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Academic literature on the topic 'Nonlinear field equations'

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Published: 18 May 2024

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Journal articles on the topic "Nonlinear field equations"

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Tan, Jinggang, Ying Wang, and Jianfu Yang. "Nonlinear fractional field equations." Nonlinear Analysis: Theory, Methods & Applications 75, no. 4 (2012): 2098–110. http://dx.doi.org/10.1016/j.na.2011.10.010.

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Fairlie, David B. "Interconnections among nonlinear field equations." Journal of Physics A: Mathematical and Theoretical 53, no. 10 (2020): 104001. http://dx.doi.org/10.1088/1751-8121/ab6f17.

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Rego-Monteiro, M. A., and F. D. Nobre. "Nonlinear quantum equations: Classical field theory." Journal of Mathematical Physics 54, no. 10 (2013): 103302. http://dx.doi.org/10.1063/1.4824129.

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Tanaka, Yosuke, Takefumi Shudo, Tetsutaro Yosinaga, and Hiroshi Kimura. "Relativistic field equations and nonlinear dynamics." Chaos, Solitons & Fractals 37, no. 4 (2008): 941–49. http://dx.doi.org/10.1016/j.chaos.2008.01.004.

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Burt, P. B. "Nonperturbative solution of nonlinear field equations." Il Nuovo Cimento B 100, no. 1 (1987): 43–52. http://dx.doi.org/10.1007/bf02829775.

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Wells, R. O. "Nonlinear field equations and twistor theory." Mathematical Intelligencer 7, no. 2 (1985): 26–32. http://dx.doi.org/10.1007/bf03024171.

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Bruce, S. A. "Nonlinear Maxwell equations and strong-field electrodynamics." Physica Scripta 97, no. 3 (2022): 035303. http://dx.doi.org/10.1088/1402-4896/ac50c2.

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Abstract We investigate two Lagrangian models of nonlinear electrodynamics (NLED). These models lead to two different sets of nonlinear (NL) Maxwell equations. The first case deals with the well-known Heisenberg-Euler (HE) model of electromagnetic (EM) self-interactions in a vacuum where only the lowest orders in EM Lorentz invariants are considered. The second instance proposes an extension of the HE model. It consists of a NL Maxwell-Dirac spinor model where the EM field modifies the dynamics of the energy-momentum operator sector of the Dirac Lagrangian instead of its rest-mass term counterpart. This work complements our recent research on NL Dirac equations in the strong EM field regime.

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Clapp, Mónica, and Tobias Weth. "Multiple Solutions of Nonlinear Scalar Field Equations." Communications in Partial Differential Equations 29, no. 9-10 (2005): 1533–54. http://dx.doi.org/10.1081/pde-200037766.

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Mederski, Jarosław. "Nonradial solutions of nonlinear scalar field equations." Nonlinearity 33, no. 12 (2020): 6349–80. http://dx.doi.org/10.1088/1361-6544/aba889.

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Liu, Jiu, Tao Liu, and Jia-Feng Liao. "A perturbation of nonlinear scalar field equations." Nonlinear Analysis: Real World Applications 45 (February 2019): 531–41. http://dx.doi.org/10.1016/j.nonrwa.2018.07.022.

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Dissertations / Theses on the topic "Nonlinear field equations"

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Chakraborty, Susanto. "Solutions of some nonlinear field equations, painleve` properties and Chaos." Thesis, University of North Bengal, 2006. http://hdl.handle.net/123456789/610.

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Dunning, Tania Clare. "Perturbed conformal field theory, nonlinear integral equations and spectral problems." Thesis, Durham University, 2000. http://etheses.dur.ac.uk/4329/.

