Conservation Laws for Pure–Cubic Optical Solitons with Self–Frequency Shift

Abstract

We investigate conservation laws for a third-order dispersion-dominated (pure-cubic) nonlinear Schrödinger equation with derivative perturbations and several forms of self-phase modulation. The multiplier approach yields local density–flux relations, and integrals of localized profiles are used to study optical power, linear momentum, and an energy-like invariant where their hypotheses are satisfied. These quantities provide diagnostics of power leakage, carrier-wave transport, and dispersive–nonlinear energy exchange. Importantly, inverse-power (anti-cubic) nonlinearities with a nonzero singular coefficient cannot support the previously proposed zero-background hyperbolic-secant profiles: the singular response diverges in their tails. Therefore, such profiles must not be interpreted as physical bright soliton solutions or used to assign conserved quantities. The distinction between formal conservation identities and admissible finite-energy waveforms is made explicit.

1. Introduction

The pulse propagation dynamics occur through the existence of a precise interplay involving chromatic dispersion (CD) together with self-phase modulation (SPM). In some regimes, this balance is compromised when the CD becomes weak enough. Under this condition, SPM is superseded by the third-order dispersive contribution (3OD). This leads toward the formation of the pure-cubic optical solitons [1,2,3,4]. However, the resulting nonlinear Schrödinger equation (NLSE) with CD replaced by 3OD is not integrable to retrieve an exact bright 1-soliton solution. Therefore, for its integrability, one must incorporate a couple of Hamiltonian correction terms, such as soliton self-frequency shifts. In such a situation, the bright 1-soliton solution has been recovered for the Kerr SPM law and also a few among the non-Kerr law structures [5]. It must be noted that the inclusion of 3OD yields soliton radiation in addition to the soliton solution [2,6]. It must be noted that the solution component from the continuous regime is neglected to just to stay focused on the discrete regime of the scattering transform.

From a physical viewpoint, the pure-cubic regime is relevant when third-order dispersive effects become comparable to, or stronger than, the residual second-order dispersion. The associated conserved quantities provide practical diagnostics for the persistence of a localized pulse under this higher-order propagation mechanism.

Related studies have considered dispersion–compensation strategies in optical communication systems [7], stationary solitons with nonlinear dispersion and general refractive responses [8], and quiescent optical solitons under nonlinear chromatic dispersion [9].

Recent extensions of the optical-envelope framework include stochastic generalized nonlinear Schrödinger models with fourth-order dispersion and comparably weak nonlocality [10], and nonlinear dispersive and refractive-index responses with multiplicative Itô noise [11]. The related perturbed Gerdjikov–Ivanov equation has also been used to examine the effects of derivative nonlinearity and multiplicative noise on soliton dynamics [12]. These studies address different stochastic or higher-order models; the studies provide helpful context but do not establish conservation laws for the deterministic pure-cubic equation considered here.

The present study retrieves the associated conservation laws for the perturbed pure-cubic optical solitary waves modeled by the NLSE with CD replaced by 3OD. The multipliers approach has made this recovery possible [13,14]. Subsequently, the reported localized one-soliton profiles have enabled the evaluation of global invariants using the established conserved density expressions. The three classes of conserved quantities that have been derived are the power (P), linear momentum (M) together with the Hamiltonian (H). Further details are presented throughout the remainder of this article for quite a wide variety of SPM structures apart from the usual Kerr law.

The optical significance of these conservation laws is as follows. The integrated intensity $P={\int }_{\mathbb{R}}{\left|q\right|}^{2}dx$, with $q(x,t)$ a complex-valued field dependent on independent space (x) and time (t) coordinates, measures the pulse power (or energy in a retarded-time envelope convention) up to normalization; invariance of P is a practical test for power leakage in reduced propagation models and numerical simulations. The linear momentum $M={\int }_{\mathbb{R}}\mathrm{Im}\left(q^{*}{q}_x\right)dx$, with the asteriks denoting complex conjugate and the subscript denoting the partial derivative over the coordinate, measures the phase-gradient-weighted intensity, so for an unchirped traveling profile with carrier slope $-\kappa$, one has $M=-\kappa P$. The energy-like invariant H combines dispersive and nonlinear contributions and is more sensitive than P to the response law and to the far-field behavior. Conservation of each quantity requires the corresponding local conservation identity, suitable decay or background conditions, and an actual solution of the governing equation; inserting a trial function into an integral alone does not establish a conserved observable.

These conservation laws are a great asset in optoelectronics. The laws aid in the study of four-wave mixing, as well as collision-induced timing jitter. Additionally, one can address the soliton perturbation theory to study the optical soliton cooling effect. Finally, additive stochastic perturbation of optical solitons can be studied with the conservation laws using the Langevin equations.

The relevance of soliton cooling is discussed in the related optical study [15]. For recent complementary results on perturbed pure-cubic optical solitons with Kerr and power-law responses, see [5]. The applications above are prospective uses of suitable conservation laws; four-wave-mixing efficiencies, collision-induced timing jitter, cooling rates, and stochastic Langevin dynamics are not computed in this study.

Recent study on Kerr-type response in engineered geometries, including solitons in composite linear–nonlinear moiré lattices [16], illustrates how the interplay between the spatial environment and nonlinear phase response affects localization. The present uniform, third-order-dispersive envelope model instead tests the applicability of conservation diagnostics under changes in the SPM function; the two physical settings are not identified with one another.

Governing Model

$$\begin{array}{c}\hfill i{q}_t+ia{q}_{xxx}+F\left({\left|q\right|}^2\right)q=i\left[c_1{\left|q\right|}^2{q}_x+c_2{\left({\left|q\right|}^2\right)}_{x}q\right]\end{array}$$

(1)

Here, in Equation (1), $q(x,t)$ gives the wave profile, and a is the coefficient of third order dispersion. The leading term represents the linear propagation term whose coefficient equals $i=\sqrt{-1}$. The next contribution is the 3OD contribution, whereas the third term models self-phase modulation having a functional F that accounts for non-Kerr laws of nonlinear response. On the right-hand side of Equation (1), the constants $c_1$ and $c_2$ are from soliton self-frequency shift. Typically when CD runs relatively low, it is discarded and replaced by 3OD. This constitutes pure-cubic optical solitons.

The two derivative nonlinearities on the right-hand side of Equation (1) describe intensity-dependent frequency shifts and therefore modify pulse transport without changing the basic envelope interpretation of q. Their contribution to the fluxes is essential when assessing whether the optical power, momentum, and Hamiltonian remain globally bounded.

For clarity, let $\rho ={\left|q\right|}^2$ and consider the common derivative contribution $i[c_1{\rho }^n{q}_x+c_2{\left({\rho }^n\right)}_{x}q]$ ($n>0$). Multiplying the evolution equation by $q^{*}$, adding the complex conjugate, and using that $F\left(\rho \right)$ is real gives the exact local intensity balance

$${\mathsf{\partial }}_t\rho +{\mathsf{\partial }}_x{J}_P=0,J_P=a\left(q^{*}{q}_{xx}+q{q}_{xx}^{*}-{\left|q_x\right|}^2\right)-{\displaystyle \frac{c_1+2n{c}_2}{n+1}}{\rho }^{n+1},$$

(2)

where ${\mathsf{\partial }}_s\equiv \mathsf{\partial }/\mathsf{\partial }s$.

In particular, on $[-L,L]$, $d{\int }_{-L}^L\rho dx/dt=J_P(-L,t)-J_P(L,t)$. Vanishing boundary flux gives constant power on the whole line. This identity describes a measurable transport balance, not merely an algebraic property of a chosen ansatz. The momentum and energy-like expressions stated below must likewise be used only on solutions for which their respective identities hold.

2. Application to Multiple SPM Structures

The conservation laws are extracted with the application of the multipliers approach for several forms of SPM. The recovered and reported bright 1-soliton solution structure is subsequently implemented for evaluating the invariant measures. The study is segmented into a number of separate considerations based on the type of nonlinearity.

For the regular SPM laws, the common sequence is to report a candidate localized profile, state local conserved density–flux pairs, and evaluate the associated integrals where the profile satisfies the field equation and the boundary terms vanish. The final two inverse-power SPM laws are instead examined separately for admissibility: a formal density–flux identity need not yield a physically bright soliton. This distinction is necessary before interpreting an integral as a conserved quantity.

2.1. Kerr Law

Considering a Kerr-type nonlinearity, this pure-cubic NLSE with soliton self-frequency shift is structured as

$$\begin{array}{c}\hfill i{q}_t+ia{q}_{xxx}+b{\left|q\right|}^{2}q=i\left[c_1{\left|q\right|}^2{q}_x+c_2{\left({\left|q\right|}^2\right)}_{x}q\right].\end{array}$$

(3)

A localized one-soliton profile takes the form [5]

$$\begin{array}{c}\hfill q(x,t)=A\mathrm{sech}\left[B(x-vt)\right]e^{i\left(-\kappa x+\omega t+\theta \right)}.\end{array}$$

(4)

Here, A denotes the soliton amplitude, while B specifies the reciprocal width with v representing its velocity with which the pulse travels. Within the phase factor, $\kappa$ denotes the carrier frequency, whereas $\omega$ specifies the soliton wavenumber. Finally, $\theta$ denotes the phase constant.

The hyperbolic-secant envelope represents a spatially localized optical pulse whose intensity decays exponentially away from its center. Hence, the resulting conserved integrals are finite whenever the parameter relations required by the governing equation are satisfied.

Substituting this envelope into the Kerr model and separating real and imaginary components gives a transparent compatibility check:

$$\begin{array}{cccc}\hfill \omega & =3a\kappa B^2-a{\kappa }^3,\hfill & \hfill v& =a(B^2-3{\kappa }^2),\hfill \\ \hfill b-c_1\kappa & ={\displaystyle \frac{6a\kappa B^2}{A^2}},\hfill & \hfill c_1+2c_2& =-{\displaystyle \frac{6a{B}^2}{A^2}}.\hfill \end{array}$$

(5)

These conditions are required for this particular constant-phase-gradient sech pulse in the stated evolution equation; they are not implied solely by convergence of P and M. Combining the final two conditions also yields $b=-2\kappa c_2$, which restricts the admissible parameter set. For $aB\ne 0$, the profile is therefore not an arbitrary solution of the Kerr model.

For a concrete nondimensional example, take $a=\kappa =B=A=1$, $b=2$, $c_1=-4$, and $c_2=-1$. Equation (5) then gives $v=-2$ and $\omega =2$. Direct substitution verifies the exact pulse $q(x,t)=\mathrm{sech}(x+2t)e^{i(-x+2t+\theta )}$; its integrated intensity and phase-gradient momentum are $P=2$ and $M=-2$. Inserting the pulse into Equation (2) gives $J_P=-2{\mathrm{sech}}^2(x+2t)=v{\left|q\right|}^2$, confirming the local transport identity and zero net power loss at spatial infinity. This is an analytical consistency demonstration, not an experiment or a numerical stability test.

