−0.4,α=0.2,ξ1(0)=ζ1(0)=0. (a) 3D plot. (b) 2D plot (a) at t = 4 (red dashed curve), t = 2 (green solid curve) and t = −2 (blue dot-dashed curve).

4.3. Explicit solutions of the NNLSE-II and dynamics

4.3.1. Formulation of solitons

4.3.2. One-soliton solution

Figure 4. The shape and motion of stationary one-soliton solution given by (4.25) for c1=0.5,β=0.04,ξ1(0)=ζ1(0)=0. (a) 3D plot. (b) 2D plot of (a) at t = −6 (blue dashed curve), t = 0 (green solid curve) and t = 5 (red dot-dashed curve).

4.3.3. Two-soliton solution

Figure 5. The shape and motion of the envelope of two-soliton solution (4.28) for c1=−0.2−0.2i,c2=0.2+0.1i, β=0.1,ξ1(0)=ζ1(0)=ξ2(0)=ζ2(0)=0. (a) 3D plot. (b) 2D plot of (a) at t = −15 (green solid curve), t = −12 (red dashed curve) and t = −6 (blue dashed curve).

4.3.4. Double-pole solution

Figure 6. The shape and motion of the envelope of the double-pole soliton (4.29) for c1=4,β=0.07,ξ1(0)=3,ζ1(0)=0. (a) 3D plot. (b) 2D plot (a) at t = 24 (red dot-dashed curve), t = 20 (green solid curve) and t = 15 (blue dashed curve).

4.4. Explicit solution of the NNLSE-III and dynamics

4.4.1. Formulation of solitons

4.4.2. One-soliton solution

4.4.3. Two-soliton solution

Figure 8. The shape and motion of the envelope of two-soliton solution (4.36) for (a) c1=−1.7+1.5i,c2=−1.7−2.5i,ξ1(0)=ζ1(0)=ξ2(0)=ζ2(0)=−1. (b) c1=−4−0.5i,c2=0.5−i,ξ1(0)=ζ1(0)=ξ2(0)=ζ2(0)=0.

4.4.4. Double-pole solution

5. Conclusions

Appendix A. Solutions to (2.3) with triangular Toeplitz matrices

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References

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Footnotes

Acknowledgments

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RIGHTS & PERMISSIONS

Figure 1. The shape and motion of the envelope of one-soliton solution given by (4.13) for $c_{1} = 0.4, α = 0.2, ξ_{1}^{(0)} = ζ_{1}^{(0)} = 0$ . (a) 3D plot. (b) 2D plot of (a) at t = 5 (red dot-dashed curve), t = −3 (blue dashed curve) and t = −4 (green solid curve).

Figure 3. The shape and motion of the double-pole soliton ∣q₁∣² with (4.19) for $c_{1} = - 0.4, α = 0.2, ξ_{1}^{(0)} = ζ_{1}^{(0)} = 0$ . (a) 3D plot. (b) 2D plot (a) at t = 4 (red dashed curve), t = 2 (green solid curve) and t = −2 (blue dot-dashed curve).

Figure 4. The shape and motion of stationary one-soliton solution given by (4.25) for $c_{1} = 0.5, β = 0.04, ξ_{1}^{(0)} = ζ_{1}^{(0)} = 0$ . (a) 3D plot. (b) 2D plot of (a) at t = −6 (blue dashed curve), t = 0 (green solid curve) and t = 5 (red dot-dashed curve).

Figure 6. The shape and motion of the envelope of the double-pole soliton (4.29) for $c_{1} = 4, β = 0.07, ξ_{1}^{(0)} = 3, ζ_{1}^{(0)} = 0$ . (a) 3D plot. (b) 2D plot (a) at t = 24 (red dot-dashed curve), t = 20 (green solid curve) and t = 15 (blue dashed curve).