The integrable nonlocal nonlinear Schrödinger equation with the self-induced parity-time-symmetric potential [M. J. Ablowitz and Z. H. Musslimani, Phys. Rev. Lett. 110, 064105 (2013)] is investigated, which is an integrable extension of the standard nonlinear Schrödinger equation. Its novel higher-order rational solitons are found using the nonlocal version of the generalized perturbation (1,N-1)-fold Darboux transformation. These rational solitons illustrate abundant wave structures for the distinct choices of parameters (e.g., the strong and weak interactions of bright and dark rational solitons). Moreover, we also explore the dynamical behaviors of these higher-order rational solitons with some small noises on the basis of numerical simulations.
Colorectal cancer (CRC) is one of the most frequently occurring malignancy tumors with a high morbidity additionally, CRC patients may develop liver metastasis, which is the major cause of death. Despite significant advances in diagnostic and therapeutic techniques, the survival rate of colorectal liver metastasis (CRLM) patients remains very low. CRLM, as a complex cascade reaction process involving multiple factors and procedures, has complex and diverse molecular mechanisms. In this review, we summarize the mechanisms/pathophysiology, diagnosis, treatment of CRLM. We also focus on an overview of the recent advances in understanding the molecular basis of CRLM with a special emphasis on tumor microenvironment and promise of newer targeted therapies for CRLM, further improving the prognosis of CRLM patients.
In this paper, a novel, simple, and constructive method is presented to find the generalized perturbation (n, M )-fold Darboux transformations (DTs) of the modified nonlinear Schrödinger (MNLS) equation in terms of fractional forms of determinants. In particular, we apply the generalized perturbation (1, N − 1)-fold DTs to find its explicit multi-rogue wave solutions. The wave structures of these rogue wave solutions of the MNLS equation are discussed in detail for different parameters, which display abundant interesting wave structures including the triangle and pentagon, etc. and may be useful to study the physical mechanism of multi-rogue waves in optics. The dynamical behaviors of these multi-rogue wave solutions are illustrated using numerical simulations. The same Darboux matrix can also be used to investigate the Gerjikov-Ivanov equation such that its multirogue wave solutions and their wave structures are also found. The method can also be extended to find multi-rogue wave solutions of other nonlinear integrable equations.
An integrable system of two-component nonlinear Ablowitz-Ladik equations is used to construct complex rogue-wave (RW) solutions in an explicit form. First, the modulational instability of continuous waves is studied in the system. Then, new higher-order discrete two-component RW solutions of the system are found by means of a newly derived discrete version of a generalized Darboux transformation. Finally, the perturbed evolution of these RW states is explored in terms of systematic simulations, which demonstrates that tightly and loosely bound RWs are, respectively, nearly stable and strongly unstable solutions.
We study higher-order rogue wave (RW) solutions of the coupled integrable dispersive AB system (also called Pedlosky system), which describes the evolution of wave-packets in a marginally stable or unstable baroclinic shear flow in geophysical fluids. We propose its continuous-wave (CW) solutions and existent conditions for their modulation instability to form the rogue waves. A new generalized N-fold Darboux transformation (DT) is proposed in terms of the Taylor series expansion for the spectral parameter in the Darboux matrix and its limit procedure and applied to the CW solutions to generate multi-rogue wave solutions of the coupled AB system, which satisfy the general compatibility condition. The dynamical behaviors of these higher-order rogue wave solutions demonstrate both strong and weak interactions by modulating parameters, in which some weak interactions can generate the abundant triangle, pentagon structures, etc. Particularly, the trajectories of motion of peaks and depressions of profiles of the first-order RWs are explicitly analyzed. The generalized DT method used in this paper can be extended to other nonlinear integrable systems. These results may be useful for understanding the corresponding rogue-wave phenomena in fluid mechanics and related fields.
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