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This thesis is concerned with various aspects of perturbed conformal field theory and the methods used to calculate finite-size effects of integrable quantum field theories. Nonlinear integral equations are the main tools to find the exact ground-state energy of a quantum field theory. The thermodyamic Bethe ansatz (TBA) equations are a set of examples and are known for a large number of models. However, it is also an interesting question to find exact equations describing the excited states of integrable models. The first part of this thesis uses analytical continuation in a continuous parameter to find TBA like equations describing the spin-zero excited states of the sine-Gordon model at coupling β(^2) = 16π/3. Comparisons are then made with a further type of nonlinear integral equation which also predicts the excited state energies. Relations between the two types of equation are studied using a set of functional relations recently introduced in integrable quantum field theory. A relevant perturbation of a conformal field theory results in either a massive quantum field theory such as the sine-Gordon model, or a different massless conformal field theory. The second part of this thesis investigates flows between conformal field theories using a nonlinear integral equation. New families of flows are found which exhibit a rather unexpected behaviour. The final part of this thesis begins with a review of a connection between integrable quantum field theory and properties of certain ordinary differential equations of second- and third-order. The connection is based on functional relations which appear on both sides of the correspondence; for the second-order case these are exactly the functional relations mentioned above. The results are extended to include a correspondence between n(^th) order differential equations and Bethe ansatz system of SU(n) type. A set of nonlinear integral equations are derived to check the results.

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O'Day, Joseph Patrick. "Investigation of a coupled Duffing oscillator system in a varying potential field /." Online version of thesis, 2005. https://ritdml.rit.edu/dspace/handle/1850/1212.

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GOFFI, ALESSANDRO. "Topics in nonlinear PDEs: from Mean Field Games to problems modeled on Hörmander vector fields." Doctoral thesis, Gran Sasso Science Institute, 2019. http://hdl.handle.net/20.500.12571/9808.

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This thesis focuses on qualitative and quantitative aspects of some nonlinear PDEs arising in optimal control and differential games, ranging from regularity issues to maximum principles. More precisely, it is concerned with the analysis of some fully nonlinear second order degenerate PDEs over Hörmander vector fields that can be written in Hamilton-Jacobi-Bellman and Isaacs form and those arising in the recent theory of Mean Field Games, where the prototype model is described by a coupled system of PDEs involving a backward Hamilton-Jacobi and a forward Fokker-Planck equation. The thesis is divided in three parts. The first part is devoted to analyze strong maximum principles for fully nonlinear second order degenerate PDEs structured on Hörmander vector fields, having as a particular example fully nonlinear subelliptic PDEs on Carnot groups. These results are achieved by introducing a notion of subunit vector field for these nonlinear degenerate operators in the spirit of the seminal works on linear equations. As a byproduct, we then prove some new strong comparison principles for equations that can be written in Hamilton-Jacobi-Bellman form and Liouville theorems for some second order fully nonlinear degenerate PDEs. The second part of the thesis deals with time-dependent fractional Mean Field Game systems. These equations arise when the dynamics of the average player is described by a stable Lévy process to which corresponds a fractional Laplacian as diffusion operator. More precisely, we establish existence and uniqueness of solutions to such systems of PDEs with regularizing coupling among the equations for every order of the fractional Laplacian $sin(0,1)$. The existence of solutions is addressed via the vanishing viscosity method and we prove that in the subcritical regime the equations are satisfied in classical sense, while if $sleq1/2$ we find weak energy solutions. To this aim, we develop an appropriate functional setting based on parabolic Bessel potential spaces. We finally show uniqueness of solutions both under the Lasry-Lions monotonicity condition and for short time horizons. The last part focuses on the regularizing effect of evolutive Hamilton-Jacobi equations with Hamiltonian having superlinear growth in the gradient and unbounded right-hand side. In particular, the analysis is performed both for viscous Hamilton-Jacobi equations and its fractional counterpart in the subcritical regime via a duality method. The results are accomplished exploiting the regularity of solutions to Fokker-Planck-type PDEs with rough velocity fields in parabolic Sobolev and Bessel potential spaces respectively.

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Mulvey, Joseph Anthony. "Symmetry methods for integrable systems." Thesis, Durham University, 1996. http://etheses.dur.ac.uk/5379/.

