Relevant bibliographies by topics / Inverse Scattering Transform, Nonlinear, Integro-Differential

Academic literature on the topic 'Inverse Scattering Transform, Nonlinear, Integro-Differential'

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Published: 4 June 2021
Last updated: 10 February 2022

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Journal articles on the topic "Inverse Scattering Transform, Nonlinear, Integro-Differential"

1

TAHA, THIAB R. "NUMERICAL SIMULATION OF THE KdV-MKdV EQUATION." International Journal of Modern Physics C 05, no. 02 (April 1994): 407–10. http://dx.doi.org/10.1142/s0129183194000593.

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In this paper two numerical schemes for the numerical simulation of the nonlinear partial differential equation ut+6αuux+6βu2ux+uxxx=0 are implemented by the method of lines (MOL). The first scheme is based on the inverse scattering transform (IST), and the second scheme is a combination of the IST schemes for the Korteweg-de Vries (KdV) and modified KdV (MKdV) equations. The only difference between the two schemes is in the discretization of the nonlinear terms. Numerical experiments have shown that the first scheme is significantly more accurate than the second one. This demonstrates the importance of a proper discretization of nonlinear terms when a numerical method is designed for solving a nonlinear differential equation.
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Osborne, A. R. "Approximate asymptotic integration of a higher order water-wave equation using the inverse scattering transform." Nonlinear Processes in Geophysics 4, no. 1 (March 31, 1997): 29–53. http://dx.doi.org/10.5194/npg-4-29-1997.

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Abstract. The complete mathematical and physical characterization of nonlinear water wave dynamics has been an important goal since the fundamental partial differential equations were discovered by Euler over 200 years ago. Here I study a subset of the full solutions by considering the irrotational, unidirectional multiscale expansion of these equations in shallow-water. I seek to integrate the first higher-order wave equation, beyond the order of the Korteweg- deVries equation, using the inverse scattering transform. While I am unable to integrate this equation directly, I am instead able to integrate an analogous equation in a closely related hierarchy. This new integrable wave equation is tested for physical validity by comparing its linear dispersion relation and solitary wave solution with those of the full water wave equations and with laboratory data. The comparison is remarkably close and thus supports the physical applicability of the new equation. These results are surprising because the inverse scattering transform, long thought to be useful for solving only very special, low-order nonlinear wave equations, can now be thought of as a useful tool for approximately integrating a wide variety of physical systems to higher order. I give a simple scenario for adapting these results to the nonlinear Fourier analysis of experimentally measured wave trains.
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Vitanov, Nikolay K., Zlatinka I. Dimitrova, and Kaloyan N. Vitanov. "Simple Equations Method (SEsM): Algorithm, Connection with Hirota Method, Inverse Scattering Transform Method, and Several Other Methods." Entropy 23, no. 1 (December 23, 2020): 10. http://dx.doi.org/10.3390/e23010010.

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The goal of this article is to discuss the Simple Equations Method (SEsM) for obtaining exact solutions of nonlinear partial differential equations and to show that several well-known methods for obtaining exact solutions of such equations are connected to SEsM. In more detail, we show that the Hirota method is connected to a particular case of SEsM for a specific form of the function from Step 2 of SEsM and for simple equations of the kinds of differential equations for exponential functions. We illustrate this particular case of SEsM by obtaining the three- soliton solution of the Korteweg-de Vries equation, two-soliton solution of the nonlinear Schrödinger equation, and the soliton solution of the Ishimori equation for the spin dynamics of ferromagnetic materials. Then we show that a particular case of SEsM can be used in order to reproduce the methodology of the inverse scattering transform method for the case of the Burgers equation and Korteweg-de Vries equation. This particular case is connected to use of a specific case of Step 2 of SEsM. This step is connected to: (i) representation of the solution of the solved nonlinear partial differential equation as expansion as power series containing powers of a “small” parameter ϵ; (ii) solving the differential equations arising from this representation by means of Fourier series, and (iii) transition from the obtained solution for small values of ϵ to solution for arbitrary finite values of ϵ. Finally, we show that the much-used homogeneous balance method, extended homogeneous balance method, auxiliary equation method, Jacobi elliptic function expansion method, F-expansion method, modified simple equation method, trial function method and first integral method are connected to particular cases of SEsM.
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Xu, Bo, and Sheng Zhang. "Integrability, exact solutions and nonlinear dynamics of a nonisospectral integral-differential system." Open Physics 17, no. 1 (June 17, 2019): 299–306. http://dx.doi.org/10.1515/phys-2019-0031.

