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Published: 14 March 2023

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1

Kamvissis, Spyridon. "Long time behavior for the focusing nonlinear schroedinger equation with real spectral singularities." Communications in Mathematical Physics 180, no. 2 (October 1996): 325–41. http://dx.doi.org/10.1007/bf02099716.

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2

Benci, Vieri, Marco Ghimenti, and Anna Maria Micheletti. "The nonlinear Schroedinger equation: Solitons dynamics." Journal of Differential Equations 249, no. 12 (December 2010): 3312–41. http://dx.doi.org/10.1016/j.jde.2010.09.026.

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3

ABLOWITZ, MARK J., and CONSTANCE M. SCHOBER. "HAMILTONIAN INTEGRATORS FOR THE NONLINEAR SCHROEDINGER EQUATION." International Journal of Modern Physics C 05, no. 02 (April 1994): 397–401. http://dx.doi.org/10.1142/s012918319400057x.

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Hamiltonian integration schemes for the Nonlinear Schroedinger Equation are examined. The efficiency with respect to accuracy and integration time of an integrable scheme, a standard conservative scheme, and a symplectic method is compared.

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4

Kim, Jong Uhn. "Invariant measures for a stochastic nonlinear Schroedinger equation." Indiana University Mathematics Journal 55, no. 2 (2006): 687–718. http://dx.doi.org/10.1512/iumj.2006.55.2701.

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5

Plastino, A. R., and C. Tsallis. "Nonlinear Schroedinger equation in the presence of uniform acceleration." Journal of Mathematical Physics 54, no. 4 (April 2013): 041505. http://dx.doi.org/10.1063/1.4798999.

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6

Degasperis, A., S. V. Manakov, and P. M. Santini. "Multiple-scale perturbation beyond the nonlinear Schroedinger equation. I." Physica D: Nonlinear Phenomena 100, no. 1-2 (January 1997): 187–211. http://dx.doi.org/10.1016/s0167-2789(96)00179-0.

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7

Jeanjean, Louis, and Kazunaga Tanaka. "A positive solution for a nonlinear Schroedinger equation on R^N." Indiana University Mathematics Journal 54, no. 2 (2005): 443–64. http://dx.doi.org/10.1512/iumj.2005.54.2502.

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8

Duell, Wolf-Patrick, and Guido Schneider. "Justification of the nonlinear Schroedinger equation for a resonant Boussinesq model." Indiana University Mathematics Journal 55, no. 6 (2006): 1813–34. http://dx.doi.org/10.1512/iumj.2006.55.2824.

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9

Mel'nikov, V. K. "Integration of the nonlinear Schroedinger equation with a self-consistent source." Communications in Mathematical Physics 137, no. 2 (April 1991): 359–81. http://dx.doi.org/10.1007/bf02431884.

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10

Bountis, Tassos, and Fernando D. Nobre. "Travelling-wave and separated variable solutions of a nonlinear Schroedinger equation." Journal of Mathematical Physics 57, no. 8 (August 2016): 082106. http://dx.doi.org/10.1063/1.4960723.

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11

Boffetta, G., and A. R. Osborne. "Computation of the direct scattering transform for the nonlinear Schroedinger equation." Journal of Computational Physics 102, no. 2 (October 1992): 252–64. http://dx.doi.org/10.1016/0021-9991(92)90370-e.

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12

Schrader, D. "Explicit calculation of N-soliton solutions of the nonlinear Schroedinger equation." IEEE Journal of Quantum Electronics 31, no. 12 (1995): 2221–25. http://dx.doi.org/10.1109/3.477750.

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13

Zakharov, V. E., E. A. Kuznetsov, and S. L. Musher. "Quasi classical regime of collapse in the three-dimensional nonlinear Schroedinger equation." Physica D: Nonlinear Phenomena 28, no. 1-2 (September 1987): 221. http://dx.doi.org/10.1016/0167-2789(87)90138-2.