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This thesis discusses various properties of a number of differential equations which we will term "integrable". There are many definitions of this word, but we will confine ourselves to two possible characterisations — either an equation can be transformed by a suitable change of variables to a linear equation, or there exists an infinite number of conserved quantities associated with the equation that commute with each other via some Hamiltonian structure. Both of these definitions rely heavily on the concept of the symmetry of a differential equation, and so Chapters 1 and 2 introduce and explain this idea, based on a geometrical theory of p.d.e.s, and describe the interaction of such methods with variational calculus and Hamiltonian systems. Chapter 3 discusses a somewhat ad hoc method for solving evolution equations involving a series ansatz that reproduces well-known solutions. The method seems to be related to symmetry methods, although the precise connection is unclear. The rest of the thesis is dedicated to the so-called Universal Field Equations and related models. In Chapter 4 we look at the simplest two-dimensional cases, the Bateman and Born-lnfeld equations. By looking at their generalised symmetries and Hamiltonian structures, we can prove that these equations satisfy both the definitions of integrability mentioned above. Chapter Five contains the general argument which demonstrates the linearisability of the Bateman Universal equation by calculation of its generalised symmetries. These symmetries are helpful in analysing and generalising the Lagrangian structure of Universal equations. An example of a linearisable analogue of the Born-lnfeld equation is also included. The chapter concludes with some speculation on Hamiltoian properties.

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Hoq, Qazi Enamul. "Quantization Of Spin Direction For Solitary Waves in a Uniform Magnetic Field." Thesis, University of North Texas, 2003. https://digital.library.unt.edu/ark:/67531/metadc4210/.

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It is known that there are nonlinear wave equations with localized solitary wave solutions. Some of these solitary waves are stable (with respect to a small perturbation of initial data)and have nonzero spin (nonzero intrinsic angular momentum in the centre of momentum frame). In this paper we consider vector-valued solitary wave solutions to a nonlinear Klein-Gordon equation and investigate the behavior of these spinning solitary waves under the inﬂuence of an externally imposed uniform magnetic ﬁeld. We ﬁnd that the only stationary spinning solitary wave solutions have spin parallel or antiparallel to the magnetic ﬁeld direction.

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Nys, Manon. "Schrödinger equations with an external magnetic field: Spectral problems and semiclassical states." Doctoral thesis, Universite Libre de Bruxelles, 2015. http://hdl.handle.net/2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/216641.

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In this thesis, we study Schrödinger equations with an external magnetic field. In the first part, we are interested in an eigenvalue problem. We work in an open, bounded and simply connected domain in dimension two. We consider a magnetic potential singular at one point in the domain, and related to the magnetic field being a multiple of a Dirac delta. Those two objects are related to the Bohm-Aharonov effect, in which a charged particle is influenced by the presence of the magnetic potential although it remains in a region where the magnetic field is zero. We consider the Schrödinger magnetic operator appearing in the Schrödinger equation in presence of an external magnetic field. We want to study the spectrum of this operator, and more particularly how it varies when the singular point moves in the domain. We prove some results of continuity and differentiability of the eigenvalues when the singular point moves in the domain or approaches its boundary. Finally, in case of half-integer circulation of the magnetic potential, we study some asymptotic behaviour of the eigenvalues close to their critical points. In the second part, we study nonlinear Schrödinger equations in a cylindrically setting. We are interested in the semiclassical limit of the equation. We prove the existence of a semiclassical solution concentrating on a circle. Moreover, the radius of that circle is determined by the electric potential, but also by the magnetic potential. This result is totally new with respect to the ones before, in which the concentration is driven only by the electric potential.<br>Doctorat en Sciences<br>info:eu-repo/semantics/nonPublished

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Nowak, Derek Brant. "The Design of a Novel Tip Enhanced Near-field Scanning Probe Microscope for Ultra-High Resolution Optical Imaging." PDXScholar, 2010. https://pdxscholar.library.pdx.edu/open_access_etds/361.

