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Journal articles on the topic 'Quadratic nonlinearity'

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Published: 4 June 2021
Last updated: 6 September 2023

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1

Mosolov, A. B., and O. Yu Dinartsev. "Filtration law with quadratic nonlinearity." Journal of Engineering Physics 51, no. 5 (November 1986): 1292–95. http://dx.doi.org/10.1007/bf00870683.

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2

Defever, F., W. Troost, and Z. Hasiewicz. "Superconformal algebras with quadratic nonlinearity." Physics Letters B 273, no. 1-2 (December 1991): 51–55. http://dx.doi.org/10.1016/0370-2693(91)90552-2.

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3

TRAN, H. T. "QUADRATIC NONLINEAR SURFACE WAVES." Journal of Nonlinear Optical Physics & Materials 05, no. 01 (January 1996): 133–38. http://dx.doi.org/10.1142/s021886359600012x.

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Nonlinear surface waves at an interface between a linear medium and another medium with quadratic nonlinearity are possible due to the phenomenon of nonlinearity-induced phase matching. The waves are numerically calculated, along with their dispersion, stability, and comparison with their cubic counterparts. An extension to guided waves is also discussed.
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4

CHANDRAN, VINOD, STEVE ELGAR, and CHARLES PEZESHKI. "BISPECTRAL AND TRISPECTRAL CHARACTERIZATION OF TRANSITION TO CHAOS IN THE DUFFING OSCILLATOR." International Journal of Bifurcation and Chaos 03, no. 03 (June 1993): 551–57. http://dx.doi.org/10.1142/s021812749300043x.

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Higher-order spectral (bispectral and trispectral) analyses of numerical solutions of the Duffing equation with a cubic stiffness are used to isolate the coupling between the triads and quartets, respectively, of nonlinearly interacting Fourier components of the system. The Duffing oscillator follows a period-doubling intermittency catastrophic route to chaos. For period-doubled limit cycles, higher-order spectra indicate that both quadratic and cubic nonlinear interactions are important to the dynamics. However, when the Duffing oscillator becomes chaotic, global behavior of the cubic nonlinearity becomes dominant and quadratic nonlinear interactions are weak, while cubic interactions remain strong. As the nonlinearity of the system is increased, the number of excited Fourier components increases, eventually leading to broad-band power spectra for chaos. The corresponding higher-order spectra indicate that although some individual nonlinear interactions weaken as nonlinearity increases, the number of nonlinearly interacting Fourier modes increases. Trispectra indicate that the cubic interactions gradually evolve from encompassing a few quartets of Fourier components for period-1 motion to encompassing many quartets for chaos. For chaos, all the components within the energetic part of the power spectrum are cubically (but not quadratically) coupled to each other.
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5

Darmanyan, S. A., and M. Nevière. "Eigenmodes of waveguides with quadratic nonlinearity." Optics Communications 176, no. 1-3 (March 2000): 231–37. http://dx.doi.org/10.1016/s0030-4018(00)00479-x.

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6

Babin, A., and A. Figotin. "Nonlinear photonic crystals: I. Quadratic nonlinearity." Waves in Random Media 11, no. 2 (April 2001): R31—R102. http://dx.doi.org/10.1088/0959-7174/11/2/201.

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7

Gu, Wen-ju, Zhen Yi, Li-hui Sun, and Yan Yan. "Enhanced quadratic nonlinearity with parametric amplifications." Journal of the Optical Society of America B 35, no. 4 (March 2, 2018): 652. http://dx.doi.org/10.1364/josab.35.000652.

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8

Wang, Ke, Jing Li, Fan Dai, Mengshuai Wang, Chuanhang Wang, Qiang Wang, Chenghou Tu, Yongnan Li, and Huitian Wang. "Robust Pulse-Pumped Quadratic Soliton Assisted by Third-Order Nonlinearity." Photonics 10, no. 2 (February 2, 2023): 155. http://dx.doi.org/10.3390/photonics10020155.

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The generation of a quadratic soliton in a pulse-pumped microresonator has attracted significant interest in recent years. The strong second-order nonlinearity and high peak power of pumps offer a straightforward way to increase efficiency. In this case, the influence of the third-order nonlinearity effect becomes significant and cannot be ignored. In this paper, we study the quadratic soliton in a degenerate optical parametric oscillator driven synchronously by the pulse pump with third-order nonlinearity. Our simulations verify that the robustness of quadratic soliton generation is enhanced when the system experiences a perturbation from pump power, cavity detuning, and pump pulse width. These results represent a new way of manipulating frequency comb in resonant microphotonic structures.
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9

Zhang, Wen, Jian Zhang, and Heilong Mi. "Ground states and multiple solutions for Hamiltonian elliptic system with gradient term." Advances in Nonlinear Analysis 10, no. 1 (July 30, 2020): 331–52. http://dx.doi.org/10.1515/anona-2020-0113.