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Abstract The investigations of integrability, exact solutions and dynamics of nonlinear partial differential equations (PDEs) are vital issues in nonlinear mathematical physics. In this paper, we derive and solve a new Lax integrable nonisospectral integral-differential system. To be specific, we first generalize an eigenvalue problem and its adjoint equation by equipping it with a new time-varying spectral parameter. Based on the generalized eigenvalue problem and the adjoint equation, we then derive a new Lax integrable nonisospectral integral-differential system. Furthermore, we obtain exact solutions and their reduced forms of the derived system by extending the famous non-linear Fourier analysis method–inverse scattering transform (IST). Finally, with graphical assistance we simulate a pair of reduced solutions, the dynamical evolutions of which show that the amplitudes of solutions vary with time.
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Matviichuk, Ruslan. "Dynamics and exact solutions of the generalized Harry Dym equation." Proceedings of the International Geometry Center 12, no. 4 (April 1, 2020): 50–59. http://dx.doi.org/10.15673/tmgc.v12i4.1682.

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The Harry Dym equation is the third-order evolutionary partial differential equation. It describes a system in which dispersion and nonlinearity are coupled together. It is a completely integrable nonlinear evolution equation that may be solved by means of the inverse scattering transform. It has an infinite number of conservation laws and does not have the Painleve property. The Harry Dym equation has strong links to the Korteweg – de Vries equation and it also has many properties of soliton solutions. A connection was established between this equation and the hierarchies of the Kadomtsev – Petviashvili equation. The Harry Dym equation has applications in acoustics: with its help, finite-gap densities of the acoustic operator are constructed. The paper considers a generalization of the Harry Dym equation, for the study of which the methods of the theory of finite-dimensional dynamics are applied. The theory of finite-dimensional dynamics is a natural development of the theory of dynamical systems. Dynamics make it possible to find families that depends on a finite number of parameters among all solutions of evolutionary differential equations. In our case, this approach allows us to obtain some classes of exact solutions of the generalized equation, and also indicates a method for numerically constructing solutions.
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Myrzakulova, Zh R., K. R. Yesmakhanova, and Zh S. Zhubayeva. "EQUIVALENCE OF THE HUNTER-SAXON EQUATION AND THE GENERALIZED HEISENBERG FERROMAGNET EQUATION." PHYSICO-MATHEMATICAL SERIES 2, no. 336 (April 15, 2021): 33–38. http://dx.doi.org/10.32014/2021.2518-1726.18.

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Integrable systems play an important role in modern mathematics, theoretical and mathematical physics. The display of integrable equations with exact solutions and some special solutions can provide important guarantees for the analysis of its various properties. The Hunter-Saxton equation belongs to the family of integrable systems. The extensive and interesting mathematical theory, underlying the Hunter-Saxton equation, creates active mathematical and physical research. The Hunter-Saxton equation (HSE) is a high-frequency limit of the famous Camassa-Holm equation. The physical interpretation of HSE is the propagation of weakly nonlinear orientation waves in a massive nematic liquid crystal director field. In this paper, we propose a matrix form of the Lax representation for HSE in 𝑠𝑢ሺ𝑛 ൅ 1ሻ/𝑠ሺ𝑢ሺ1ሻ ⊕ 𝑢ሺ𝑛ሻሻ - symmetric space for the case 𝑛 ൌ 2. Lax pairs, introduced in 1968 by Peter Lax, are a tool for finding conserved quantities of integrable evolutionary differential equations. The Lax representation expands the possibilities of the equation we are considering. For example, in this paper, we will use the matrix Lax representation for the HSE to establish the gauge equivalence of this equation with the generalized Heisenberg ferromagnet equation (GHFE). The famous Heisenberg Ferromagnet Equation (HFE) is one of the classical equations integrable through the inverse scattering transform. In this paper, we will consider its generalization. Andalso the connection between the decisions of the HSE and the GHFE will be presented.
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Иванов, К. А., А. Р. Губайдуллин, and М. А. Калитеевский. "Квантование электромагнитного поля в трехмерных фотонных структурах на основе формализма матрицы рассеяния (S-квантование)." Физика и техника полупроводников 52, no. 9 (2018): 1023. http://dx.doi.org/10.21883/ftp.2018.09.46150.8796.

