Some localized wave solutions for the coupled Gerdjikov–Ivanov equation

Dong, Ming; Li, Tian; Wei, Jingdong; Wang, Yun

doi:10.1016/j.aml.2021.107483

Cited by 13 publications

(3 citation statements)

References 17 publications

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“…The nonlinear Schrödinger equation (NLSE) has been widely studied as an important class of models for mathematical physics equations. In order to analyze the effects of higher-order perturbations, scholars have started to study the modified NLSE and the original derived NLSE, which include the third kind of derivative nonlinear Schrödinger equation (DNLSE), i.e., the Gerdjikov-Ivanov equation [24][25][26][27], iq t + q xx + iq 2 q *…”

Section: Modulation Instability Localized Wave Solutions Of the Modif...mentioning

confidence: 99%

Modulation instability, localized wave solutions of the modified Gerdjikov–Ivanov equation with anomalous dispersion

et al. 2023

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Solving new integrable systems and exploring their physical applications have been a hot topic. This paper gives the modulation instability in the continuous wave background of the new modified Gerdjikov-Ivanov equation with anomalous dispersion. Based on the extend Lax pair, the localized wave solutions of the new model are obtained via generalized Darboux transformation method, various types solutions including breather, rogue waves, interaction solutions are presented and their dynamic properties are analyzed. The results obtained have certain application value for nonlinear optics and long-distance transmission.

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Section: Modulation Instability Localized Wave Solutions Of the Modif...mentioning

confidence: 99%

Modulation instability, localized wave solutions of the modified Gerdjikov–Ivanov equation with anomalous dispersion

et al. 2023

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“…In Ding and Liu-Q (2019), two types of the breathers on the periodic background, and the k-th order rogue waves were constructed for the GIE, based on the existing N-th order analytic solutions.Under suitable hypothesis for the current velocity, the GI envelope solitons were derived and discussed in Lü et al (2015). The coupled GIE was investigated by using Lax pair, Darboux transformation (Dong et al 2021). In Hassan et al (2021), the collective variable technique was used to explore the GIE.The perturbed optical solitons to the time-space fractional GIE with conformable derivatives was investigated (Younis et al 2021).…”

Section: Introductionmentioning

confidence: 99%

Approximate-analytic optical soliton solutions of a modified-Gerdjikov–Ivanov equation: modulation instability

Abdel‐Gawad

2023

Opt Quant Electron

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The Gerdjikov–Ivanov equation (GIE) occupied a remarkable area of research in the literature. In the present work, a modified GIE (MGIE) is considered which is new and was not studied in the literature. Also, the modified-unified method (MUM) is used to obtain approximate analytic solutions (AASs) of MGIE. Up to our knowledge, no AASs for non-integrable complex field equation were found up to now. Thus the AASs found, here, are novel. The UM addresses finding the exact solutions to integrable equations. In this sense as no exact solution for MGIE exists, consequently, it is not integrable. So, here, approximate analytic optical soliton solutions are invoked. The UM stands for expressing the solution of nonlinear evolution equations in polynomial and rational forms in an auxiliary function (AF) with an appropriate auxiliary equation. For finding exact solutions by the UM, the coefficients of the AF, with all powers, are set equal to zero, For a non-integrable equation, only approximate solutions are affordable. In this case, we are led to utilizing the MUM. Herein, non-zero coefficients (residue terms (RTs)) are considered as errors, which are space and time-independent. It is worth mentioning that, this is in contrast to the errors found by the different numerical methods, where they are space and time-dependent. Further, in the present case, the maximum error is controlled via an adequate choice of the parameters in the RTs. These solutions are displayed in graphs. Breather soliton, chirped soliton and M-shape soliton, among others, are observed. Furthermore, modulation instability (MI) is studied and it is found MI triggers when the coefficient of the nonlinear dispersion exceeds a critical value.

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“…The authors gave the explicit expression of eigenfunction and analyzed the dynamics of the soliton–RW solutions by choosing different parameters. We can see that the eigenfunctions have been the key ingredient for constructing new solutions to nonlinear integrable systems such as Sasa–Satsuma equation [47], coupled GI equation [48–50], and three‐wave resonance equation [51]. In this work, starting with the nonzero seed solution in the exponential form of variable

t

, we will give the eigenfunction of the

n

‐component NLS equation on the basis of the Cayley–Hamilton theorem.…”

Section: Introductionmentioning

confidence: 99%

Localized wave solutions to the vector nonlinear Schrödinger equation with nonzero backgrounds

Zhai

et al. 2023

Math Methods in App Sciences

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The vector nonlinear Schrödinger (NLS) system is studied through the generalized Darboux transformation (DT). We begin with the nonzero seed solution in the exponential form of evolution variable and explore the eigenfunction of the ‐component NLS equation. With the formulae of the eigenfunction, localized wave solutions of the two‐component and three‐component NLS equations are studied. By choosing different free parameters, some special solutions and the dynamic evolutions are presented, including breather–rogue wave, breather fission, breather fusion, and breather–soliton.

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Some localized wave solutions for the coupled Gerdjikov–Ivanov equation

Cited by 13 publications

References 17 publications

Modulation instability, localized wave solutions of the modified Gerdjikov–Ivanov equation with anomalous dispersion

Modulation instability, localized wave solutions of the modified Gerdjikov–Ivanov equation with anomalous dispersion

Approximate-analytic optical soliton solutions of a modified-Gerdjikov–Ivanov equation: modulation instability

Localized wave solutions to the vector nonlinear Schrödinger equation with nonzero backgrounds

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