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Abstract This paper is concerned with the following nonlinear Hamiltonian elliptic system with gradient term $$\begin{array}{} \displaystyle \left\{\,\, \begin{array}{ll} -{\it\Delta} u +\vec{b}(x)\cdot \nabla u+V(x)u = H_{v}(x,u,v)\,\,\hbox{in}\,\mathbb{R}^{N},\\[-0.3em] -{\it\Delta} v -\vec{b}(x)\cdot \nabla v +V(x)v = H_{u}(x,u,v)\,\,\hbox{in}\,\mathbb{R}^{N}.\\ \end{array} \right. \end{array}$$ Compared with some existing issues, the most interesting feature of this paper is that we assume that the nonlinearity satisfies a local super-quadratic condition, which is weaker than the usual global super-quadratic condition. This case allows the nonlinearity to be super-quadratic on some domains and asymptotically quadratic on other domains. Furthermore, by using variational method, we obtain new existence results of ground state solutions and infinitely many geometrically distinct solutions under local super-quadratic condition. Since we are without more global information on the nonlinearity, in the proofs we apply a perturbation approach and some special techniques.
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10

Wu, Junyao, Chunbiao Li, Xu Ma, Tengfei Lei, and Guanrong Chen. "Simplification of Chaotic Circuits With Quadratic Nonlinearity." IEEE Transactions on Circuits and Systems II: Express Briefs 69, no. 3 (March 2022): 1837–41. http://dx.doi.org/10.1109/tcsii.2021.3125680.

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11

Buryak, Alexander V., and Yuri S. Kivshar. "Spatial optical solitons governed by quadratic nonlinearity." Optics Letters 19, no. 20 (October 15, 1994): 1612. http://dx.doi.org/10.1364/ol.19.001612.

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12

Piron, R., E. Toussaere, D. Josse, S. Brasselet, and J. Zyss. "Polymer-based microcavity with photoencoded quadratic nonlinearity." Optics Letters 25, no. 17 (September 1, 2000): 1255. http://dx.doi.org/10.1364/ol.25.001255.

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13

Zhang, Guoce, and Bo Zhang. "Secondary Resonance Energy Harvesting with Quadratic Nonlinearity." Materials 13, no. 15 (July 31, 2020): 3389. http://dx.doi.org/10.3390/ma13153389.

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Piezoelectric energy harvesters can transform the mechanical strain into electrical energy. The microelectromechanical transformation device is often composed of piezoelectric cantilevers and has been largely experimented. Most resonances have been developed to harvest nonlinear vibratory energy except for combination resonances. This paper is to analyze several secondary resonances of a cantilever-type piezoelectric energy harvester with a tip magnet. The conventional Galerkin method is improved to truncate the continuous model, an integro-partial differential equation with time-dependent boundary conditions. Then, more resonances on higher-order vibration modes can be obtained. The stable steady-state response is formulated approximately but analytically for the first two subharmonic and combination resonances. The instability boundaries are discussed for these secondary resonances from quadratic nonlinearity. A small damping and a large excitation readily result in an unstable response, including the period-doubling and quasiperiodic motions that can be employed to enhance the voltage output around a wider band of working frequency. Runge–Kutta method is employed to numerically compute the time history for stable and unstable motions. The stable steady-state responses from two different methods agree well with each other. The outcome enriches structural dynamic theory on nonlinear vibration.
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14

Mak, William C. K., Boris A. Malomed, and P. L. Chu. "Solitons in coupled waveguides with quadratic nonlinearity." Physical Review E 55, no. 5 (May 1, 1997): 6134–40. http://dx.doi.org/10.1103/physreve.55.6134.

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15

Cveticanin, L., M. Zukovic, Gy Mester, I. Biro, and J. Sarosi. "Oscillators with symmetric and asymmetric quadratic nonlinearity." Acta Mechanica 227, no. 6 (March 7, 2016): 1727–42. http://dx.doi.org/10.1007/s00707-016-1582-9.

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16

Vabishchevich, P. N. "Difference schemes for problems with quadratic nonlinearity." Computational Mathematics and Modeling 4, no. 1 (1993): 16–21. http://dx.doi.org/10.1007/bf01130872.