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AbstractA technique for quantization of the electromagnetic field in photonic nanostructures with three-dimensional modulation of the dielectric constant is developed on the basis of the scattering matrix formalism ( S quantization in the three-dimensional case). Quantization is based on equating the eigenvalues of the scattering matrix to unity, which is equivalent to equating each other the sets of Fourier expansions for the fields of the waves incident on the structure and propagating away from the structure. The spatial distribution of electromagnetic fields of the modes in a photonic nanostructure is calculated on the basis of the R and T matrices describing the reflection and transmission of the Fourier components by the structure. To calculate the reflection and transmission coefficients of individual real-space and Fourier-space components, the structure is divided into parallel layers within which the dielectric constant varies as a function of two-dimensional coordinates. Using the Fourier transform, Maxwell’s equations are written in the form of a matrix connecting the Fourier components of the electric field at the boundaries of neighboring layers. Based on the calculated reflection and transmission vectors for all polarizations and Fourier components, the scattering matrix for the entire structure is formed and quantization is carried out by equating the eigenvalues of the scattering matrix to unity. The developed method makes it possible to obtain the spatial profiles of eigenmodes without solving a system of nonlinear integro-differential equations and significantly reduces the computational resources required for calculating the probability of spontaneous emission in three-dimensional systems.
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Deconinck, B., A. S. Fokas, and J. Lenells. "The implementation of the unified transform to the nonlinear Schrödinger equation with periodic initial conditions." Letters in Mathematical Physics 111, no. 1 (February 2021). http://dx.doi.org/10.1007/s11005-021-01356-7.

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AbstractThe unified transform method (UTM) provides a novel approach to the analysis of initial boundary value problems for linear as well as for a particular class of nonlinear partial differential equations called integrable. If the latter equations are formulated in two dimensions (either one space and one time, or two space dimensions), the UTM expresses the solution in terms of a matrix Riemann–Hilbert (RH) problem with explicit dependence on the independent variables. For nonlinear integrable evolution equations, such as the celebrated nonlinear Schrödinger (NLS) equation, the associated jump matrices are computed in terms of the initial conditions and all boundary values. The unknown boundary values are characterized in terms of the initial datum and the given boundary conditions via the analysis of the so-called global relation. In general, this analysis involves the solution of certain nonlinear equations. In certain cases, called linearizable, it is possible to bypass this nonlinear step. In these cases, the UTM solves the given initial boundary value problem with the same level of efficiency as the well-known inverse scattering transform solves the initial value problem on the infinite line. We show here that the initial boundary value problem on a finite interval with x-periodic boundary conditions (which can alternatively be viewed as the initial value problem on a circle) belongs to the linearizable class. Indeed, by employing certain transformations of the associated RH problem and by using the global relation, the relevant jump matrices can be expressed explicitly in terms of the so-called scattering data, which are computed in terms of the initial datum. Details are given for NLS, but similar considerations are valid for other well-known integrable evolution equations, including the Korteweg–de Vries (KdV) and modified KdV equations.
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Ahmed, Shazad Sh, Shokhan Ahmed Hama Salih, and Mariwan R. Ahmed. "Laplace Adomian and Laplace Modified Adomian Decomposition Methods for Solving Nonlinear Integro-Fractional Differential Equations of the Volterra-Hammerstein Type." Iraqi Journal of Science, October 28, 2019, 2207–22. http://dx.doi.org/10.24996/ijs.2019.60.10.15.