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14

Osborne, A. R. "The Hyperelliptic Inverse Scattering Transform for the Periodic, Defocusing Nonlinear Schroedinger Equation." Journal of Computational Physics 109, no. 1 (November 1993): 93–107. http://dx.doi.org/10.1006/jcph.1993.1202.

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15

Van, Cao Long. "Propagation of Ultrashort Pulses in Nonlinear Media." Communications in Physics 26, no. 4 (March 10, 2017): 301. http://dx.doi.org/10.15625/0868-3166/26/4/9184.

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In this paper, a general propagation equation of ultrashort pulses in an arbitrary dispersive nonlinear medium derived in [9] has been used for the case of Kerr media. This equation which is called Generalized Nonlinear Schroedinger Equation usually has very complicated form and looking for its solutions is usually a very difficult task. Theoretical methods reviewed in this paper to solve this equation are effective only for some special cases. As an example we describe the method of developed elliptic Jacobi function expansion and its expended form: F-expansion Method. Several numerical methods of finding approximate solutions are briefly discussed. We concentrate mainly on the methods: Split-Step, Runge-Kutta and Imaginary-time algorithms. Some numerical experiments are implemented for soliton propagation and interacting high order solitons. We consider also an interesting phenomenon, namely the collapse of solitons, where the variational formalism has been used.

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16

Tian, Huiping, Zhonghao Li, and Guosheng Zhou. "Stable propagation of ultrashort optical pulses in modified higher-order nonlinear Schroedinger equation." Optics Communications 205, no. 1-3 (April 2002): 221–26. http://dx.doi.org/10.1016/s0030-4018(02)01316-0.

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17

Dudko, G. M., Yu A. Filimonov, A. A. Galishnikov, R. Marcelli, and S. A. Nikitov. "Nonlinear Schroedinger equation analysis of MSSW pulse propagation in ferrite-dielectric-metal structure." Journal of Magnetism and Magnetic Materials 272-276 (May 2004): 999–1000. http://dx.doi.org/10.1016/j.jmmm.2003.12.673.

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18

Meškauskas, T., and F. Ivanauskas. "Initial Boundary-Value Problems for Derivative Nonlinear Schroedinger Equation. Justification of Two-Step Algorithm." Nonlinear Analysis: Modelling and Control 7, no. 2 (December 5, 2002): 69–104. http://dx.doi.org/10.15388/na.2002.7.2.15195.

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We investigate two different initial boundary-value problems for derivative nonlinear Schrödinger equation. The boundary conditions are Dirichlet or generalized periodic ones. We propose a two-step algorithm for numerical solving of this problem. The method consists of Bäcklund type transformations and difference scheme. We prove the convergence and stability in C and H1 norms of Crank–Nicolson finite difference scheme for the transformed problem. There are no restrictions between space and time grid steps. For the derivative nonlinear Schrödinger equation, the proposed numerical algorithm converges and is stable in C1 norm.

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19

Marshall, Ian, and Michael Semenov-Tian-Shansky. "Poisson Groups and Differential Galois Theory of Schroedinger Equation on the Circle." Communications in Mathematical Physics 284, no. 2 (June 24, 2008): 537–52. http://dx.doi.org/10.1007/s00220-008-0539-9.

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20

Abdou, M. A. "New exact travelling wave solutions for the generalized nonlinear Schroedinger equation with a source." Chaos, Solitons & Fractals 38, no. 4 (November 2008): 949–55. http://dx.doi.org/10.1016/j.chaos.2007.01.027.

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21

LeMesurier, B. J., G. Papanicolaou, C. Sulem, and P. L. Sulem. "Focusing and multi-focusing solutions of the nonlinear Schrödinger equation." Physica D: Nonlinear Phenomena 31, no. 1 (May 1988): 78–102. http://dx.doi.org/10.1016/0167-2789(88)90015-2.