Conservation Laws

The space and time components of the conserved flow $(T^t,T^x)$ that satisfies $D_t{T}^t+D_x{T}^x=0$ along the solution of the system (where $D_s$ denotes the total derivative in s), when $q=u+iv$ are separated are:

• Power:

$$\begin{array}{cc}\hfill & T^t={\displaystyle \frac{1}{2}}(u^2+v^2),\hfill \\ \hfill & T^x=-{\displaystyle \frac{c_1{u}^4}{4}}-{\displaystyle \frac{c_1{u}^2{v}^2}{2}}-{\displaystyle \frac{c_1{v}^4}{4}}-{\displaystyle \frac{c_2{u}^4}{2}}-c_2{u}^2{v}^2-{\displaystyle \frac{c_2{v}^4}{2}}+au{u}_{\mathit{xx}}-{\displaystyle \frac{a{u}_x^2}{2}}+av{v}_{\mathit{xx}}-{\displaystyle \frac{a{v}_x^2}{2}}\hfill \end{array}$$

(6)

so that the components of the conserved flow $({\mathsf{\Phi }}^t,{\mathsf{\Phi }}^x)$ that satisfy $D_t{\mathsf{\Phi }}^t+D_x{\mathsf{\Phi }}^x=0$ of Equation (2) are

$$\begin{array}{cc}\hfill & {\mathsf{\Phi }}^t={\displaystyle \frac{1}{2}}{\left|q\right|}^2,\hfill \\ \hfill & {\mathsf{\Phi }}^x=-{\displaystyle \frac{1}{4}}{c}_1{\left|q\right|}^4-{\displaystyle \frac{1}{2}}{c}_2{\left|q\right|}^4+a\Re \left(q{q}_{xx}^{*}\right)-{\displaystyle \frac{a}{2}}{\left|q_x\right|}^2,\hfill \end{array}$$

(7)

where ℜ denotes the real part. It is worth noting that ${\mathsf{\Phi }}^t$ and ${\mathsf{\Phi }}^x$ is the conserved density and flux, respectively.

• Linear Momentum ($c_2=0$):

$$\begin{array}{cc}\hfill & T^t={\displaystyle \frac{v_{x}u}{2}}-{\displaystyle \frac{u_{x}v}{2}},\hfill \\ \hfill & T^x=-\left(-{\displaystyle \frac{b{u}^4}{4}}-{\displaystyle \frac{b{u}^2{v}^2}{2}}-{\displaystyle \frac{b{v}^4}{4}}+a{v}_{\mathit{xx}}{u}_x-a{u}_{\mathit{xx}}{v}_x+{\displaystyle \frac{u{v}_t}{2}}-{\displaystyle \frac{u_{t}v}{2}}\right)\hfill \end{array}$$

(8)

or

$$\begin{array}{cc}\hfill & {\mathsf{\Phi }}^t=\Im \left(q^{*}{q}_x\right),\hfill \\ \hfill & {\mathsf{\Phi }}^x=-[-{\displaystyle \frac{1}{4}}{b\left|q\right|}^4+a\Im \left(q_x^{*}{q}_{xx}\right)+\Im \left(q^{*}{q}_t\right)],\hfill \end{array}$$

(9)

where ℑ denotes the imaginary part.

• Hamiltonian:

$$\begin{array}{cc}\hfill & T^t=-\left(-{\displaystyle \frac{b{u}^4}{4}}-{\displaystyle \frac{b{u}^2{v}^2}{2}}-{\displaystyle \frac{b{v}^4}{4}}-{\displaystyle \frac{c_1{u}^3{v}_x}{4}}+{\displaystyle \frac{c_1{u}^2{u}_{x}v}{4}}-{\displaystyle \frac{c_{1}u{v}^2{v}_x}{4}}+{\displaystyle \frac{c_1{u}_x{v}^3}{4}}+{\displaystyle \frac{au{v}_{\mathit{xxx}}}{2}}-{\displaystyle \frac{a{u}_{\mathit{xxx}}v}{2}}\right),\hfill \\ \hfill & T^x=-{\displaystyle \frac{c_1{u}^3{v}_t}{4}}+{\displaystyle \frac{c_1{u}^2{u}_{t}v}{4}}-{\displaystyle \frac{c_{1}u{v}^2{v}_t}{4}}+{\displaystyle \frac{c_1{u}_t{v}^3}{4}}+{\displaystyle \frac{au{v}_{xxt}}{2}}-{\displaystyle \frac{a{v}_{\mathit{xt}}{u}_x}{2}}-{\displaystyle \frac{a{u}_t{v}_{\mathit{xx}}}{2}}+{\displaystyle \frac{a{v}_t{u}_{\mathit{xx}}}{2}}+{\displaystyle \frac{a{u}_{\mathit{xt}}{v}_x}{2}}-{\displaystyle \frac{a{u}_{\mathit{xxt}}v}{2}}\hfill \end{array}$$

(10)

or

$$\begin{array}{cc}\hfill & {\mathsf{\Phi }}^t=-{\displaystyle \frac{b}{4}}{\left|q\right|}^4-{\displaystyle \frac{c_1}{4}}{\left|q\right|}^2\Im \left(q^{*}{q}_x\right)-{\displaystyle \frac{1}{2}}a\Im \left(q^{*}{q}_{xxx}\right),\hfill \\ \hfill & {\mathsf{\Phi }}^x=-{\displaystyle \frac{c_1}{4}}{\left|q\right|}^2\Im \left(q^{*}{q}_t\right)+{\displaystyle \frac{a}{2}}{D}_x\Im \left(q^{*}{q}_{xt}\right).\hfill \end{array}$$

(11)

The density–flux representation separates the local balance mechanism from the global invariant obtained after integration. Under the usual vanishing boundary conditions, the flux contributions at spatial infinity disappear, leaving time-independent integral measures of the localized pulse.

Therefore, the conserved quantities, respectively, are

$$\begin{array}{c}\hfill P=2{\int }_{-\infty }^{\infty }{\mathsf{\Phi }}^{t}dx={\int }_{-\infty }^{\infty }{\left|q\right|}^{2}dx={\displaystyle \frac{2A^2}{B}},\end{array}$$

(12)

$$\begin{array}{c}\hfill M={\int }_{-\infty }^{\infty }{\mathsf{\Phi }}^{t}dx={\displaystyle \frac{1}{2i}}{\int }_{-\infty }^{\infty }\left(q^{*}{q}_x-q{q}_x^{*}\right)dx=-{\displaystyle \frac{2\kappa A^2}{B}}\end{array}$$

(13)

and

$$\begin{array}{ccc}\hfill H& =& {\int }_{-\infty }^{\infty }{\mathsf{\Phi }}^{t}dx={\int }_{-\infty }^{\infty }\left[b{\left|q\right|}^4+{\displaystyle \frac{c_1}{2i}}{\left|q\right|}^2\left\{q^{*}{q}_x-q{q}_x^{*}\right\}+{\displaystyle \frac{a}{i}}\left(q^{*}{q}_{xxx}-q{q}_{xxx}^{*}\right)\right]dx\hfill \\ & =& {\displaystyle \frac{4A^2}{3b}}\left[A^2\left(b-\kappa c_1\right)+3a\kappa \left(B^2+{\kappa }^2\right)\right].\hfill \end{array}$$

(14)

2.2. Power Law

Adopting a power-law response, the perturbed pure-cubic NLSE is

$$\begin{array}{c}\hfill i{q}_t+ia{q}_{xxx}+b{\left|q\right|}^{2n}q=i\left[c_1{\left|q\right|}^{2n}{q}_x+c_2{\left({\left|q\right|}^{2n}\right)}_{x}q\right].\end{array}$$

(15)

Here, n is the power-law nonlinearity parameter. A localized one-soliton structure takes the form

$$\begin{array}{c}\hfill q(x,t)=A{\mathrm{sech}}^{{\displaystyle \frac{1}{n}}}\left[B(x-vt)\right]e^{i\left(-\kappa x+\omega t+\theta \right)}.\end{array}$$

(16)

Here, the exponent $1/n$ controls the degree of pulse localization. Consequently, the power-law index changes not only the nonlinear response but also the width dependence of the global invariants, as reflected by the Gamma functions in formulas below.

Conservation Laws

• Power:

$$\begin{array}{cc}\hfill T^t=& {\displaystyle \frac{1}{2}}(u^2+v^2),\hfill \\ \hfill T^x=& -{\displaystyle \frac{1}{2(n+1)}}[-2anu{u}_{\mathit{xx}}+an{u}_x^2-2anv{v}_{\mathit{xx}}+an{v}_x^2\hfill \\ \hfill & +2{\left(u^2+v^2\right)}^{1+n}{c}_{2}n-2au{u}_{\mathit{xx}}+a{u}_x^2-2av{v}_{\mathit{xx}}\hfill \\ \hfill & +a{v}_x^2+{\left(u^2+v^2\right)}^{1+n}{c}_1]\hfill \end{array}$$

(17)

so that the components of the conserved flow $({\mathsf{\Phi }}^t,{\mathsf{\Phi }}^x)$ that satisfy $D_t{\mathsf{\Phi }}^t+D_x{\mathsf{\Phi }}^x=0$ of Equation (2) are

$$\begin{array}{cc}\hfill & {\mathsf{\Phi }}^t={\displaystyle \frac{1}{2}}{\left|q\right|}^2,\hfill \\ \hfill & {\mathsf{\Phi }}^x={\displaystyle \frac{a}{2}}{\left(\right|q|}^2{)}_{xx}-{\displaystyle \frac{3a}{2}}|q_x{|}^2-{\displaystyle \frac{2c_{2}n+c_1}{2(n+1)}}{\left|q\right|}^{2(n+1)}.\hfill \end{array}$$

(18)

• Linear Momentum ($c_2=0$):

$$\begin{array}{cc}\hfill T^t=& {\displaystyle \frac{v_{x}u}{2}}-{\displaystyle \frac{u_{x}v}{2}},\hfill \\ \hfill T^x=& -{\displaystyle \frac{1}{2(n+1)}}[2an{u}_x{v}_{\mathit{xx}}-2an{u}_{\mathit{xx}}{v}_x+2a{v}_{\mathit{xx}}{u}_x\hfill \\ \hfill & -2a{u}_{\mathit{xx}}{v}_x+nu{v}_t-n{u}_{t}v-{\left(u^2+v^2\right)}^{1+n}b+u{v}_t-u_{t}v]\hfill \end{array}$$

(19)

or

$$\begin{array}{cc}\hfill & {\mathsf{\Phi }}^t=\Im \left(q^{*}{q}_x\right),\hfill \\ \hfill & {\mathsf{\Phi }}^x=-a\Im \left(q_x^{*}{q}_{xx}\right)-{\displaystyle \frac{1}{2}}\Im \left(q^{*}{q}_t\right)+{\displaystyle \frac{b}{2(n+1)}}{\left|q\right|}^{2(n+1)}.\hfill \end{array}$$

(20)