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17

Ryabov, Vladimir G. "Maximally nonlinear functions over finite fields." Discrete Mathematics and Applications 33, no. 1 (February 1, 2023): 41–53. http://dx.doi.org/10.1515/dma-2023-0005.

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Abstract An n-place function over a field F q $ \mathbf {F}_q $ with q elements is called maximally nonlinear if it has the largest nonlinearity among all q-valued n-place functions. We show that, for even n=2, a function is maximally nonlinear if and only if its nonlinearity is q n − 1 ( q − 1 ) − q n 2 − 1 $ q^{n-1}(q - 1) - q^{\frac n2-1} $ ; for n=1, the corresponding criterion for maximal nonlinearity is q − 2. For q > 2 $ q \gt 2 $ and even n=2, we describe the set of all maximally nonlinear quadratic functions and find its cardinality. In this case, all maximally nonlinear quadratic functions are quadratic bent functions and their number is smaller than the halved number of the bent functions.
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18

Li, Yaning, Quanguo Zhang, and Baoyan Sun. "Existence of Solutions for Fractional Boundary Value Problems with a Quadratic Growth of Fractional Derivative." Journal of Function Spaces 2020 (May 13, 2020): 1–10. http://dx.doi.org/10.1155/2020/7139795.

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In this paper, we deal with two fractional boundary value problems which have linear growth and quadratic growth about the fractional derivative in the nonlinearity term. By using variational methods coupled with the iterative methods, we obtain the existence results of solutions. To the best of the authors’ knowledge, there are no results on the solutions to the fractional boundary problem which have quadratic growth about the fractional derivative in the nonlinearity term.
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19

Jeong, Hyunjo, Hyojeong Shin, Shuzeng Zhang, and Xiongbing Li. "Measurement and In-Depth Analysis of Higher Harmonic Generation in Aluminum Alloys with Consideration of Source Nonlinearity." Materials 16, no. 12 (June 18, 2023): 4453. http://dx.doi.org/10.3390/ma16124453.

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Harmonic generation measurement is recognized as a promising tool for inspecting material state or micro-damage and is an ongoing research topic. Second harmonic generation is most frequently employed and provides the quadratic nonlinearity parameter (β) that is calculated by the measurement of fundamental and second harmonic amplitudes. The cubic nonlinearity parameter (β2), which dominates the third harmonic amplitude and is obtained by third harmonic generation, is often used as a more sensitive parameter in many applications. This paper presents a detailed procedure for determining the correct β2 of ductile polycrystalline metal samples such as aluminum alloys when there exists source nonlinearity. The procedure includes receiver calibration, diffraction, and attenuation correction and, more importantly, source nonlinearity correction for third harmonic amplitudes. The effect of these corrections on the measurement of β2 is presented for aluminum specimens of various thicknesses at various input power levels. By correcting the source nonlinearity of the third harmonic and further verifying the approximate relationship between the cubic nonlinearity parameter and the square of the quadratic nonlinearity parameter (β∗β), β2≈β∗β, the cubic nonlinearity parameters could be accurately determined even with thinner samples and lower input voltages.
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20

Darmanyan, S., A. Kamchatnov, and F. Lederer. "Optical shock waves in media with quadratic nonlinearity." Physical Review E 58, no. 4 (October 1, 1998): R4120—R4123. http://dx.doi.org/10.1103/physreve.58.r4120.

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21

Berezansky, Leonid, Jaromír Baštinec, Josef Diblík, and Zdeněk Šmarda. "On a delay population model with quadratic nonlinearity." Advances in Difference Equations 2012, no. 1 (2012): 230. http://dx.doi.org/10.1186/1687-1847-2012-230.

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22

Liu, Q. M. "Exact solutions to nonlinear equations with quadratic nonlinearity." Journal of Physics A: Mathematical and General 34, no. 24 (June 7, 2001): 5083–88. http://dx.doi.org/10.1088/0305-4470/34/24/306.

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23

Buryak, Alexander V., and Yuri S. Kivshar. "Spatial optical solitons governed by quadratic nonlinearity: erratum." Optics Letters 20, no. 9 (May 1, 1995): 1080. http://dx.doi.org/10.1364/ol.20.001080.

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24

Cveticanin, L. "Vibrations of the nonlinear oscillator with quadratic nonlinearity." Physica A: Statistical Mechanics and its Applications 341 (October 2004): 123–35. http://dx.doi.org/10.1016/j.physa.2004.04.123.