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22

Lee, T. D. "A New Approach to Solve the Low-lying States of the Schroedinger Equation." Journal of Statistical Physics 121, no. 5-6 (December 2005): 1015–71. http://dx.doi.org/10.1007/s10955-005-5476-9.

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23

Strampp, W., and W. Oevel. "A Nonlinear Derivative Schroedinger-Equation: Its Bi-Hamilton Structures, Their Inverses, Nonlocal Symmetries and Mastersymmetries." Progress of Theoretical Physics 74, no. 4 (October 1, 1985): 922–25. http://dx.doi.org/10.1143/ptp.74.922.

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24

Nassif, Cláudio, and P. R. Silva. "Anomalous coalescence from a nonlinear Schroedinger equation with a quintic term: interpretation through Thompson's approach." Physica A: Statistical Mechanics and its Applications 334, no. 3-4 (March 2004): 335–42. http://dx.doi.org/10.1016/j.physa.2003.11.019.

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25

Azzollini, A., and A. Pomponio. "On the Schroedinger equation in $\mathbb{R}^{N}$ under the effect of a general nonlinear term." Indiana University Mathematics Journal 58, no. 3 (2009): 1361–78. http://dx.doi.org/10.1512/iumj.2009.58.3576.

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26

Tajiri, Masayoshi, and Yosuke Watanabe. "Breather solutions to the focusing nonlinear Schrödinger equation." Physical Review E 57, no. 3 (March 1, 1998): 3510–19. http://dx.doi.org/10.1103/physreve.57.3510.

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27

Aktosun, Tuncay, Francesco Demontis, and Cornelis van der Mee. "Exact solutions to the focusing nonlinear Schrödinger equation." Inverse Problems 23, no. 5 (September 11, 2007): 2171–95. http://dx.doi.org/10.1088/0266-5611/23/5/021.

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28

Liu, Zhongxuan, Qi Feng, Chengyou Lin, Zhaoyang Chen, and Yingchun Ding. "Bipolar solitons of the focusing nonlinear Schrödinger equation." Physica B: Condensed Matter 501 (November 2016): 117–22. http://dx.doi.org/10.1016/j.physb.2016.08.015.

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29

Sulem, Catherine, and Pierre-Louis Sulem. "Focusing nonlinear schrödinger equation and wave-packet collapse." Nonlinear Analysis: Theory, Methods & Applications 30, no. 2 (December 1997): 833–44. http://dx.doi.org/10.1016/s0362-546x(96)00168-x.

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30

Liu, Xiao, Gideon Simpson, and Catherine Sulem. "Focusing singularity in a derivative nonlinear Schrödinger equation." Physica D: Nonlinear Phenomena 262 (November 2013): 48–58. http://dx.doi.org/10.1016/j.physd.2013.07.011.

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31

Kamvissis, Spyridon. "Focusing nonlinear Schrödinger equation with infinitely many solitons." Journal of Mathematical Physics 36, no. 8 (August 1995): 4175–80. http://dx.doi.org/10.1063/1.530953.

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32

Fibich, G. "Self-Focusing in the Damped Nonlinear Schrödinger Equation." SIAM Journal on Applied Mathematics 61, no. 5 (January 2001): 1680–705. http://dx.doi.org/10.1137/s0036139999362609.

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33

Chen, Yu, Jing Lu, and Fanfei Meng. "Focusing nonlinear Hartree equation with inverse‐square potential." Mathematische Nachrichten 293, no. 12 (September 21, 2020): 2271–98. http://dx.doi.org/10.1002/mana.201900331.

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34

IBRAHIM, SLIM. "GEOMETRIC-OPTICS FOR NONLINEAR CONCENTRATING WAVES IN FOCUSING AND NON-FOCUSING TWO GEOMETRIES." Communications in Contemporary Mathematics 06, no. 01 (February 2004): 1–23. http://dx.doi.org/10.1142/s0219199704001239.