• Hamiltonian:

$$\begin{array}{cc}\hfill T^t=& {\displaystyle \frac{1}{4(n+1)}}[c_{1}n{u}^3{v}_x-c_{1}n{u}^2{u}_{x}v+c_{1}nu{v}^2{v}_x\hfill \\ \hfill & -c_{1}n{u}_x{v}^3+c_1{u}^3{v}_x-c_1{u}^2{u}_{x}v+c_{1}u{v}^2{v}_x\hfill \\ \hfill & -c_1{u}_x{v}^3\hfill \\ \hfill & +2{\left(u^2+v^2\right)}^{n}b{u}^2+2{\left(u^2+v^2\right)}^{n}b{v}^2-2au{v}_{\mathit{xxx}}n\hfill \\ \hfill & +2a{u}_{\mathit{xxx}}vn-2au{v}_{\mathit{xxx}}+2a{u}_{\mathit{xxx}}v],\hfill \\ \hfill T^x=& -{\displaystyle \frac{c_1{u}^3{v}_t}{4}}+{\displaystyle \frac{c_1{u}^2{u}_{t}v}{4}}-{\displaystyle \frac{c_{1}u{v}^2{v}_t}{4}}\hfill \\ \hfill & +{\displaystyle \frac{c_1{u}_t{v}^3}{4}}+{\displaystyle \frac{au{v}_{1,1,2}}{2}}-{\displaystyle \frac{a{v}_{\mathit{xt}}{u}_x}{2}}\hfill \\ \hfill & -{\displaystyle \frac{a{u}_t{v}_{\mathit{xx}}}{2}}+{\displaystyle \frac{a{v}_t{u}_{\mathit{xx}}}{2}}+{\displaystyle \frac{a{u}_{\mathit{xt}}{v}_x}{2}}\hfill \\ \hfill & -{\displaystyle \frac{a{u}_{\mathit{xxt}}v}{2}}\hfill \end{array}$$

(21)

or

$$\begin{array}{cc}\hfill & {\mathsf{\Phi }}^t=-{\displaystyle \frac{a}{2}}\Im \left(q^{*}{q}_{xxx}\right)+{\displaystyle \frac{c_1}{4}}{\left|q\right|}^2\Im \left(q^{*}{q}_x\right)+{\displaystyle \frac{b}{2(n+1)}}{\left|q\right|}^{2(n+1)},\hfill \\ \hfill & {\mathsf{\Phi }}^x={\displaystyle \frac{a}{2}}{D}_x\Im \left(q^{*}{q}_{xt}\right)-{\displaystyle \frac{c_1}{4}}{\left|q\right|}^2\Im \left(q^{*}{q}_t\right).\hfill \end{array}$$

(22)

Therefore, the resulting invariants, respectively, take the forms

$$\begin{array}{c}\hfill P=2{\int }_{-\infty }^{\infty }{\mathsf{\Phi }}^{t}dx={\int }_{-\infty }^{\infty }{\left|q\right|}^{2}dx={\displaystyle \frac{A^2}{B}}{\displaystyle \frac{\mathsf{\Gamma }\left(\frac{1}{n}\right)\mathsf{\Gamma }\left(\frac{1}{2}\right)}{\mathsf{\Gamma }\left(\frac{1}{n}+\frac{1}{2}\right)}},\end{array}$$

(23)

$$\begin{array}{c}\hfill M={\int }_{-\infty }^{\infty }{\mathsf{\Phi }}^{t}dx={\displaystyle \frac{1}{2i}}{\int }_{-\infty }^{\infty }\left(q^{*}{q}_x-q{q}_x^{*}\right)dx=-{\displaystyle \frac{\kappa A^2}{B}}·{\displaystyle \frac{\mathsf{\Gamma }\left(\frac{1}{n}\right)\mathsf{\Gamma }\left(\frac{1}{2}\right)}{\mathsf{\Gamma }\left(\frac{1}{n}+\frac{1}{2}\right)}}\end{array}$$

(24)

and

$$\begin{array}{ccc}\hfill H& =& {\int }_{-\infty }^{\infty }{\mathsf{\Phi }}^{t}dx={\int }_{-\infty }^{\infty }\left({\displaystyle \frac{a}{i}}\Im \left(q^{*}{q}_{xxx}\right)+{\displaystyle \frac{c_1}{2i}}{\left|q\right|}^2\Im \left(q^{*}{q}_x\right)+{\displaystyle \frac{b}{n+1}}{\left|q\right|}^{2(n+1)}\right)dx\hfill \\ & =& {\displaystyle \frac{1}{B}}\left\{2a\kappa A^2\left[{\kappa }^2+{\displaystyle \frac{3B^2}{n(n+2)}}\right]+{\displaystyle \frac{2b{A}^{2n+2}}{(n+1)(n+2)}}{\displaystyle \frac{\mathsf{\Gamma }\left(\frac{1}{2}\right)\mathsf{\Gamma }\left(\frac{1}{n}\right)}{\mathsf{\Gamma }\left(\frac{1}{n}+\frac{1}{2}\right)}}-{\displaystyle \frac{c_1\kappa A^4}{B}}{\displaystyle \frac{\mathsf{\Gamma }\left(\frac{1}{2}\right)\mathsf{\Gamma }\left(\frac{2}{n}\right)}{\mathsf{\Gamma }\left(\frac{2}{n}+\frac{1}{2}\right)}}\right\}.\hfill \end{array}$$

(25)

2.3. Dual-Power Law

Here, the perturbed pure-cubic NLSE is given as

$$\begin{array}{c}\hfill i{q}_t+ia{q}_{xxx}+\left(b_1{\left|q\right|}^{2n}+b_2{\left|q\right|}^{2n+2}\right)q=i\left[c_1{\left|q\right|}^{2n}{q}_x+c_2{\left({\left|q\right|}^{2n}\right)}_{x}q\right].\end{array}$$

(26)

A localized one-soliton structure takes the form

$$\begin{array}{c}\hfill q(x,t)=A{\mathrm{sech}}^{{\displaystyle \frac{1}{n+1}}}\left[B(x-vt)\right]e^{i\left(-\kappa x+\omega t+\theta \right)}.\end{array}$$

(27)

These solitons exist whenever the following two parameter constraints hold:

$$\begin{array}{c}\hfill c_1+2n{c}_2=0\end{array}$$

(28)

and

$$\begin{array}{c}\hfill b_1+2na{c}_2=0.\end{array}$$

(29)

The restrictions (28) and (29) are compatibility conditions for the exact localized waveform. Physically, they balance the competing nonlinear phase response, third-order dispersion, and self-frequency-shift contributions so that a stationary pulse shape can propagate without secular deformation.

Conservation Laws

• Power:

$$\begin{array}{cc}\hfill & T^t={\displaystyle \frac{1}{2}}(u^2+v^2),\hfill \\ \hfill & T^x=-{\displaystyle \frac{1}{2(n+1)}}[-2anu{u}_{\mathit{xx}}+an{u}_x^2-2anv{v}_{\mathit{xx}}+an{v}_x^2+2{\left(u^2+v^2\right)}^{1+n}{c}_{2}n-2au{u}_{\mathit{xx}}+a{u}_x^2\hfill \\ \hfill & -2av{v}_{\mathit{xx}}+a{v}_x^2+{\left(u^2+v^2\right)}^{1+n}{c}_1]\hfill \end{array}$$

(30)

so that the components of the conserved flow $({\mathsf{\Phi }}^t,{\mathsf{\Phi }}^x)$ that satisfy $D_t{\mathsf{\Phi }}^t+D_x{\mathsf{\Phi }}^x=0$ of Equation (2) are

$$\begin{array}{cc}\hfill & {\mathsf{\Phi }}^t={\displaystyle \frac{1}{2}}{\left|q\right|}^2,\hfill \\ \hfill & {\mathsf{\Phi }}^x={\displaystyle \frac{a}{2}}{({|q|}^2)}_{xx}-{\displaystyle \frac{3a}{2}}|q_x{|}^2-{\displaystyle \frac{2c_{2}n+c_1}{2(n+1)}}{\left|q\right|}^{2(n+1)}.\hfill \end{array}$$

(31)

• Linear Momentum ($c_2=0$):

$$\begin{array}{cc}\hfill T^t=& {\displaystyle \frac{v_{x}u}{2}}-{\displaystyle \frac{u_{x}v}{2}},\hfill \\ \hfill T^x=& {\displaystyle \frac{1}{2(n^2+3n+2)}}[-{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{2}}n{u}^4\hfill \\ \hfill & -2{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{2}}n{u}^2{v}^2-{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{2}}n{v}^4\hfill \\ \hfill & -{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{2}}{u}^4\hfill \\ \hfill & -2{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{2}}{u}^2{v}^2-{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{2}}{v}^4\hfill \\ \hfill & -{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{1}}n{u}^2-{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{1}}n{v}^2\hfill \\ \hfill & +2a{n}^2{u}_x{v}_{\mathit{xx}}-2a{n}^2{u}_{\mathit{xx}}{v}_x\hfill \\ \hfill & -2{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{1}}{u}^2-2{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{1}}{v}^2\hfill \\ \hfill & +6an{u}_x{v}_{\mathit{xx}}-6an{u}_{\mathit{xx}}{v}_x+n^{2}u{v}_t-n^2{u}_{t}v+4a{v}_{\mathit{xx}}{u}_x\hfill \\ \hfill & -4a{u}_{\mathit{xx}}{v}_x\hfill \\ \hfill & +3nu{v}_t-3n{u}_{t}v+2u{v}_t-2u_{t}v]\hfill \end{array}$$

(32)

or

$$\begin{array}{cc}\hfill & {\mathsf{\Phi }}^t=\Im \left(q^{*}{q}_x\right),\hfill \\ \hfill & {\mathsf{\Phi }}^x={\displaystyle \frac{{\left|q\right|}^{2(n+1)}\left[-b_2(n+1){\left|q\right|}^2-b_1(n+2)\right]}{2(n+1)(n+2)}}+a\Im \left(q_x^{*}{q}_{xx}\right)+{\displaystyle \frac{1}{2}}\Im \left(q^{*}{q}_t\right).\hfill \end{array}$$

(33)