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Traditional light microscopy suffers from the diffraction limit, which limits the spatial resolution to λ/2. The current trend in optical microscopy is the development of techniques to bypass the diffraction limit. Resolutions below 40 nm will make it possible to probe biological systems by imaging the interactions between single molecules and cell membranes. These resolutions will allow for the development of improved drug delivery mechanisms by increasing our understanding of how chemical communication within a cell occurs. The materials sciences would also benefit from these high resolutions. Nanomaterials can be analyzed with Raman spectroscopy for molecular and atomic bond information, or with fluorescence response to determine bulk optical properties with tens of nanometer resolution. Near-field optical microscopy is one of the current techniques, which allows for imaging at resolutions beyond the diffraction limit. Using a combination of a shear force microscope (SFM) and an inverted optical microscope, spectroscopic resolutions below 20 nm have been demonstrated. One technique, in particular, has been named tip enhanced near-field optical microscopy (TENOM). The key to this technique is the use of solid metal probes, which are illuminated in the far field by the excitation wavelength of interest. These probes are custom-designed using finite difference time domain (FDTD) modeling techniques, then fabricated with the use of a focused ion beam (FIB) microscope. The measure of the quality of probe design is based directly on the field enhancement obtainable. The greater the field enhancement of the probe, the more the ratio of near-field to far-field background contribution will increase. The elimination of the far-field signal by a decrease of illumination power will provide the best signal-to-noise ratio in the near-field images. Furthermore, a design that facilitates the delocalization of the near-field imaging from the far-field will be beneficial. Developed is a novel microscope design that employs two-photon non-linear excitation to allow the imaging of the fluorescence from almost any visible fluorophore at resolutions below 30 nm without changing filters or excitation wavelength. The ability of the microscope to image samples at atmospheric pressure, room temperature, and in solution makes it a very promising tool for the biological and materials science communities. The microscope demonstrates the ability to image topographical, optical, and electronic state information for single-molecule identification. A single computer, simple custom control circuits, field programmable gate array (FPGA) data acquisition, and a simplified custom optical system controls the microscope are thoroughly outlined and documented. This versatility enables the end user to custom-design experiments from confocal far-field single molecule imaging to high resolution scanning probe microscopy imaging. Presented are the current capabilities of the microscope, most importantly, high-resolution near-field images of J-aggregates with PIC dye. Single molecules of Rhodamine 6G dye and quantum dots imaged in the far-field are presented to demonstrate the sensitivity of the microscope. A comparison is made with the use of a mode-locked 50 fs pulsed laser source verses a continuous wave laser source on single molecules and J-aggregates in the near-field and far-field. Integration of an intensified CCD camera with a high-resolution monochromator allows for spectral information about the sample. The system will be disseminated as an open system design.

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Ruy, Danilo Virges [UNESP]. "Estrutura hamiltoniana da hierarquia PIV." Universidade Estadual Paulista (UNESP), 2011. http://hdl.handle.net/11449/92036.

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Made available in DSpace on 2014-06-11T19:25:34Z (GMT). No. of bitstreams: 0 Previous issue date: 2011-02-18Bitstream added on 2014-06-13T19:53:27Z : No. of bitstreams: 1 ruy_dv_me_ift.pdf: 500107 bytes, checksum: fef6f049175c290422f569aa7ad7e26e (MD5)<br>Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)<br>Esta dissertação trata da construção de hierarquias compatíveis com a equação PIV a partir dos modelos: AKNS, dois bósons e dois bósons quadráticos. Também são construidos os problema linear de Jimbo-Miwa dos três modelos e discutimos a hamiltoniana correspondente a equação PIV a partir do formalismo lagrangiano<br>This dissertation contains the construction of compatible hierarchies with the PIV equation from the models: AKNS, two-boson and quadratic two-boson. Also it is build the Jimbo-Miwa linear problem for the three models and we discuss the hamiltonian corresponding to fouth Painlevé equation from the Lagrangian formalism

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Ruy, Danilo Virges. "Estrutura hamiltoniana da hierarquia PIV /." São Paulo : [s.n.], 2011. http://hdl.handle.net/11449/92036.

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Orientador: Abraham Hirsz Zimerman<br>Banca: Iberê Luiz Caldas<br>Banca: Roberto André Kraenkel<br>Resumo: Esta dissertação trata da construção de hierarquias compatíveis com a equação PIV a partir dos modelos: AKNS, dois bósons e dois bósons quadráticos. Também são construidos os problema linear de Jimbo-Miwa dos três modelos e discutimos a hamiltoniana correspondente a equação PIV a partir do formalismo lagrangiano<br>Abstract: This dissertation contains the construction of compatible hierarchies with the PIV equation from the models: AKNS, two-boson and quadratic two-boson. Also it is build the Jimbo-Miwa linear problem for the three models and we discuss the hamiltonian corresponding to fouth Painlevé equation from the Lagrangian formalism<br>Mestre

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Books on the topic "Nonlinear field equations"

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Benci, Vieri, and Donato Fortunato. Variational Methods in Nonlinear Field Equations. Springer International Publishing, 2014. http://dx.doi.org/10.1007/978-3-319-06914-2.

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V, Murthy M. K., Spagnolo S, and Workshop on Nonlinear Hyperbolic Equations and Field Theory (1990 : Lake Como, Italy), eds. Nonlinear hyperbolic equations and field theory. Longman Scientific & Technical, 1992.