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25

Skryabin, Dmitry V. "Coupled-mode theory for microresonators with quadratic nonlinearity." Journal of the Optical Society of America B 37, no. 9 (August 14, 2020): 2604. http://dx.doi.org/10.1364/josab.397015.

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26

Marikhin, V. G. "Integrable systems with quadratic nonlinearity in Fourier space." Physics Letters A 310, no. 1 (April 2003): 60–66. http://dx.doi.org/10.1016/s0375-9601(03)00246-9.

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27

Torchigin, V. P., and A. V. Torchigin. "Manifestation of optical quadratic nonlinearity in gas mixtures." Doklady Physics 49, no. 10 (October 2004): 553–55. http://dx.doi.org/10.1134/1.1815412.

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28

Khviengia, Z., and E. Sezgin. "BRST operator for superconformal algebras with quadratic nonlinearity." Physics Letters B 326, no. 3-4 (May 1994): 243–50. http://dx.doi.org/10.1016/0370-2693(94)91317-x.

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29

Kurosu, Megumi, Daiki Hatanaka, Hajime Okamoto, and Hiroshi Yamaguchi. "Buckling-induced quadratic nonlinearity in silicon phonon waveguide structures." Japanese Journal of Applied Physics 61, SD (May 20, 2022): SD1025. http://dx.doi.org/10.35848/1347-4065/ac5532.

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Abstract We fabricated and characterized a single-crystal silicon phonon waveguide structure with lead zirconate titanate (PZT) piezoelectric transducers. The compressive stress in a silicon-on-insulator wafer causes a membrane waveguide to buckle, leading to the quadratic nonlinearity. The PZT transducer integrated in an on-chip configuration enables us to excite high-intensity mechanical vibration, which allows the characterization of nonlinear behavior. We observed a softening nonlinear response as a function of the drive power and demonstrated the mode shift and frequency conversion. This is the first report of the nonlinear behavior caused by the quadratic nonlinearity in a buckled phonon waveguide structure. This study provides a method to control the sign and the order of nonlinearity in a phonon waveguide by utilizing the internal stress, which allows the precise manipulation of elastic waves in phononic integrated circuits.
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30

Salas, Alvaro H., S. A. El-Tantawy, and Noufe H. Aljahdaly. "An Exact Solution to the Quadratic Damping Strong Nonlinearity Duffing Oscillator." Mathematical Problems in Engineering 2021 (January 18, 2021): 1–8. http://dx.doi.org/10.1155/2021/8875589.

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The nonlinear equations of motion such as the Duffing oscillator equation and its family are seldom addressed in intermediate instruction in classical dynamics; this one is problematic because it cannot be solved in terms of elementary functions before. Thus, in this work, the stability analysis of quadratic damping higher-order nonlinearity Duffing oscillator is investigated. Hereinafter, some new analytical solutions to the undamped higher-order nonlinearity Duffing oscillator in the form of Weierstrass elliptic function are obtained. Posteriorly, a novel exact analytical solution to the quadratic damping higher-order nonlinearity Duffing equation under a certain condition (not arbitrary initial conditions) and in the form of Weierstrass elliptic function is derived in detail for the first time. Furthermore, the obtained solutions are camped to the Runge–Kutta fourth-order (RK4) numerical solution.
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31

Grimshaw, Roger, Efim Pelinovsky, and Tatjana Talipova. "Solitary wave transformation in a medium with sign-variable quadratic nonlinearity and cubic nonlinearity." Physica D: Nonlinear Phenomena 132, no. 1-2 (July 1999): 40–62. http://dx.doi.org/10.1016/s0167-2789(99)00045-7.

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32

Bose, Anil K., Alan S. Cover, and James A. Reneke. "On point dissipativeN-dimensional systems of differential equations with quadratic nonlinearity." International Journal of Mathematics and Mathematical Sciences 16, no. 1 (1993): 139–48. http://dx.doi.org/10.1155/s016117129300016x.

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Known sufficient conditions for quadratic dynamical systemx′=Ax+f(x)to be point dissipative given in terms ofAandffor dimensions2and3are extended to allow for more general forms for the nonlinear termf(x). Furthermore, the conditions extend tondimensions whenfis quadratic with zero set an(n−1)-dimensional hyperplane.
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33

Komissarova, M. V., T. M. Lysak, I. G. Zakharova, and A. A. Kalinovich. "Parametric gap solitons in PT-symmetric optical structures." Journal of Physics: Conference Series 2249, no. 1 (April 1, 2022): 012008. http://dx.doi.org/10.1088/1742-6596/2249/1/012008.