• Hamiltonian:

$$\begin{array}{cc}\hfill T^t=& {\displaystyle \frac{1}{4(n^2+3n+2)}}[c_1{n}^{2}u{v}^2{v}_x-c_1{n}^2{u}^2{u}_{x}v\hfill \\ \hfill & -4au{v}_{\mathit{xxx}}+3c_{1}nu{v}^2{v}_x+4{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{2}}n{u}^2{v}^2\hfill \\ \hfill & -2c_1{u}^2{u}_{x}v+2c_{1}u{v}^2{v}_x\hfill \\ \hfill & +4{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{1}}{v}^2+2{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{2}}n{u}^4\hfill \\ \hfill & +2{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{2}}{u}^4+2{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{2}}{v}^4\hfill \\ \hfill & +4{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{1}}{u}^2\hfill \\ \hfill & +2{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{2}}n{v}^4+4{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{2}}{u}^2{v}^2\hfill \\ \hfill & +2{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{1}}n{u}^2+2{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{1}}n{v}^2\hfill \\ \hfill & -2a{n}^{2}u{v}_{\mathit{xxx}}+2a{n}^2{u}_{\mathit{xxx}}v\hfill \\ \hfill & -{\mathit{c}}_{\mathit{1}}{n}^2{u}_x{v}^3+c_1{n}^2{u}^3{v}_x-2c_1{u}_x{v}^3\hfill \\ \hfill & +4a{u}_{\mathit{xxx}}v+2c_1{u}^3{v}_x+3c_{1}n{u}^3{v}_x-3c_{1}n{u}_x{v}^3\hfill \\ \hfill & -6au{v}_{\mathit{xxx}}n+6a{u}_{\mathit{xxx}}vn-3c_{1}n{u}^2{u}_{x}v],\hfill \\ \hfill T^x=& -{\displaystyle \frac{c_1{u}^3{v}_t}{4}}+{\displaystyle \frac{c_1{u}^2{u}_{t}v}{4}}-{\displaystyle \frac{c_{1}u{v}^2{v}_t}{4}}\hfill \\ \hfill & +{\displaystyle \frac{c_1{u}_t{v}^3}{4}}+{\displaystyle \frac{au{v}_{1,1,2}}{2}}-{\displaystyle \frac{a{v}_{\mathit{xt}}{u}_x}{2}}\hfill \\ \hfill & -{\displaystyle \frac{a{u}_t{v}_{\mathit{xx}}}{2}}+{\displaystyle \frac{a{v}_t{u}_{\mathit{xx}}}{2}}+{\displaystyle \frac{a{u}_{\mathit{xt}}{v}_x}{2}}\hfill \\ \hfill & -{\displaystyle \frac{a{u}_{\mathit{xxt}}v}{2}}\hfill \end{array}$$

(34)

or

$$\begin{array}{cc}\hfill & {\mathsf{\Phi }}^t={\displaystyle \frac{b_2}{2(n+2)}}{\left|q\right|}^{2n+4}+{\displaystyle \frac{b_1}{2(n+1)}}{\left|q\right|}^{2n+2}+{\displaystyle \frac{c_1}{4}}{\left|q\right|}^2\Im \left(q^{*}{q}_x\right)-{\displaystyle \frac{a}{2}}\Im \left(q^{*}{q}_{xxx}\right),\hfill \\ \hfill & {\mathsf{\Phi }}^x={\displaystyle \frac{a}{2}}{D}_x\Im \left(q^{*}{q}_{xt}\right)-{\displaystyle \frac{c_1}{4}}{\left|q\right|}^2\Im \left(q^{*}{q}_t\right).\hfill \end{array}$$

(35)

Therefore, the conserved quantities, respectively, are

$$\begin{array}{c}\hfill P=2{\int }_{-\infty }^{\infty }{\mathsf{\Phi }}^{t}dx={\int }_{-\infty }^{\infty }{\left|q\right|}^{2}dx=A^2{\displaystyle \frac{\mathsf{\Gamma }\left(\frac{1}{2}\right)\mathsf{\Gamma }\left(\frac{1}{n+1}\right)}{B\mathsf{\Gamma }\left(\frac{1}{2}+\frac{1}{n+1}\right)}},\end{array}$$

(36)

$$\begin{array}{c}\hfill M={\int }_{-\infty }^{\infty }{\mathsf{\Phi }}^{t}dx={\displaystyle \frac{1}{2i}}{\int }_{-\infty }^{\infty }\left(q^{*}{q}_x-q{q}_x^{*}\right)dx=-\kappa A^2{\displaystyle \frac{\mathsf{\Gamma }\left(\frac{1}{2}\right)\mathsf{\Gamma }\left(\frac{1}{n+1}\right)}{B\mathsf{\Gamma }\left(\frac{1}{2}+\frac{1}{n+1}\right)}}\end{array}$$

(37)

and

$$\begin{array}{ccc}\hfill H& =& {\int }_{-\infty }^{\infty }{\mathsf{\Phi }}^{t}dx\hfill \\ & =& {\int }_{-\infty }^{\infty }\left[b_2{\left|q\right|}^{2n+4}+b_1{\left|q\right|}^{2n+2}+{\displaystyle \frac{c_1}{2i}}{\left|q\right|}^2\left(q^{*}{q}_x-q{q}_x^{*}\right)+{\displaystyle \frac{a}{i}}\Im \left(q^{*}{q}_{xxx}\right)\right]dx\hfill \\ & =& {\displaystyle \frac{1}{B}}{\displaystyle \frac{\mathsf{\Gamma }\left(\frac{1}{2}\right)\mathsf{\Gamma }\left(\frac{1}{n+1}\right)}{\mathsf{\Gamma }\left(\frac{1}{n+1}+\frac{1}{2}\right)}}\left\{2a\kappa A^2\left[{\kappa }^2+{\displaystyle \frac{3B^2}{(n+1)(n+3)}}\right]+{\displaystyle \frac{2b_2{A}^{2n+4}}{n+3}}\right\}\hfill \\ & -& {\displaystyle \frac{c_1\kappa A^4}{B}}{\displaystyle \frac{\mathsf{\Gamma }\left(\frac{1}{2}\right)\mathsf{\Gamma }\left(\frac{2}{n+1}\right)}{\mathsf{\Gamma }\left(\frac{2}{n+1}+\frac{1}{2}\right)}}+{\displaystyle \frac{2b_1{A}^{2n+2}}{B}}.\hfill \end{array}$$

(38)

2.4. Parabolic Law

This is the special case of dual-power law and is recovered by setting $n=1$ within the dual-power formulation. The resulting governing model can thus be written as

$$\begin{array}{c}\hfill i{q}_t+ia{q}_{xxx}+\left(b_1{\left|q\right|}^2+b_2{\left|q\right|}^4\right)q=i\left[c_1{\left|q\right|}^2{q}_x+c_2{\left({\left|q\right|}^2\right)}_{x}q\right].\end{array}$$

(39)

A localized one-soliton structure takes the form

$$\begin{array}{c}\hfill q(x,t)=A{\mathrm{sech}}^{{\displaystyle \frac{1}{2}}}\left[B(x-vt)\right]e^{i\left(-\kappa x+\omega t+\theta \right)}.\end{array}$$

(40)

These solitons exist for

$$\begin{array}{c}\hfill c_1+2c_2=0.\end{array}$$

(41)

This parabolic response provides a convenient intermediate case between the Kerr and more general dual-power models. Its reduced exponent permits a direct comparison of how the same balance laws scale with the pulse amplitude and inverse width.

Conservation Laws

Therefore, the resulting quantities, respectively, are

$$\begin{array}{c}\hfill P=2{\int }_{-\infty }^{\infty }{\mathsf{\Phi }}^{t}dx={\int }_{-\infty }^{\infty }{\left|q\right|}^{2}dx={\displaystyle \frac{\pi A^2}{B}},\end{array}$$

(42)

$$\begin{array}{c}\hfill M={\int }_{-\infty }^{\infty }{\mathsf{\Phi }}^{t}dx={\displaystyle \frac{1}{2i}}{\int }_{-\infty }^{\infty }\left(q^{*}{q}_x-q{q}_x^{*}\right)dx=-{\displaystyle \frac{\kappa \pi A^2}{B}}\end{array}$$

(43)

and

$$\begin{array}{c}\hfill H={\displaystyle \frac{A^2\pi }{2B\mathsf{\Gamma }\left(\frac{2}{3}\right)}}\left[\left({\displaystyle \frac{b_2{A}^4+a\kappa B^2}{2}}\right)+{\displaystyle \frac{2A^4}{B}}(b_1-c_1)+{\displaystyle \frac{2a\kappa \pi }{B}}\left({\displaystyle \frac{B^2}{4}}+{\kappa }^2\right)\right].\end{array}$$

(44)

2.5. Triple-Power Law

The perturbed pure-cubic NLSE here is given by

$$\begin{array}{c}\hfill i{q}_t+ia{q}_{xxx}+\left(b_1{\left|q\right|}^{2n}+b_2{\left|q\right|}^{2n+2}+b_3{\left|q\right|}^{2n+4}\right)=i\left[c_1{\left|q\right|}^{2n}{q}_x+c_2{\left({\left|q\right|}^{2n}\right)}_{x}q\right].\end{array}$$

(45)

A localized one-soliton structure takes the form

$$\begin{array}{c}\hfill q(x,t)=A{\mathrm{sech}}^{{\displaystyle \frac{1}{n+2}}}\left[B(x-vt)\right]e^{i\left(-\kappa x+\omega t+\theta \right)}.\end{array}$$

(46)

These solitons exist whenever the following two parameter constraints hold:

$$\begin{array}{c}\hfill c_1+2n{c}_2=0,\end{array}$$

(47)

$$\begin{array}{c}\hfill b_1+2na{c}_2=0\end{array}$$

(48)

and

$$\begin{array}{c}\hfill b_2=0.\end{array}$$

(49)

This signifies that the governing model compresses to

$$\begin{array}{c}\hfill i{q}_t+ia{q}_{xxx}+\left(b_1{\left|q\right|}^{2n}+b_3{\left|q\right|}^{2n+4}\right)q=i\left[c_1{\left|q\right|}^{2n}{q}_x+c_2{\left({\left|q\right|}^{2n}\right)}_{x}q\right].\end{array}$$

(50)

The disappearance of the intermediate nonlinear coefficient $b_2$ shows that the exact soliton occupies a reduced parameter manifold of the full triple-power model. On this manifold, the remaining nonlinear orders cooperate with the perturbation terms to sustain localization.