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In this work, we will combine the Laplace transform method with the Adomian decomposition method and modified Adomian decomposition method for semi-analytic treatments of the nonlinear integro-fractional differential equations of the Volterra-Hammerstein type with difference kernel and such a problem which the kernel has a first order simple degenerate kind which the higher-multi fractional derivative is described in the Caputo sense. In these methods, the solution of a functional equation is considered as the sum of infinite series of components after applying the inverse of Laplace transformation usually converging to the solution, where a closed form solution is not obtainable, a truncated number of terms is usually used for numerical purposes. Finally, examples are prepared to illustrate these considerations.
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Dissertations / Theses on the topic "Inverse Scattering Transform, Nonlinear, Integro-Differential"

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Scoufis, George. "An Application of the Inverse Scattering Transform to some Nonlinear Singular Integro-Differential Equations." University of Sydney, Mathematics and Statistics, 1999. http://hdl.handle.net/2123/412.

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ABSTRACT The quest to model wave propagation in various physical systems has produced a large set of diverse nonlinear equations. Nonlinear singular integro-differential equations rank amongst the intricate nonlinear wave equations available to study the classical problem of wave propagation in physical systems. Integro-differential equations are characterized by the simultaneous presence of integration and differentiation in a single equation. Substantial interest exists in nonlinear wave equations that are amenable to the Inverse Scattering Transform (IST). The IST is an adroit mathematical technique that delivers analytical solutions of a certain type of nonlinear equation: soliton equation. Initial value problems of numerous physically significant nonlinear equations have now been solved through elegant and novel implementations of the IST. The prototype nonlinear singular integro-differential equation receptive to the IST is the Intermediate Long Wave (ILW) equation, which models one-dimensional weakly nonlinear internal wave propagation in a density stratified fluid of finite total depth. In the deep water limit the ILW equation bifurcates into a physically significant nonlinear singular integro-differential equation known as the 'Benjamin-Ono' (BO) equation; the shallow water limit of the ILW equation is the famous Korteweg-de Vries (KdV) equation. Both the KdV and BO equations have been solved by dissimilar implementations of the IST. The Modified Korteweg-de Vries (MKdV) equation is a nonlinear partial differential equation, which was significant in the historical development of the IST. Solutions of the MKdV equation are mapped by an explicit nonlinear transformation known as the 'Miura transformation' into solutions of the KdV equation. Historically, the Miura transformation manifested the intimate connection between solutions of the KdV equation and the inverse problem for the one-dimensional time independent Schroedinger equation. In light of the MKdV equation's significance, it is natural to seek 'modified' versions of the ILW and BO equations. Solutions of each modified nonlinear singular integro-differential equation should be mapped by an analogue of the original Miura transformation into solutions of the 'unmodified' equation. In parallel with the limiting cases of the ILW equation, the modified version of the ILW equation should reduce to the MKdV equation in the shallow water limit and to the modified version of the BO equation in the deep water limit. The Modified Intermediate Long Wave (MILW) and Modified Benjamin-Ono (MBO) equations are the two nonlinear singular integro-differential equations that display all the required attributes. Several researchers have shown that the MILW and MBO equations exhibit the signature characteristic of soliton equations. Despite the significance of the MILW and MBO equations to soliton theory, and the possible physical applications of the MILW and MBO equations, the initial value problems for these equations have not been solved. In this thesis we use the IST to solve the initial value problems for the MILW and MBO equations on the real-line. The only restrictions that we place on the initial values for the MILW and MBO equations are that they be real-valued, sufficiently smooth and decay to zero as the absolute value of the spatial variable approaches large values.
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李達明 and Tad-ming Lee. "Isospectral transformations between soliton-solutions of the Korteweg-de Vries equation." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 1994. http://hub.hku.hk/bib/B29866261.

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Lee, Tad-ming. "Isospectral transformations between soliton-solutions of the Korteweg-de Vries equation /." [Hong Kong : University of Hong Kong], 1994. http://sunzi.lib.hku.hk/hkuto/record.jsp?B1359753X.

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Scoufis, George. "An application of the inverse scattering transform to some nonlnear singular integro-differential equations." Connect to full text, 1999. http://hdl.handle.net/2123/412.