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Abstract It is well known that quadratic nonlinearity and feedback through Bragg periodicity are the basis for parametric gap solitons. The major part of the relevant investigations refers to passive systems. At the same time, optical systems supplemented with active elements can demonstrate unusual properties. Asymmetry intrinsic to structures with parity-time (PT) symmetry is a bright confirmation of this statement. The interplay of nonlinearity, Bragg reflection and gain/loss profile can lead to the complicated pattern of wave interactions and novel results. In this study we address the properties of two-color solitons in complex PT symmetric periodic structures with quadratic nonlinearity. We focus on the case of single Bragg resonance. We reveal the region of parameters where stable parametric solitons may exist. We demonstrate that characteristics of forming solitons depend on the order of alteration of amplifying and absorbing layers.
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34

Cormack, John M., and Mark F. Hamilton. "Nonlinear propagation of quasiplanar shear wave beams in soft elastic media with transverse isotropy." Journal of the Acoustical Society of America 153, no. 5 (May 1, 2023): 2887. http://dx.doi.org/10.1121/10.0019358.

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Model equations are developed for shear wave propagation in a soft elastic material that include effects of nonlinearity, diffraction, and transverse isotropy. A theory for plane wave propagation by Cormack [J. Acoust. Soc. Am. 150, 2566 (2021)] is extended to include leading order effects of wavefront curvature by assuming that the motion is quasiplanar, which is consistent with other paraxial model equations in nonlinear acoustics. The material is modeled using a general expansion of the strain energy density to fourth order in strain that comprises thirteen terms defining the elastic moduli. Equations of motion for the transverse displacement components are obtained using Hamilton's principle. The coupled equations of motion describe diffraction, anisotropy of the wave speeds, quadratic and cubic plane wave nonlinearity, and quadratic nonlinearity associated with wavefront curvature. Two illustrative special cases are investigated. Spatially varying shear vertical wave motion in the fiber direction excites a quadratic nonlinear interaction unique to transversely isotropic soft solids that results in axial second harmonic motion with longitudinal polarization. Shear horizontal wave motion in the fiber plane reveals effects of anisotropy on third harmonic generation, such as beam steering and dependence of harmonic generation efficiency on the propagation and fiber directions.
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35

Li, Mengmeng, Alexey Lomonosov, Zhonghua Shen, Hogeon Seo, Kyung-Young Jhang, Vitalyi Gusev, and Chenyin Ni. "Monitoring of Thermal Aging of Aluminum Alloy via Nonlinear Propagation of Acoustic Pulses Generated and Detected by Lasers." Applied Sciences 9, no. 6 (March 21, 2019): 1191. http://dx.doi.org/10.3390/app9061191.

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Nonlinear acoustic techniques are established tools for the characterization of micro-inhomogeneous materials with higher sensitivity, compared to linear ultrasonic techniques. In particular, the evaluation of material elastic quadratic nonlinearity via the detection of the second harmonic generation by acoustic waves is known to provide an assessment of the state variation of heat treated micro-structured materials. We report on the first application for non-destructive diagnostics of material thermal aging of finite-amplitude longitudinal acoustic pulses generated and detected by lasers. Finite-amplitude longitudinal pulses were launched in aluminum alloy samples by deposited liquid-suspended carbon particles layer irradiated by a nanosecond laser source. An out-of-plane displacement at the epicenter of the opposite sample surface was measured by an interferometer. This laser ultrasonic technique provided an opportunity to study the propagation in aluminum alloys of finite-amplitude acoustic pulses with a strain up to 5 × 10−3. The experiments revealed a signature of the hysteretic quadratic nonlinearity of micro-structured material manifested in an increase of the duration of detected acoustic pulses with an increase of their amplitude. The parameter of the hysteretic quadratic nonlinearity of the aluminum alloy (Al6061) was found to be of the order of 100 and to exhibit more than 50% variations in the process of the alloy thermal aging. By comparing the measured parameter of the hysteretic quadratic nonlinearity in aluminum alloys that were subjected to heat-treatment at 220 °C for different times (0 min, 20 min, 40 min, 1 h, 2 h, 10 h, 100 h, and 1000 h), with measurements of yield strength in same samples, it was established that the extrema in the dependence of the hysteretic nonlinearity and of the yield strength of this alloy on heat treatment time are correlated. This experimental observation provides the background for future research with the application goal of suggested nonlinear laser ultrasonic techniques for non-destructive evaluation of alloys’ strength and rigidity in the process of their heat treatment.
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36

Komissarova, M. V., I. G. Zakharova, T. M. Lysak, and A. A. Kalinovich. "Wave Beams in Active Periodic Structures with Quadratic Nonlinearity." Bulletin of the Russian Academy of Sciences: Physics 85, no. 12 (December 2021): 1370–76. http://dx.doi.org/10.3103/s1062873821120108.