Conservation Laws

• Power:

$$\begin{array}{cc}\hfill & T^t={\displaystyle \frac{1}{2}}(u^2+v^2),\hfill \\ \hfill & T^x={\displaystyle \frac{1}{2(n+1)}}[2anu{u}_{\mathit{xx}}-an{u}_x^2+2anv{v}_{\mathit{xx}}-an{v}_x^2-2{\left(u^2+v^2\right)}^{1+n}{c}_{2}n+2au{u}_{\mathit{xx}}-a{u}_x^2\hfill \\ \hfill & +2av{v}_{\mathit{xx}}-a{v}_x^2-{\left(u^2+v^2\right)}^{1+n}{c}_1]\hfill \end{array}$$

(51)

so that the components of the conserved flow $({\mathsf{\Phi }}^t,{\mathsf{\Phi }}^x)$ that satisfy $D_t{\mathsf{\Phi }}^t+D_x{\mathsf{\Phi }}^x=0$ of Equation (2) are

$$\begin{array}{cc}\hfill & {\mathsf{\Phi }}^t={\displaystyle \frac{1}{2}}{\left|q\right|}^2,\hfill \\ \hfill & {\mathsf{\Phi }}^x={\displaystyle \frac{a}{2}}\left(q{q}_{xx}^{*}+q^{*}{q}_{xx}-q_x{q}_x^{*}\right)-{\displaystyle \frac{2n{c}_2+c_1}{2(n+1)}}{\left(q{q}^{*}\right)}^{n+1}.\hfill \end{array}$$

(52)

• Linear Momentum ($c_2=0$):

$$\begin{array}{cc}\hfill T^t=& {\displaystyle \frac{v_{x}u}{2}}-{\displaystyle \frac{u_{x}v}{2}},\hfill \\ \hfill T^x=& {\displaystyle \frac{1}{n^3+6n^2+11n+6}}[9{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{3}}n{u}^2{v}^4-6u{v}_t+6u_{t}v+22an{u}_{\mathit{xx}}{v}_x\hfill \\ \hfill & +3{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{3}}n{v}^6\hfill \\ \hfill & -12a{v}_{\mathit{xx}}{u}_x+12a{u}_{\mathit{xx}}{v}_x+2{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{2}}{n}^2{u}^2{v}^2\hfill \\ \hfill & +3{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{2}}{u}^4+3{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{2}}{v}^4\hfill \\ \hfill & +6{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{1}}{u}^2+{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{2}}{n}^2{v}^4\hfill \\ \hfill & +6{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{3}}{u}^4{v}^2+3{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{3}}n{u}^6\hfill \\ \hfill & +8{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{2}}n{u}^2{v}^2\hfill \\ \hfill & +2{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{3}}{u}^6+{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{1}}{n}^2{u}^2\hfill \\ \hfill & +{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{1}}{n}^2{v}^2+{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{3}}{n}^2{u}^6\hfill \\ \hfill & +2{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{3}}{v}^6\hfill \\ \hfill & -6n^{2}u{v}_t+6n^2{u}_{t}v+4{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{2}}n{u}^4+6{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{1}}{v}^2\hfill \\ \hfill & +4{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{2}}n{v}^4+6{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{3}}{u}^2{v}^4\hfill \\ \hfill & +6{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{2}}{u}^2{v}^2+5{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{1}}n{u}^2\hfill \\ \hfill & +5{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{1}}n{v}^2+3{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{3}}{n}^2{u}^2{v}^4-n^{3}u{v}_t\hfill \\ \hfill & +n^3{u}_{t}v\hfill \\ \hfill & -22an{u}_x{v}_{\mathit{xx}}+{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{2}}{n}^2{u}^4+{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{3}}{n}^2{v}^6\hfill \\ \hfill & -12a{n}^2{u}_x{v}_{\mathit{xx}}+12a{n}^2{u}_{\mathit{xx}}{v}_x+9{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{3}}n{u}^4{v}^2\hfill \\ \hfill & -11nu{v}_t+11n{u}_{t}v+3{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{3}}{n}^2{u}^4{v}^2-2a{n}^3{u}_x{v}_{\mathit{xx}}\hfill \\ \hfill & +2a{n}^3{u}_{\mathit{xx}}{v}_x]\hfill \end{array}$$

(53)

or

$$\begin{array}{c}{\mathsf{\Phi }}^t=\Im \left(q^{*}{q}_x\right),\hfill \\ {\mathsf{\Phi }}^x=-\Im \left(q^{*}{q}_t\right)-2a\Im \left(q_x^{*}{q}_{xx}\right)+{\displaystyle \frac{b_1}{n+1}}{\left|q\right|}^{2n+2}+{\displaystyle \frac{b_2}{n+2}}{\left|q\right|}^{2n+4}+{\displaystyle \frac{b_3}{n+3}}{\left|q\right|}^{2n+6}.\hfill \end{array}$$

(54)

• Hamiltonian:

$$\begin{array}{cc}\hfill T^t=& {\displaystyle \frac{1}{4(n^3+6n^2+11n+6)}}[12{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{3}}{u}^2{v}^4-c_1{n}^3{u}^2{u}_{x}v\hfill \\ \hfill & +6{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{3}}n{v}^6+12{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{1}}{v}^2\hfill \\ \hfill & +6{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{3}}n{u}^6+12{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{3}}{u}^4{v}^2\hfill \\ \hfill & +4{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{2}}{n}^2{u}^2{v}^2+2{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{2}}{n}^2{v}^4\hfill \\ \hfill & -6c_1{n}^2{u}^2{u}_{x}v+16{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{2}}n{u}^2{v}^2\hfill \\ \hfill & +4{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{3}}{v}^6-12au{v}_{\mathit{xxx}}+18{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{3}}n{u}^2{v}^4\hfill \\ \hfill & +6c_1{n}^{2}u{v}^2{v}_x\hfill \\ \hfill & +6c_{1}u{v}^2{v}_x+6{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{2}}{v}^4+8{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{2}}n{u}^4\hfill \\ \hfill & +4{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{3}}{u}^6+12{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{1}}{u}^2\hfill \\ \hfill & -6c_1{u}^2{u}_{x}v+6{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{2}}{u}^4+c_1{n}^{3}u{v}^2{v}_x\hfill \\ \hfill & -12a{n}^{2}u{v}_{\mathit{xxx}}+12a{n}^2{u}_{\mathit{xxx}}v\hfill \\ \hfill & -6c_1{n}^2{u}_x{v}^3+6c_1{n}^2{u}^3{v}_x+10{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{1}}n{v}^2\hfill \\ \hfill & +10{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{1}}n{u}^2-6c_1{u}_x{v}^3+12a{u}_{\mathit{xxx}}v\hfill \\ \hfill & +6{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{3}}{n}^2{u}^4{v}^2+2{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{3}}{n}^2{v}^6\hfill \\ \hfill & +18{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{3}}n{u}^4{v}^2+6{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{3}}{n}^2{u}^2{v}^4\hfill \\ \hfill & +6c_1{u}^3{v}_x+2{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{3}}{n}^2{u}^6+8{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{2}}n{v}^4\hfill \\ \hfill & +11c_{1}n{u}^3{v}_x\hfill \\ \hfill & -11c_{1}n{u}_x{v}^3-22au{v}_{\mathit{xxx}}n+22a{u}_{\mathit{xxx}}vn+2a{n}^3{u}_{\mathit{xxx}}v\hfill \\ \hfill & +c_1{n}^3{u}^3{v}_x-2a{n}^{3}u{v}_{\mathit{xxx}}-c_1{n}^3{u}_x{v}^3\hfill \\ \hfill & +2{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{2}}{n}^2{u}^4-11c_{1}n{u}^2{u}_{x}v+11c_{1}nu{v}^2{v}_x\hfill \\ \hfill & +12{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{2}}{u}^2{v}^2\hfill \\ \hfill & +2{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{1}}{n}^2{v}^2+2{\left(u^2+v^2\right)}^n{\mathit{b}}_{\mathit{1}}{n}^2{u}^2],\hfill \\ \hfill T^x=& -{\displaystyle \frac{c_1{u}^3{v}_t}{4}}+{\displaystyle \frac{c_1{u}^2{u}_{t}v}{4}}-{\displaystyle \frac{c_{1}u{v}^2{v}_t}{4}}\hfill \\ \hfill & +{\displaystyle \frac{c_1{u}_t{v}^3}{4}}+{\displaystyle \frac{au{v}_{1,1,2}}{2}}-{\displaystyle \frac{a{v}_{\mathit{xt}}{u}_x}{2}}-{\displaystyle \frac{a{u}_t{v}_{\mathit{xx}}}{2}}\hfill \\ \hfill & +{\displaystyle \frac{a{v}_t{u}_{\mathit{xx}}}{2}}+{\displaystyle \frac{a{u}_{\mathit{xt}}{v}_x}{2}}-{\displaystyle \frac{a{u}_{\mathit{xxt}}v}{2}}\hfill \end{array}$$

(55)

or

$$\begin{array}{cc}\hfill & {\mathsf{\Phi }}^t={\displaystyle \frac{c_1}{4}}{\left|q\right|}^2\Im \left(q^{*}{q}_x\right)-{\displaystyle \frac{a}{2}}\Im \left(q^{*}{q}_{xxx}\right)+{\displaystyle \frac{2b_1}{n+1}}{\left|q\right|}^{2n+2}+{\displaystyle \frac{2b_2}{n+2}}{\left|q\right|}^{2n+4}+{\displaystyle \frac{2b_3}{n+3}}{\left|q\right|}^{2n+6},\hfill \\ \hfill & {\mathsf{\Phi }}^x=-{\displaystyle \frac{c_1}{4}}{\left|q\right|}^2\Im \left(q^{*}{q}_t\right)+{\displaystyle \frac{a}{2}}{D}_x\Im \left(q^{*}{q}_{xt}\right).\hfill \end{array}$$

(56)

Therefore, the conserved quantities, respectively, are

$$\begin{array}{c}\hfill P=2{\int }_{-\infty }^{\infty }{\mathsf{\Phi }}^{t}dx={\int }_{-\infty }^{\infty }{\left|q\right|}^{2}dx={\displaystyle \frac{A^2\mathsf{\Gamma }\left(\frac{1}{2}\right)\mathsf{\Gamma }\left(\frac{1}{n+2}\right)}{B\mathsf{\Gamma }\left(\frac{1}{n+2}+\frac{1}{2}\right)}},\end{array}$$

(57)

$$\begin{array}{c}\hfill M={\int }_{-\infty }^{\infty }{\mathsf{\Phi }}^{t}dx={\displaystyle \frac{1}{2i}}{\int }_{-\infty }^{\infty }\left(q^{*}{q}_x-q{q}_x^{*}\right)dx=-{\displaystyle \frac{\kappa A^2\mathsf{\Gamma }\left(\frac{1}{2}\right)\mathsf{\Gamma }\left(\frac{1}{n+2}\right)}{B\mathsf{\Gamma }\left(\frac{1}{n+2}+\frac{1}{2}\right)}}\end{array}$$

(58)

and

$$\begin{array}{ccc}\hfill H& =& {\int }_{-\infty }^{\infty }{\mathsf{\Phi }}^{t}dx\hfill \\ & =& {\int }_{-\infty }^{\infty }\left[{\displaystyle \frac{c_1}{2i}}{\left|q\right|}^2\Im \left(q^{*}{q}_x\right)+{\displaystyle \frac{a}{i}}\Im \left(q^{*}{q}_{xxx}\right)+b_1{\left|q\right|}^{2n+2}+b_2{\left|q\right|}^{2n+4}+b_3{\left|q\right|}^{2n+6}\right]dx\hfill \\ & =& -{\displaystyle \frac{c_1\kappa A^4}{B}}{\displaystyle \frac{\mathsf{\Gamma }\left(\frac{1}{2}\right)\mathsf{\Gamma }\left(\frac{2}{n+2}\right)}{\mathsf{\Gamma }\left(\frac{1}{2}+\frac{2}{n+2}\right)}}+{\displaystyle \frac{2a{A}^2\kappa }{B}}\left[{\kappa }^2+{\left({\displaystyle \frac{B}{n+2}}\right)}^2\right]{\displaystyle \frac{\mathsf{\Gamma }\left(\frac{1}{2}\right)\mathsf{\Gamma }\left(\frac{1}{n+2}\right)}{\mathsf{\Gamma }\left(\frac{1}{2}+\frac{1}{n+2}\right)}}\hfill \\ & +& {\displaystyle \frac{4a\kappa (n+1)B^2{A}^2}{{(n+2)}^2(n+4)}}{\displaystyle \frac{\mathsf{\Gamma }\left(\frac{1}{2}\right)\mathsf{\Gamma }\left(\frac{1}{n+2}\right)}{\mathsf{\Gamma }\left(\frac{1}{2}+\frac{1}{n+2}\right)}}+{\displaystyle \frac{b_1{A}^{2n+2}}{B}}{\displaystyle \frac{\mathsf{\Gamma }\left(\frac{1}{2}\right)\mathsf{\Gamma }\left(\frac{n+1}{n+2}\right)}{\mathsf{\Gamma }\left(\frac{1}{2}+\frac{n+1}{n+2}\right)}}\hfill \\ & +& {\displaystyle \frac{2b_2{A}^{2n+4}}{B}}+{\displaystyle \frac{2b_3{A}^{2n+6}}{(n+4)B}}{\displaystyle \frac{\mathsf{\Gamma }\left(\frac{1}{2}\right)\mathsf{\Gamma }\left(\frac{1}{n+2}\right)}{\mathsf{\Gamma }\left(\frac{1}{2}+\frac{1}{n+2}\right)}}.\hfill \end{array}$$