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Selfdual gauge field vortices: An analytical approach. Birkhäuser, 2007.

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LeFloch, Philippe G. The hyperboloidal foliation method. World Scientific, 2015.

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Tarantello, Gabriella. Selfdual gauge field vortices: An analytical approach. Birkhäuser, 2007.

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Tarantello, Gabriella. Selfdual gauge field vortices: An analytical approach. Birkhäuser, 2007.

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Prikarpatskiĭ, A. K. Algebraicheskie aspekty integriruemosti nelineĭnykh dinamicheskikh sistem na mnogoobrazii͡a︡kh. Nauk. dumka, 1991.

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Philippe, Blanchard, Dias J. P. 1944-, and Stubbe J. 1959-, eds. New methods and results in non-linear field equations: Proceedings of a conference held at the University of Bielefeld, Fed. Rep. of Germany, 7-10 July 1987. Springer-Verlag, 1989.

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Korsunskiĭ, S. V. Nonlinear waves in dispersive and dissipative systems with coupled fields. Longman, 1997.

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Gatica, Gabriel N. Boundary-field equation methods for a class of nonlinear problems. Longman, 1995.

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Book chapters on the topic "Nonlinear field equations"

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Ambrosio, Vincenzo. "Fractional Scalar Field Equations." In Nonlinear Fractional Schrödinger Equations in R^N. Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-60220-8_3.

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Nachman, Adrian I. "Multidimensional inverse scattering and nonlinear equations." In Field Theory, Quantum Gravity and Strings. Springer Berlin Heidelberg, 1988. http://dx.doi.org/10.1007/3-540-16452-9_18.

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Agüero Granados, M. A. "Coherent State Theory and the Field Lattice Model." In Nonlinear Evolution Equations and Dynamical Systems. Springer Berlin Heidelberg, 1991. http://dx.doi.org/10.1007/978-3-642-76172-0_45.

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Winternitz, P. "Lie Groups and Solutions of Nonlinear Partial Differential Equations." In Integrable Systems, Quantum Groups, and Quantum Field Theories. Springer Netherlands, 1993. http://dx.doi.org/10.1007/978-94-011-1980-1_11.

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Dafermos, C. M. "Equivalence of Referential and Spatial Field Equations in Continuum Physics." In Nonlinear Hyperbolic Problems: Theoretical, Applied, and Computational Aspects. Vieweg+Teubner Verlag, 1993. http://dx.doi.org/10.1007/978-3-322-87871-7_21.

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Krichever, I., O. Lipan, P. Wiegmann, and A. Zabrodin. "Quantum Integrable Systems and Elliptic Solutions of Classical Discrete Nonlinear Equations." In Low-Dimensional Applications of Quantum Field Theory. Springer US, 1997. http://dx.doi.org/10.1007/978-1-4899-1919-9_16.

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Klainerman, S. "On the Regularity of Classical Field Theories in Minkowski Space-Time R3+1." In Nonlinear Partial Differential Equations in Geometry and Physics. Birkhäuser Basel, 1997. http://dx.doi.org/10.1007/978-3-0348-8895-0_2.

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Esteban, Maria J., and Pierre-Louis Lions. "Stationary Solutions of Nonlinear Schrödinger Equations with an External Magnetic Field." In Partial Differential Equations and the Calculus of Variations. Birkhäuser Boston, 1989. http://dx.doi.org/10.1007/978-1-4684-9196-8_18.

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Esteban, Maria J., and Pierre-Louis Lions. "Stationary Solutions of Nonlinear Schrödinger Equations with an External Magnetic Field." In Partial Differential Equations and the Calculus of Variations. Birkhäuser Boston, 1989. http://dx.doi.org/10.1007/978-1-4615-9828-2_18.

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Antontsev, S. N., J. I. Díaz, and H. B. de Oliveira. "Stopping a Viscous Fluid by a Feedback Dissipative Field: Thermal Effects without Phase Changing." In Progress in Nonlinear Differential Equations and Their Applications. Birkhäuser Basel, 2005. http://dx.doi.org/10.1007/3-7643-7317-2_1.