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With the methods used in [1] and [4], we prove that in the absence of focus, nonlinear geometrical optics of the critical wave equation with variable coefficients, is reduced to linear geometrical optics combined with wave operators for the critical wave equation with coefficients fixed on concentrating points. On the odd-dimensional spheres, we prove that passing through a focus is generated by a modified scattering operator.

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35

Saanouni, Tarek. "Remarks on the critical nonlinear high-order heat equation." Arab Journal of Mathematical Sciences 26, no. 1/2 (March 15, 2019): 127–52. http://dx.doi.org/10.1016/j.ajmsc.2019.03.002.

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The initial value problem for a semi-linear high-order heat equation is investigated. In the focusing case, global well-posedness and exponential decay are obtained. In the focusing sign, global and non global existence of solutions are discussed via the potential well method.

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36

Ibrahim, Slim, Nader Masmoudi, and Kenji Nakanishi. "Scattering threshold for the focusing nonlinear Klein–Gordon equation." Analysis & PDE 4, no. 3 (December 28, 2011): 405–60. http://dx.doi.org/10.2140/apde.2011.4.405.

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37

Wright, Otis C. "Near homoclinic orbits of the focusing nonlinear Schrödinger equation." Nonlinearity 12, no. 5 (August 13, 1999): 1277–87. http://dx.doi.org/10.1088/0951-7715/12/5/304.

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38

Chen, Jinbing, and Dmitry E. Pelinovsky. "Rogue periodic waves of the focusing nonlinear Schrödinger equation." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 474, no. 2210 (February 2018): 20170814. http://dx.doi.org/10.1098/rspa.2017.0814.

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Rogue periodic waves stand for rogue waves on a periodic background. The nonlinear Schrödinger equation in the focusing case admits two families of periodic wave solutions expressed by the Jacobian elliptic functions dn and cn . Both periodic waves are modulationally unstable with respect to long-wave perturbations. Exact solutions for the rogue periodic waves are constructed by using the explicit expressions for the periodic eigenfunctions of the Zakharov–Shabat spectral problem and the Darboux transformations. These exact solutions generalize the classical rogue wave (the so-called Peregrine’s breather). The magnification factor of the rogue periodic waves is computed as a function of the elliptic modulus. Rogue periodic waves constructed here are compared with the rogue wave patterns obtained numerically in recent publications.

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39

Fang, DaoYuan, Jian Xie, and Thierry Cazenave. "Scattering for the focusing energy-subcritical nonlinear Schrödinger equation." Science China Mathematics 54, no. 10 (October 2011): 2037–62. http://dx.doi.org/10.1007/s11425-011-4283-9.

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40

Arora, Anudeep Kumar, Svetlana Roudenko, and Kai Yang. "On the focusing generalized Hartree equation." Mathematics in Applied Sciences and Engineering 9999, no. 9999 (December 16, 2020): 1–20. http://dx.doi.org/10.5206/mase/10855.

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In this paper we give a review of the recent progress on the focusing generalized Hartree equation, which is a nonlinear Schrodinger-type equation with the nonlocal nonlinearity, expressed as a convolution with the Riesz potential. We describe the local well-posedness in H1 and Hs settings, discuss the extension to the global existence and scattering, or finite time blow-up. We point out different techniques used to obtain the above results, and then show the numerical investigations of the stable blow-up in the L2 -critical setting. We finish by showing known analytical results about the stable blow-up dynamics in the L2 -critical setting.

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41

Lugiato, L. A., F. Prati, M. L. Gorodetsky, and T. J. Kippenberg. "From the Lugiato–Lefever equation to microresonator-based soliton Kerr frequency combs." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 376, no. 2135 (November 12, 2018): 20180113. http://dx.doi.org/10.1098/rsta.2018.0113.