(59)

2.6. Polynomial Law

This law is recovered from triple power law upon setting $n=1$. The governing model therefore is

$$\begin{array}{c}\hfill i{q}_t+ia{q}_{xxx}+\left(b_1{\left|q\right|}^2+b_2{\left|q\right|}^4+b_3{\left|q\right|}^6\right)q=i\left[c_1{\left|q\right|}^2{q}_x+c_2{\left({\left|q\right|}^2\right)}_{x}q\right].\end{array}$$

(60)

A localized one-soliton structure takes the form

$$\begin{array}{c}\hfill q(x,t)=A{\mathrm{sech}}^{{\displaystyle \frac{1}{3}}}\left[B(x-vt)\right]e^{i\left(-\kappa x+\omega t+\theta \right)}.\end{array}$$

(61)

These solitons exist whenever the following parameter constraints hold:

$$\begin{array}{c}\hfill c_1+2c_2=0,\end{array}$$

(62)

$$\begin{array}{c}\hfill b_1+2a{c}_2=0\end{array}$$

(63)

and

$$\begin{array}{c}\hfill b_2=0.\end{array}$$

(64)

This signifies that the governing model compresses to

$$\begin{array}{c}\hfill i{q}_t+ia{q}_{xxx}+\left(b_1{\left|q\right|}^2+b_3{\left|q\right|}^6\right)q=i\left[c_1{\left|q\right|}^2{q}_x+c_2{\left({\left|q\right|}^2\right)}_{x}q\right].\end{array}$$

(65)

For the polynomial specialization, the cubic and septic intensity contributions form a transparent hierarchy of nonlinear corrections. The associated integral quantities therefore indicate how higher optical intensities redistribute the effective energy while preserving the localized envelope. Closely related strongly enough dispersive solitary waves governed by cubic and cubic–quintic–septic responses were investigated in Ref. [17].

Conservation Laws

The resulting invariants, respectively, take the forms

$$\begin{array}{c}\hfill P=2{\int }_{-\infty }^{\infty }{\mathsf{\Phi }}^{t}dx={\int }_{-\infty }^{\infty }{\left|q\right|}^{2}dx={\displaystyle \frac{A^2\mathsf{\Gamma }\left(\frac{1}{2}\right)\mathsf{\Gamma }\left(\frac{1}{3}\right)}{B\mathsf{\Gamma }\left(\frac{5}{6}\right)}},\end{array}$$

(66)

$$\begin{array}{c}\hfill M={\int }_{-\infty }^{\infty }{\mathsf{\Phi }}^{t}dx={\displaystyle \frac{1}{2i}}{\int }_{-\infty }^{\infty }\left(q^{*}{q}_x-q{q}_x^{*}\right)dx=-{\displaystyle \frac{\kappa A^2\mathsf{\Gamma }\left(\frac{1}{2}\right)\mathsf{\Gamma }\left(\frac{1}{3}\right)}{B\mathsf{\Gamma }\left(\frac{5}{6}\right)}}\end{array}$$

(67)

and

$$\begin{array}{ccc}\hfill H& =& {\displaystyle \frac{2a{\kappa }^3{A}^2\mathsf{\Gamma }\left(\frac{1}{2}\right)\mathsf{\Gamma }\left(\frac{1}{3}\right)}{B\mathsf{\Gamma }\left(\frac{5}{6}\right)}}+{\displaystyle \frac{2a\kappa B{A}^2\mathsf{\Gamma }\left(\frac{1}{2}\right)\mathsf{\Gamma }\left(\frac{1}{3}\right)}{5\mathsf{\Gamma }\left(\frac{5}{6}\right)}}+{\displaystyle \frac{2b_3{A}^8\mathsf{\Gamma }\left(\frac{1}{2}\right)\mathsf{\Gamma }\left(\frac{1}{3}\right)}{5B\mathsf{\Gamma }\left(\frac{5}{6}\right)}}\hfill \\ & +& {\displaystyle \frac{6b_3{A}^4\mathsf{\Gamma }\left(\frac{1}{2}\right)\mathsf{\Gamma }\left(\frac{2}{3}\right)}{B\mathsf{\Gamma }\left(\frac{1}{6}\right)}}-{\displaystyle \frac{6c_1\kappa A^4\mathsf{\Gamma }\left(\frac{1}{2}\right)\mathsf{\Gamma }\left(\frac{2}{3}\right)}{B\mathsf{\Gamma }\left(\frac{1}{6}\right)}}.\hfill \end{array}$$

(68)

2.7. Generalized Quadratic–Cubic Law

The pure-cubic NLSE with perturbation terms is

$$\begin{array}{c}\hfill i{q}_t+ia{q}_{xxx}+\left(b_1{\left|q\right|}^n+b_2{\left|q\right|}^{n+1}\right)q=i\left[c_1{\left|q\right|}^{2n}{q}_x+c_2{\left({\left|q\right|}^{2n}\right)}_{x}q\right].\end{array}$$

(69)

A localized one-soliton structure takes the form

$$\begin{array}{c}\hfill q(x,t)=A{\mathrm{sech}}^{{\displaystyle \frac{1}{n}}}\left[B(x-vt)\right]e^{i\left(-\kappa x+\omega t+\theta \right)}\end{array}$$

(70)

for

$$\begin{array}{c}\hfill c_1+2n{c}_2=0\end{array}$$

(71)

and

$$\begin{array}{c}\hfill b_2=0.\end{array}$$

(72)

This signifies that the governing model collapses to

$$\begin{array}{c}\hfill i{q}_t+ia{q}_{xxx}+b_1{\left|q\right|}^{n}q=i\left[c_1{\left|q\right|}^{2n}{q}_x+c_2{\left({\left|q\right|}^{2n}\right)}_{x}q\right].\end{array}$$

(73)

Although the initial response contains two nonlinear orders, the localized solution requires the higher contribution to vanish. The reduced model still retains a nontrivial competition between generalized SPM and the derivative perturbations.

Conservation Laws

• Power:

$$\begin{array}{cc}\hfill T^t=& {\displaystyle \frac{1}{2}}(u^2+v^2),\hfill \\ \hfill T^x=& -{\displaystyle \frac{1}{2(n+1)}}[-2anu{u}_{\mathit{xx}}+an{u}_x^2-2anv{v}_{\mathit{xx}}+an{v}_x^2\hfill \\ \hfill & +2{\left(u^2+v^2\right)}^{1+n}{c}_{2}n-2au{u}_{\mathit{xx}}+a{u}_x^2-2av{v}_{\mathit{xx}}\hfill \\ \hfill & +a{v}_x^2+{\left(u^2+v^2\right)}^{1+n}{c}_1]\hfill \end{array}$$

(74)

so that the components of the conserved flow $({\mathsf{\Phi }}^t,{\mathsf{\Phi }}^x)$ that satisfy $D_t{\mathsf{\Phi }}^t+D_x{\mathsf{\Phi }}^x=0$ Equation (2) are

$$\begin{array}{cc}\hfill & {\mathsf{\Phi }}^t={\displaystyle \frac{1}{2}}{\left|q\right|}^2,\hfill \\ \hfill & {\mathsf{\Phi }}^x={\displaystyle \frac{a}{2}}{\left(\right|q|}^2{)}_{xx}-{\displaystyle \frac{3a}{2}}|q_x{|}^2-{\displaystyle \frac{2c_{2}n+c_1}{2(n+1)}}{\left|q\right|}^{2(n+1)}.\hfill \end{array}$$

(75)

• Linear Momentum ($c_2=0$):

$$\begin{array}{cc}\hfill T^t=& {\displaystyle \frac{v_{x}u}{2}}-{\displaystyle \frac{u_{x}v}{2}},\hfill \\ \hfill T^x=& {\displaystyle \frac{1}{2(n^2+5n+6)}}[-2a{n}^2{u}_x{v}_{\mathit{xx}}+2a{n}^2{u}_{\mathit{xx}}{v}_x\hfill \\ \hfill & -10an{u}_x{v}_{\mathit{xx}}+10an{u}_{\mathit{xx}}{v}_x-n^{2}u{v}_t+n^2{u}_{t}v\hfill \\ \hfill & +2{\left(u^2+v^2\right)}^{1+{\displaystyle \frac{n}{2}}}{\mathit{b}}_{\mathit{1}}n\hfill \\ \hfill & +2{\left(u^2+v^2\right)}^{{\displaystyle \frac{3}{2}}+{\displaystyle \frac{n}{2}}}{\mathit{b}}_{\mathit{2}}n-12a{v}_{\mathit{xx}}{u}_x\hfill \\ \hfill & +12a{u}_{\mathit{xx}}{v}_x-5nu{v}_t+5n{u}_{t}v+6{\left(u^2+v^2\right)}^{1+{\displaystyle \frac{n}{2}}}{\mathit{b}}_{\mathit{1}}\hfill \\ \hfill & +4{\left(u^2+v^2\right)}^{{\displaystyle \frac{3}{2}}+{\displaystyle \frac{n}{2}}}{\mathit{b}}_{\mathit{2}}-6u{v}_t+6u_{t}v]\hfill \end{array}$$

(76)

or

$$\begin{array}{cc}\hfill & {\mathsf{\Phi }}^t=\Im \left(q^{*}{q}_x\right),\hfill \\ \hfill & {\mathsf{\Phi }}^x=-a\Im \left(q_x^{*}{q}_{xx}\right)-{\displaystyle \frac{1}{2}}\Im \left(q^{*}{q}_t\right)+{\displaystyle \frac{b_1}{n+2}}{\left|q\right|}^{n+2}+{\displaystyle \frac{b_2}{n+3}}{\left|q\right|}^{n+3}.\hfill \end{array}$$

(77)