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Conference papers on the topic "Nonlinear field equations"

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Verweij, Martin D. "Nonlinear and dissipative constitutive equations for coupled first-order acoustic field equations that are consistent with the generalized Westervelt equation." In INNOVATIONS IN NONLINEAR ACOUSTICS: ISNA17 - 17th International Symposium on Nonlinear Acoustics including the International Sonic Boom Forum. AIP, 2006. http://dx.doi.org/10.1063/1.2210354.

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Etrich, C., Paul Mandel, and Kenju Otsuka. "Laser rate equations with phase-sensitive interactions." In Nonlinear Dynamics in Optical Systems. Optica Publishing Group, 1992. http://dx.doi.org/10.1364/nldos.1992.tuc7.

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We derive the following set of equations describing a two-mode semiconductor laser for the case of a Fabry-Perot configuration, taking into account the holes burned into the amplifying medium by the standing field pattern and phase-sensitive interactions: (1) where κ = κ2/κ1 is the ratio of the decay rates of the electric fields E1 and E2. It is fixed to be larger than unity.

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Ghiner, A. V., and G. I. Surdutovich. "Method of integral equations and extinction theorem in volumetric and surface phenomena in nonlinear optics." In Nonlinear Optics. Optica Publishing Group, 1992. http://dx.doi.org/10.1364/nlo.1992.tud8.

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The method of integral equations of molecular optics [1] is based on the representation of the medium as the system of discrete oscillators and gives self-consistent description of the electromagnetic phenomena in medium without using operation of averaging when one comes from micro to macro-scopic field. In [2] the attempt has been made to apply method of integral equations to the nonlinear optics within the limits of the conception of local (microscopic) field. For justification of the corner-stone assumption that non-lineal polarization satisfies the wave equation it is necessary to assume the number of rather strict restrictions — plane pumping wave, plane interface between the media, approximation of given field and so on. In [3] this problem was solved in a general case without the above-mentioned assumptions but only for nonmagnetic medium with dipole mechanism of nonlinearity.

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Chavanis, P. H., Sumiyoshi Abe, Hans Herrmann, Piero Quarati, Andrea Rapisarda, and Constantino Tsallis. "General properties of nonlinear mean field Fokker-Planck equations." In COMPLEXITY, METASTABILITY, AND NONEXTENSIVITY: An International Conference. AIP, 2007. http://dx.doi.org/10.1063/1.2828726.

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García-Salcedo, R., and Aarón V. B. Arellano. "Nonlinear Electrodynamics and Wormhole Type Solutions for Einstein Field Equations." In ADVANCED SUMMER SCHOOL IN PHYSICS 2006: Frontiers in Contemporary Physics: EAV06. AIP, 2007. http://dx.doi.org/10.1063/1.2563177.

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Goorjian, Peter M., Rose M. Joseph, and Allen Taflove. "Calculations of Femtosecond Temporal Solitons and Spatial Solitons Using the Vector Maxwell's Equations." In Nonlinear Guided-Wave Phenomena. Optica Publishing Group, 1993. http://dx.doi.org/10.1364/nlgwp.1993.tub.12.

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Experimentalists have produced all-optical switches capable of 100-fs responses [1]. Also, there are experimental observations [2] and theoretical calculations [3] of spatial soliton interactions. To adequately model such effects, nonlinearities in optical materials [4] (both instantaneous and dispersive) must be included. In principle, the behavior of electromagnetic fields in nonlinear dielectrics can be determined by solving Maxwell's equations subject to the assumption that the electric polarization has a nonlinear relation to the electric field. However, until our previous work [5 - 8], the resulting nonlinear Maxwell's equations have not been solved directly. Rather, approximations have been made that result in a class of generalized nonlinear Schrodinger equations (GNLSE) [9] that solve only for the envelope of the optical pulses.

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ORTNER, J., and V. M. RYLYUK. "THE USE OF RELATIVISTIC ACTION IN STRONG-FIELD NONLINEAR PHOTOIONIZATION." In Proceedings of the Conference “Kadanoff-Baym Equations: Progress and Perspectives for Many-Body Physics”. WORLD SCIENTIFIC, 2000. http://dx.doi.org/10.1142/9789812793812_0022.

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Mukamel, Shaul, and Jasper Knoester. "Nonlinear Optical Susceptibilities; Beyond the Local Field Approximation." In Nonlinear Optical Properties of Materials. Optica Publishing Group, 1988. http://dx.doi.org/10.1364/nlopm.1988.mb3.