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The model, that is usually called the Lugiato–Lefever equation (LLE), was introduced in 1987 with the aim of providing a paradigm for dissipative structure and pattern formation in nonlinear optics. This model, describing a driven, detuned and damped nonlinear Schroedinger equation, gives rise to dissipative spatial and temporal solitons. Recently, the rather idealized conditions, assumed in the LLE, have materialized in the form of continuous wave driven optical microresonators, with the discovery of temporal dissipative Kerr solitons (DKS). These experiments have revealed that the LLE is a perfect and exact description of Kerr frequency combs—first observed in 2007, i.e. 20 years after the original formulation of the LLE—and in particular describe soliton states. Observed to spontaneously form in Kerr frequency combs in crystalline microresonators in 2013, such DKS are preferred state of operation, offering coherent and broadband optical frequency combs, whose bandwidth can be extended exploiting soliton-induced broadening phenomena. Combined with the ability to miniaturize and integrate on-chip, microresonator-based soliton Kerr frequency combs have already found applications in self-referenced frequency combs, dual-comb spectroscopy, frequency synthesis, low noise microwave generation, laser frequency ranging, and astrophysical spectrometer calibration, and have the potential to make comb technology ubiquitous. As such, pattern formation in driven, dissipative nonlinear optical systems is becoming the central Physics of soliton micro-comb technology. This article is part of the theme issue ‘Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 2)’.

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42

Zhong, Wei-Ping, Zhengping Yang, Milivoj Belić, and WenYe Zhong. "Breather solutions of the nonlocal nonlinear self-focusing Schrödinger equation." Physics Letters A 395 (April 2021): 127228. http://dx.doi.org/10.1016/j.physleta.2021.127228.

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43

Miller, Peter D., and Spyridon Kamvissis. "On the semiclassical limit of the focusing nonlinear Schrödinger equation." Physics Letters A 247, no. 1-2 (October 1998): 75–86. http://dx.doi.org/10.1016/s0375-9601(98)00565-9.

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44

Brydges, David C., and Gordon Slade. "Statistical mechanics of the 2-dimensional focusing nonlinear Schrödinger equation." Communications in Mathematical Physics 182, no. 2 (December 1996): 485–504. http://dx.doi.org/10.1007/bf02517899.

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45

Borghese, Michael, Robert Jenkins, and Kenneth D. T. R. McLaughlin. "Long time asymptotic behavior of the focusing nonlinear Schrödinger equation." Annales de l'Institut Henri Poincaré C, Analyse non linéaire 35, no. 4 (July 2018): 887–920. http://dx.doi.org/10.1016/j.anihpc.2017.08.006.

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46

Lyng, Gregory D., and Peter D. Miller. "TheN-soliton of the focusing nonlinear Schrödinger equation forN large." Communications on Pure and Applied Mathematics 60, no. 7 (2007): 951–1026. http://dx.doi.org/10.1002/cpa.20162.

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47

Luo, Yongming. "Sharp scattering for the cubic-quintic nonlinear Schrödinger equation in the focusing-focusing regime." Journal of Functional Analysis 283, no. 1 (July 2022): 109489. http://dx.doi.org/10.1016/j.jfa.2022.109489.

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48

Serkin, Vladimir N., E. M. Schmidt, T. L. Belyaeva, E. Marti-Panameno, and H. Salazar. "Femtosecond Maxwellian solitons. II. Verification of a model of the nonlinear Schroedinger equation in the theory of optical solitons." Quantum Electronics 27, no. 11 (November 30, 1997): 940–43. http://dx.doi.org/10.1070/qe1997v027n11abeh001123.

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49

Masaki, Satoshi. "A sharp scattering condition for focusing mass-subcritical nonlinear Schrödinger equation." Communications on Pure & Applied Analysis 14, no. 4 (2015): 1481–531. http://dx.doi.org/10.3934/cpaa.2015.14.1481.

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50

Ibrahim, Slim, Nader Masmoudi, and Kenji Nakanishi. "Correction to “Scattering threshold for the focusing nonlinear Klein–Gordon equation”." Analysis & PDE 9, no. 2 (March 24, 2016): 503–14. http://dx.doi.org/10.2140/apde.2016.9.503.

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