• Hamiltonian, for $n=1$:

$$\begin{array}{cc}\hfill T^t=& {\displaystyle \frac{-1}{2\sqrt{u^2+v^2}}}[\left(\begin{array}{c}-{\displaystyle \frac{{\mathit{b}}_{\mathit{2}}{u}^4}{2}}-{\displaystyle \frac{c_1{u}^3{v}_x}{2}}+\left(-{\mathit{b}}_{\mathit{2}}{v}^2+{\displaystyle \frac{1}{2}}{c}_{1}v{u}_x\right)u^2\hfill \\ +\left(-{\displaystyle \frac{c_1{v}^2{v}_x}{2}}+a{v}_{\mathit{xxx}}\right)u-{\displaystyle \frac{{\mathit{b}}_{\mathit{2}}{v}^4}{2}}+{\displaystyle \frac{c_1{u}_x{v}^3}{2}}-a{u}_{\mathit{xxx}}v\hfill \end{array}\right)\sqrt{u^2+v^2}\hfill \\ \hfill & -{\displaystyle \frac{2{\mathit{b}}_{\mathit{1}}{\left(u^2+v^2\right)}^2}{3}}],\hfill \\ \hfill T^x=& -{\displaystyle \frac{\left(\begin{array}{c}-2u{v}_{xxt}+2v{u}_{\mathit{xxt}}+2u_t{v}_{\mathit{xx}}\hfill \\ +2u_x{v}_{\mathit{xt}}-2u_{\mathit{xt}}{v}_x-2u_{\mathit{xx}}{v}_t\hfill \end{array}\right)a}{4}}\hfill \\ \hfill & -{\displaystyle \frac{c_1\left(u^2+v^2\right)\left(u{v}_t-u_{t}v\right)}{4}}\hfill \end{array}$$

(78)

or

$$\begin{array}{cc}\hfill & {\mathsf{\Phi }}^t={\displaystyle \frac{b_2}{4}}{\left|q\right|}^4+{\displaystyle \frac{c_1}{4}}{\left|q\right|}^2\Im \left(q^{*}{q}_x\right)+{\displaystyle \frac{a}{2}}\Im \left(q^{*}{q}_{xxx}\right)+{\displaystyle \frac{b_1}{3}}{\left|q\right|}^3,\hfill \\ \hfill & {\mathsf{\Phi }}^x=-{\displaystyle \frac{a}{2}}{D}_x\Im \left(q^{*}{q}_{xt}\right)-{\displaystyle \frac{c_1}{4}}{\left|q\right|}^2\Im \left(q^{*}{q}_t\right).\hfill \end{array}$$

(79)

Therefore, the resulting invariants, respectively, take the forms

$$\begin{array}{c}\hfill P=2{\int }_{-\infty }^{\infty }{\mathsf{\Phi }}^{t}dx={\int }_{-\infty }^{\infty }{\left|q\right|}^{2}dx={\displaystyle \frac{A^2}{B}}{\displaystyle \frac{\mathsf{\Gamma }\left(\frac{1}{n}\right)\mathsf{\Gamma }\left(\frac{1}{2}\right)}{\mathsf{\Gamma }\left(\frac{1}{n}+\frac{1}{2}\right)}},\end{array}$$

(80)

$$\begin{array}{c}\hfill M={\int }_{-\infty }^{\infty }{\mathsf{\Phi }}^{t}dx={\displaystyle \frac{1}{2i}}{\int }_{-\infty }^{\infty }\left(q^{*}{q}_x-q{q}_x^{*}\right)dx=-{\displaystyle \frac{\kappa A^2}{B}}·{\displaystyle \frac{\mathsf{\Gamma }\left(\frac{1}{n}\right)\mathsf{\Gamma }\left(\frac{1}{2}\right)}{\mathsf{\Gamma }\left(\frac{1}{n}+\frac{1}{2}\right)}},\end{array}$$

(81)

$$\begin{array}{c}\hfill H={\displaystyle \frac{4a\kappa A^2\left(B^2+{\kappa }^2\right)}{B}}-{\displaystyle \frac{4c_1\kappa A^4}{3B}}+{\displaystyle \frac{\pi b_1{A}^3}{6B}}.\end{array}$$

(82)

2.8. Quadratic–Cubic Law

When $n=1$, the generalized quadratic–cubic formulation collapses to the quadratic–cubic specialization. This yields a governing model of the form

$$\begin{array}{c}\hfill i{q}_t+ia{q}_{xxx}+\left(b_1\left|q\right|+b_2{\left|q\right|}^2\right)q=i\left[c_1{\left|q\right|}^2{q}_x+c_2{\left({\left|q\right|}^2\right)}_{x}q\right].\end{array}$$

(83)

A localized one-soliton structure takes the form

$$\begin{array}{c}\hfill q(x,t)=A\mathrm{sech}\left[B(x-vt)\right]e^{i\left(-\kappa x+\omega t+\theta \right)}\end{array}$$

(84)

provided

$$\begin{array}{c}\hfill c_1+2c_2=0.\end{array}$$

(85)

In this case, the quadratic contribution introduces an asymmetric intensity correction relative to the standard cubic response. The compatibility condition ensures that this mixed response remains consistent with the self-frequency-shift perturbation for the stated pulse profile. Nonlinearity-managed quadratic–cubic media are also known to support complex soliton dynamics [18].

Conservation Laws

The resulting invariants, respectively, take the forms

$$\begin{array}{c}\hfill P=2{\int }_{-\infty }^{\infty }{\mathsf{\Phi }}^{t}dx={\int }_{-\infty }^{\infty }{\left|q\right|}^{2}dx={\displaystyle \frac{2A^2}{B}},\end{array}$$

(86)

$$\begin{array}{c}\hfill M={\int }_{-\infty }^{\infty }{\mathsf{\Phi }}^{t}dx={\displaystyle \frac{1}{2i}}{\int }_{-\infty }^{\infty }\left(q^{*}{q}_x-q{q}_x^{*}\right)dx=-{\displaystyle \frac{2\kappa A^2}{B}},\end{array}$$

(87)

$$\begin{array}{c}\hfill H={\displaystyle \frac{4a\kappa A^2\left(B^2+{\kappa }^2\right)}{B}}-{\displaystyle \frac{4c_1\kappa A^4}{3B}}+{\displaystyle \frac{\pi b_1{A}^3}{6B}}\end{array}$$

(88)

2.9. Generalized Anti-Cubic Law

This inverse-power response requires separate scrutiny before bright soliton conservation integrals can be evaluated. For $n>0$ the governing equation is

$$i{q}_t+ia{q}_{xxx}+\left(b_1{\left|q\right|}^{-2n-2}+b_2{\left|q\right|}^{2n}+b_3{\left|q\right|}^{2n+2}\right)q=i\left[c_1{\left|q\right|}^{2n}{q}_x+c_2{\left({\left|q\right|}^{2n}\right)}_{x}q\right].$$

(89)

The zero-background candidate proposed above was $q=A{\mathrm{sech}}^{1/(n+1)}\left[B(x-vt)\right]e^{i(-\kappa x+\omega t+\theta )}$ with $A,B>0$. This profile approaches zero exponentially as $\left|x\right|\to \infty$, together with its ordinary spatial and temporal derivatives. In contrast, the magnitude of the singular SPM contribution is

$$\left|b_1{\left|q\right|}^{-2n-2}q\right|=|b_1{\left|\right|q|}^{-2n-1}=\left|b_1\right|A^{-2n-1}{\cosh }^{{\displaystyle \frac{2n+1}{n+1}}}\left[B(x-vt)\right]⟶\infty (b_1\ne 0).$$

(90)

The linear and regular nonlinear terms, including both derivative perturbations, vanish in these tails. Hence, those terms cannot cancel this divergence: the proposed profile does not solve Equation (89) for $b_1\ne 0$, regardless of the values of $c_1$ and $c_2$. Any algebraic relation such as $c_1+2n{c}_2=0$ fails to remove this obstruction. The formal integral of ${\left|q\right|}^2$, although finite, is therefore not the conserved power of a solution to the singular model; nor can one assign a physically meaningful bright-pulse momentum or Hamiltonian on this basis.

The proper alternatives are to set $b_1=0$ (thereby removing the singular response and returning to a regular SPM model) or to formulate a nonzero-background problem with boundary conditions and, if necessary, background-subtracted conserved quantities. The latter problem lies outside the zero-background bright soliton framework considered here.

2.10. Anti-Cubic Law

For $n=1$ the specialization is

$$i{q}_t+ia{q}_{xxx}+\left(b_1{\left|q\right|}^{-4}+b_2{\left|q\right|}^2+b_3{\left|q\right|}^4\right)q=i\left[c_1{\left|q\right|}^2{q}_x+c_2{\left({\left|q\right|}^2\right)}_{x}q\right].$$

(91)

For the above-stated $A{\mathrm{sech}}^{1/2}\left[B(x-vt)\right]e^{i(-\kappa x+\omega t+\theta )}$ candidate, the inverse-power term has magnitude $|b_1|A^{-3}{\cosh }^{3/2}\left[B(x-vt)\right]$, which diverges exponentially for $b_1\ne 0$. Therefore, the candidate cannot be a zero-background bright soliton of Equation (91); formal values such as $P=\pi A^2/B$ and $M=-\kappa \pi A^2/B$ would characterize only the trial function, not an exact solution or invariants of that equation. In particular, reporting $H\to \infty$ as the Hamiltonian of a bright soliton is misleading, because the asserted soliton is not admissible in the first place.

The study of Ref. [19] concerns an anti-cubic model with a nonvanishing pedestal and should not be cited as evidence for a vanishing-background ${\mathrm{sech}}^{1/2}$ soliton. A correct physical treatment of such backgrounds would need its own asymptotic conditions and renormalized integrals. We therefore exclude both nonzero-$b_1$ anti-cubic cases from the catalog of admissible bright soliton conserved quantities.

2.11. Parabolic Law with Non-Local Nonlinearity

The perturbed pure-cubic NLSE here is

$$\begin{array}{c}\hfill i{q}_t+ia{q}_{xxx}+\left\{b_1{\left|q\right|}^2+b_2{\left|q\right|}^4+b_3{\left({\left|q\right|}^2\right)}_{xx}\right\}q=i\left[c_1{\left|q\right|}^2{q}_x+c_2{\left({\left|q\right|}^2\right)}_{x}q\right].\end{array}$$

(92)

A localized one-soliton structure takes the form

$$\begin{array}{c}\hfill q(x,t)=A\mathrm{sech}\left[B(x-vt)\right]e^{i\left(-\kappa x+\omega t+\theta \right)}\end{array}$$

(93)

provided

$$\begin{array}{c}\hfill c_1+2c_2=0.\end{array}$$

(94)

The nonlocal coefficient $b_3$ couples the envelope to spatial variations in its own intensity. This mechanism broadens the energetic content of the model by adding a gradient-dependent contribution while leaving the optical power and momentum in their familiar localized forms [20].