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The Bloch-Maxwell equations are usually derived for isolated molecules interacting with an external electromagnetic field. This is justified in the limit of low molecular density, where intermolecular forces may be neglected. The problem which we address in this work is how to extend these equations in a systematic way to incorporate properly intermolecular forces. We thus develop a systematic microscopic basis for the calculation of nonlinear response functions and susceptibilities. Such a theory is essential for relating microscopic, single-molecule, polarizabilities to the macroscopic susceptibilities of optical materials. The local field approximation [1,2] is a mean-field procedure which is widely used in the calculation of molecular susceptibilities at finite densities, when intermolecular forces are important. The local field model provides a simple way to relate the polarizabilities of isolated molecules to the macroscopic susceptibilities. It is clear, however, that this procedure is not rigorous. It fails to take properly into account the correlated dynamics of the interacting many-body system, i.e., correlations among the molecules, as well as correlations between the molecules and the radiation field. Short-range forces (e.g., exchange) are totally neglected in this procedure. Moreover, even the dipoledipole forces are not fully taken into account. The resulting susceptibilities do not depend at all on the wavevectors (apart from the local field contribution) but just on the frequencies. This indicates that processes such as exciton migration and energy transfer and transport (e.g., the Forster transfer) are neglected in this procedure. Such processes are often added phenomelogically in order to interpret transient grating spectroscopy[3], which is a four-wave mixing technique that measures transport processes by following the wavevector dependence of the susceptibilities. The common derivation of the local field approximation cannot be extended to include these processes, since it is intrinsically a mean-field single molecule theory.

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Statman, David, Karl Gass, and Bruce W. Liby. "Periodic Behavior in Photorefractive Two Beam Coupling." In Nonlinear Optics. Optica Publishing Group, 1992. http://dx.doi.org/10.1364/nlo.1992.md8.

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Abstract:

It is well known that the behavior of dynamical processes described by non-linear equations may lead to period doubling bifurcations and quasi-periodic behavior. Krolikowski, et.al. (1) have predicted that for four wave mixing, in the presence of an applied external field, periodic behavior can be observed. These predictions show that the observed period is constant in time for a given set of parameters. We have completed two beam coupling experiments using a helium neon laser as a source, and barium titanate as the photorefractive material. In the absense of an applied electric field we have observed periodic behavior during these experiments.

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Wang, Yuefang, Ganyun Sun, and Lihua Huang. "Magnetic Field-Induced Nonlinear Vibration of an Unbalanced Rotor." In ASME 2003 International Mechanical Engineering Congress and Exposition. ASMEDC, 2003. http://dx.doi.org/10.1115/imece2003-42498.

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The free and forced flexural vibrations are investigated for rotors of electric motors operating in unsymmetrical magnetic fields. The magnetic potential energy reserved in the air-gap is analytically derived and the unbalanced magnetic pull is obtained through the law of energy conservation. With this excitation, the equations of motion of the unbalanced rotor are developed for nonlinear displacements response. For small dynamic eccentricities, the equations of motion are simplified and the rotor is compared to a free Duffing oscillatory system. The analytic solution for forced vibrations subject to residual mass-unbalance excitations is also obtained. Jump phenomenon in the solution is pointed out, and the effects of initial eccentricity and flux density on the natural frequency are also investigated.

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Reports on the topic "Nonlinear field equations"

1

Davidson, R. C., W. W. Lee, and P. Stoltz. Statistically-averaged rate equations for intense nonneutral beam propagation through a periodic solenoidal focusing field based on the nonlinear Vlasov-Maxwell equations. Office of Scientific and Technical Information (OSTI), 1997. http://dx.doi.org/10.2172/304184.

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2

Davidson, R. C., and C. Chen. Kinetic description of intense nonneutral beam propagation through a periodic solenoidal focusing field based on the nonlinear Vlasov-Maxwell equations. Office of Scientific and Technical Information (OSTI), 1997. http://dx.doi.org/10.2172/304185.

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3

Saxena, Avadh. Solitary waves in nonlinear Dirac equation. From field theory to Dirac materials. Office of Scientific and Technical Information (OSTI), 2015. http://dx.doi.org/10.2172/1225286.

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