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37

Darmanyan, S., A. Kobyakov, and F. Lederer. "Strongly localized modes in discrete systems with quadratic nonlinearity." Physical Review E 57, no. 2 (February 1, 1998): 2344–49. http://dx.doi.org/10.1103/physreve.57.2344.

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38

Mak, William C. K., Boris A. Malomed, and P. L. Chu. "Three-wave gap solitons in waveguides with quadratic nonlinearity." Physical Review E 58, no. 5 (November 1, 1998): 6708–22. http://dx.doi.org/10.1103/physreve.58.6708.

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39

Sazonov, S. V., I. G. Zakharova, and A. A. Kalinovich. "Generating Optic Terahertz Pulses in Waveguides with Quadratic Nonlinearity." Bulletin of the Russian Academy of Sciences: Physics 84, no. 2 (February 2020): 177–79. http://dx.doi.org/10.3103/s106287382002032x.

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40

Conforti, Matteo. "Exact cascading nonlinearity in quasi-phase-matched quadratic media." Optics Letters 39, no. 8 (April 10, 2014): 2427. http://dx.doi.org/10.1364/ol.39.002427.

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41

Kalocsai, Andre G., and Joseph W. Haus. "Nonlinear Schrödinger equation for optical media with quadratic nonlinearity." Physical Review A 49, no. 1 (January 1, 1994): 574–85. http://dx.doi.org/10.1103/physreva.49.574.

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42

Subačius, L., I. Kašalynas, M. Vingelis, R. Aleksiejūnas, and K. Jarašiūnas. "High-Speed Quadratic Electrooptic Nonlinearity in dc-Biased InP." Acta Physica Polonica A 107, no. 2 (February 2005): 280–85. http://dx.doi.org/10.12693/aphyspola.107.280.

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43

Horinouchi, S., H. Imai, G. J. Zhang, K. Mito, and K. Sasaki. "Optical quadratic nonlinearity in multilayer corona‐poled glass films." Applied Physics Letters 68, no. 25 (June 17, 1996): 3552–54. http://dx.doi.org/10.1063/1.116634.

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44

Malomed, B. A., T. Wagenknecht, A. R. Champneys, and M. J. Pearce. "Accumulation of embedded solitons in systems with quadratic nonlinearity." Chaos: An Interdisciplinary Journal of Nonlinear Science 15, no. 3 (September 2005): 037116. http://dx.doi.org/10.1063/1.1938433.

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45

Hamaya, Yoshihiro. "Global attractivity in a discrete model with quadratic nonlinearity." Applicable Analysis 69, no. 3-4 (July 1998): 257–63. http://dx.doi.org/10.1080/00036819808840661.

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46

Shekoyan, A. V. "Wave beams in crystals with dislocations and quadratic nonlinearity." Technical Physics Letters 35, no. 4 (April 2009): 337–39. http://dx.doi.org/10.1134/s1063785009040142.

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47

Peschel, T., U. Peschel, F. Lederer, and B. A. Malomed. "Solitary waves in Bragg gratings with a quadratic nonlinearity." Physical Review E 55, no. 4 (April 1, 1997): 4730–39. http://dx.doi.org/10.1103/physreve.55.4730.

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48

Gusev, Vitalyi. "Self-modulation instability in materials with hysteretic quadratic nonlinearity." Wave Motion 33, no. 2 (February 2001): 145–53. http://dx.doi.org/10.1016/s0165-2125(00)00052-4.

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49

Mak, William C. K., Boris A. Malomed, and P. L. Chu. "Solitary waves in asymmetric coupled waveguides with quadratic nonlinearity." Optics Communications 154, no. 1-3 (August 1998): 145–51. http://dx.doi.org/10.1016/s0030-4018(98)00293-4.

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Golubkov, A. A., and Vladimir A. Makarov. "Spectroscopy of one-dimensionally inhomogeneous media with quadratic nonlinearity." Quantum Electronics 41, no. 11 (November 30, 2011): 968–75. http://dx.doi.org/10.1070/qe2011v041n11abeh014710.

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