Conservation Laws

• Power:

$$\begin{array}{cc}& T^t={\displaystyle \frac{1}{2}}(u^2+v^2),\hfill \\ \hfill & T^x=-{\displaystyle \frac{c_1{u}^4}{4}}-{\displaystyle \frac{c_1{u}^2{v}^2}{2}}-{\displaystyle \frac{c_1{v}^4}{4}}-{\displaystyle \frac{c_2{u}^4}{2}}-c_2{u}^2{v}^2-{\displaystyle \frac{c_2{v}^4}{2}}+au{u}_{\mathit{xx}}-{\displaystyle \frac{a{u}_x^2}{2}}+av{v}_{\mathit{xx}}-{\displaystyle \frac{a{v}_x^2}{2}}\hfill \end{array}$$

(95)

so that the components of the conserved flow $({\mathsf{\Phi }}^t,{\mathsf{\Phi }}^x)$ that satisfy $D_t{\mathsf{\Phi }}^t+D_x{\mathsf{\Phi }}^x=0$ of Equation (2) are

$$\begin{array}{cc}\hfill & {\mathsf{\Phi }}^t={\displaystyle \frac{1}{2}}{\left|q\right|}^2,\hfill \\ \hfill & {\mathsf{\Phi }}^x=-\left({\displaystyle \frac{c_1}{4}}+{\displaystyle \frac{c_2}{2}}\right){\left|q\right|}^4+{\displaystyle \frac{a}{2}}{\left(\right|q|}^2{)}_{xx}-{\displaystyle \frac{3a}{2}}{\left|q_x\right|}^2.\hfill \end{array}$$

(96)

• Linear Momentum ($c_2=0$):

$$\begin{array}{cc}\hfill & T^t={\displaystyle \frac{v_{x}u}{2}}-{\displaystyle \frac{u_{x}v}{2}},\hfill \\ \hfill & T^x={\displaystyle \frac{{\mathit{b}}_{\mathit{2}}{u}^6}{6}}+{\displaystyle \frac{{\mathit{b}}_{\mathit{2}}{u}^4{v}^2}{2}}+{\displaystyle \frac{{\mathit{b}}_{\mathit{2}}{u}^2{v}^4}{2}}+{\displaystyle \frac{{\mathit{b}}_{\mathit{2}}{v}^6}{6}}+{\displaystyle \frac{{\mathit{b}}_{\mathit{1}}{u}^4}{4}}+{\displaystyle \frac{{\mathit{b}}_{\mathit{1}}{u}^2{v}^2}{2}}+{\displaystyle \frac{{\mathit{b}}_{\mathit{1}}{v}^4}{4}}+{\mathit{b}}_{\mathit{3}}{u}^2{u}_x^2+2{\mathit{b}}_{\mathit{3}}u{u}_{x}v{v}_x+{\mathit{b}}_{\mathit{3}}{v}^2{v}_x^2\hfill \\ \hfill & -a{v}_{\mathit{xx}}{u}_x+a{u}_{\mathit{xx}}{v}_x-{\displaystyle \frac{u{v}_t}{2}}+{\displaystyle \frac{u_{t}v}{2}}\hfill \end{array}$$

(97)

or

$$\begin{array}{cc}\hfill & {\mathsf{\Phi }}^t=\Im \left(q^{*}{q}_x\right),\hfill \\ \hfill & {\mathsf{\Phi }}^x={\displaystyle \frac{b_1}{4}}{\left|q\right|}^4+{\displaystyle \frac{b_2}{6}}{\left|q\right|}^6+{\displaystyle \frac{b_3}{4}}{\left({\left(\right|q|}^2{)}_x\right)}^2-a\Im \left(q_x^{*}{q}_{xx}\right)-{\displaystyle \frac{1}{2}}\Im \left(q^{*}{q}_t\right).\hfill \end{array}$$

(98)

• Energy:

$$\begin{array}{cc}\hfill T^t=& -{\displaystyle \frac{{\mathit{b}}_{\mathit{2}}{v}^6}{6}}-{\displaystyle \frac{\left(6{\mathit{b}}_{\mathit{2}}{u}^2+3{\mathit{b}}_{\mathit{1}}\right)v^4}{12}}\hfill \\ \hfill & -{\displaystyle \frac{\left(6{\mathit{b}}_{\mathit{3}}{v}_{\mathit{xx}}-3c_1{u}_x\right)v^3}{12}}\hfill \\ \hfill & -{\displaystyle \frac{\left(\begin{array}{c}\left(6u{u}_{\mathit{xx}}+6u_x^2+6v_x^2\right){\mathit{b}}_{\mathit{3}}\hfill \\ +6{\mathit{b}}_{\mathit{2}}{u}^4+6{\mathit{b}}_{\mathit{1}}{u}^2+3c_{1}u{v}_x\hfill \end{array}\right)v^2}{12}}\hfill \\ \hfill & -{\displaystyle \frac{\left(6{\mathit{b}}_{\mathit{3}}{u}^2{v}_{\mathit{xx}}-3c_1{u}^2{u}_x+6a{u}_{\mathit{xxx}}\right)v}{12}}\hfill \\ \hfill & +{\displaystyle \frac{u\left(\begin{array}{c}-u\left(u{u}_{\mathit{xx}}+u_x^2+v_x^2\right){\mathit{b}}_{\mathit{3}}-\frac{{\mathit{b}}_{\mathit{2}}{u}^5}{3}-\frac{{\mathit{b}}_{\mathit{1}}{u}^3}{2}\hfill \\ -\frac{c_1{u}^2{v}_x}{2}+a{v}_{\mathit{xxx}}\hfill \end{array}\right)}{2}},\hfill \\ \hfill T^x=& -{\displaystyle \frac{\left(\begin{array}{c}\left(-2u^2{u}_x-4uv{v}_x+2v^2{u}_x\right)u_t\hfill \\ +\left(2u^2{v}_x-4uv{u}_x-2v^2{v}_x\right)v_t\hfill \\ +2\left(u^2+v^2\right)\left(u{u}_{\mathit{xt}}+v{v}_{\mathit{xt}}\right)\hfill \end{array}\right){\mathit{b}}_{\mathit{3}}}{4}}\hfill \\ \hfill & -{\displaystyle \frac{\left(-c_1{u}^{2}v-c_1{v}^3+2a{v}_{\mathit{xx}}\right)u_t}{4}}\hfill \\ \hfill & -{\displaystyle \frac{\left(c_1{u}^3+c_{1}u{v}^2-2a{u}_{\mathit{xx}}\right)v_t}{4}}\hfill \\ \hfill & +{\displaystyle \frac{a\left(u{v}_{1,1,2}-v{u}_{\mathit{xxt}}-u_x{v}_{\mathit{xt}}+u_{\mathit{xt}}{v}_x\right)}{2}}\hfill \end{array}$$

(99)

or

$$\begin{array}{cc}\hfill & {\mathsf{\Phi }}^t={\displaystyle \frac{b_1}{4}}{\left|q\right|}^4+{\displaystyle \frac{b_2}{6}}{\left|q\right|}^6+{\displaystyle \frac{b_3}{4}}{\left({\left(\right|q|}^2{)}_x\right)}^2-a\Im \left(q_x^{*}{q}_{xx}\right)-{\displaystyle \frac{1}{2}}\Im \left(q^{*}{q}_t\right)-{\displaystyle \frac{c_1}{4}}{\left|q\right|}^2\Im \left(q^{*}{q}_x\right),\hfill \\ \hfill & {\mathsf{\Phi }}^x=-{\displaystyle \frac{b_3}{4}}{D}_t\left({\left|q\right|}^2{\left|q_x\right|}^2\right)-{\displaystyle \frac{c_1}{4}}{\left|q\right|}^2\Im \left(q^{*}{q}_t\right).\hfill \end{array}$$

(100)

Therefore, the conserved quantities, respectively, are

$$\begin{array}{c}\hfill P=2{\int }_{-\infty }^{\infty }{\mathsf{\Phi }}^{t}dx={\int }_{-\infty }^{\infty }{\left|q\right|}^{2}dx={\displaystyle \frac{2A^2}{B}},\end{array}$$

(101)

$$\begin{array}{c}\hfill M={\int }_{-\infty }^{\infty }{\mathsf{\Phi }}^{t}dx={\displaystyle \frac{1}{2i}}{\int }_{-\infty }^{\infty }\left(q^{*}{q}_x-q{q}_x^{*}\right)dx=-{\displaystyle \frac{2\kappa A^2}{B}}\end{array}$$

(102)

and

$$\begin{array}{ccc}\hfill H& =& {\int }_{-\infty }^{\infty }{\mathsf{\Phi }}^{t}dx={\displaystyle \frac{A^2}{B}}\left[\omega -2a\kappa \left(B^2+{\kappa }^2\right)\right]+{\displaystyle \frac{4A^4}{3B}}\left[b_1-c_1\kappa +{\displaystyle \frac{4}{5}}{b}_3{B}^2\right]+{\displaystyle \frac{16b_2{A}^6}{15B}}\hfill \end{array}$$

(103)

Taken together, the regular-response formulas show that integrated optical power depends mainly on envelope localization while momentum additionally records the carrier-frequency parameter. The energy-like invariant is more sensitive to the details of the SPM and dispersive terms. In the singular inverse-power cases, however, the proposed zero-background bright profiles fail the field equation; their formal power and momentum integrals must not be reported as conserved properties of the singular model.

3. Conclusions

Conservation laws for a perturbed pure-cubic nonlinear Schrödinger equation were examined for eleven representative SPM response structures, with particular attention to the hypotheses needed to interpret local balances as global optical invariants. The intensity balance provides a direct connection to propagation physics: on a spatial interval, changes in integrated power are exactly accounted for by the boundary power flux; hence, a finite-power localized pulse retains its integrated intensity when the net flux vanishes.

The three measures are governed by different aspects of the wave. First, P depends principally on the envelope amplitude, width, and localization exponent ($P\propto A^2/B$ times a shape factor for sech-type profiles). Second, for the unchirped traveling profiles considered here, $M=-\kappa P$, so the carrier phase gradient determines the signed momentum once the power is known. Third, the energy-like quantity H depends on the detailed nonlinear response and higher-order dispersive and derivative terms, making its existence and value more sensitive to compatibility conditions and far-field behavior. Finite integrals of a trial profile should not, by themselves, be taken as proof that it solves the nonlinear evolution equation.

For both generalized anti-cubic and anti-cubic laws, we corrected an important admissibility issue: if the coefficient of the singular inverse-power response is nonzero, the proposed zero-background sech-type profiles cannot solve the field equation. Consequently, these profiles do not support physically meaningful bright soliton values of P, M, or H for the singular model. The previously claimed divergent Hamiltonian must not be presented as a property of a soliton whose existence is contradicted by the tails. This restriction distinguishes genuinely applicable conservation diagnostics from purely formal expressions.

Future studies may formulate inverse-power nonlinearities on nonzero backgrounds, including background-subtracted conserved quantities and perturbation dynamics, and may use direct PDE simulations to test boundary-flux identities and spectral stability in physically admissible parameter regimes. These extensions are not assumed or claimed to have been completed here.