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Thesis (Ph. D.)--University of Sydney, 1999.
Title from title screen (viewed Apr. 21, 2008). Submitted in fulfilment of the requirements for the degree of Doctor of Philosophy to the School of Mathematics and Statistics, Faculty of Science. Includes bibliography. Also available in print form.
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Books on the topic "Inverse Scattering Transform, Nonlinear, Integro-Differential"

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Ablowitz, Mark J. Solitons, nonlinear evolution equations and inverse scattering. Cambridge: Cambridge University Press, 1991.

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Centre national de la recherche scientifique (France). Recherche coopérative sur programme 264. Some topics on inverse problems: Proceedings of the XVIth Workshop on Interdisciplinary Study of Inverse Problems, Montpellier, France, Nov. 30-Dec. 4, 1987. Edited by Sabatier Pierre Célestin 1935-. Singapore: World Scientific, 1988.

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1935-, Sabatier Pierre Célestin, ed. Some topics on inverse problems: Proceedings of the XVIth Workshop on Interdisciplinary Study of Inverse Problems : Nov. 30-Dec. 4 1987. Singapore: World Scientific, 1988.

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Asano, N. Algebraic and spectral methods for nonlinear wave equations. Harlow, Essex, England: Longman Scientific & Technical, 1990.

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1935-, Sabatier Pierre Celestin, and Centre national de la recherche scientifique (France), eds. Inverse problems: An interdisciplinary study. London: Academic Press, 1987.

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Belokolos, E. D., A. I. Bobenko, V. Z. Enol'Skii, A. R. Its, and V. B. Mateveev. Algebro-Geometrical Approach to Nonlinear Evolution Equations (Springer Series in Nonlinear Dynamics). Springer-Verlag Berlin and Heidelberg GmbH & Co. K, 1994.

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Algebro-geometric approach to nonlinear integrable equations. Berlin: Springer-Verlag, 1994.

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Asano, N., and Y. Kato. Algebraic and Spectral Methods for Non-Linear Wave Equations (Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 49). Longman Sc & Tech, 1991.

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Book chapters on the topic "Inverse Scattering Transform, Nonlinear, Integro-Differential"

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Debnath, Lokenath. "Solitons and the Inverse Scattering Transform." In Nonlinear Partial Differential Equations for Scientists and Engineers, 331–404. Boston, MA: Birkhäuser Boston, 1997. http://dx.doi.org/10.1007/978-1-4899-2846-7_9.

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Debnath, Lokenath. "Solitons and the Inverse Scattering Transform." In Nonlinear Partial Differential Equations for Scientists and Engineers, 425–533. Boston: Birkhäuser Boston, 2012. http://dx.doi.org/10.1007/978-0-8176-8265-1_9.

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"Solitons and the Inverse Scattering Transform." In Nonlinear Partial Differential Equations for Scientists and Engineers, 417–514. Boston, MA: Birkhäuser Boston, 2005. http://dx.doi.org/10.1007/0-8176-4418-0_9.

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"Inverse Scattering for Integro-Differential Equations." In Solitons, Nonlinear Evolution Equations and Inverse Scattering, 163–94. Cambridge University Press, 1991. http://dx.doi.org/10.1017/cbo9780511623998.005.

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Conference papers on the topic "Inverse Scattering Transform, Nonlinear, Integro-Differential"

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Osborne, Alfred R. "Theory of Nonlinear Fourier Analysis: The Construction of Quasiperiodic Fourier Series for Nonlinear Wave Motion." In ASME 2020 39th International Conference on Ocean, Offshore and Arctic Engineering. American Society of Mechanical Engineers, 2020. http://dx.doi.org/10.1115/omae2020-18850.

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Abstract I give a description of nonlinear water wave dynamics using a recently discovered tool of mathematical physics I call nonlinear Fourier analysis (NLFA). This method is based upon and is an application of a theorem due to Baker [1897, 1907] and Mumford [1984] in the field of algebraic geometry and from additional sources by the author [Osborne, 2010, 2018, 2019]. The theory begins with the Kadomtsev-Petviashvili (KP) equation, a two dimensional generalization of the Korteweg-deVries (KdV) equation: Here the NLFA method is derived from the complete integrability of the equation by finite gap theory or the inverse scattering transform for periodic/quasiperiodic boundary conditions. I first show, for a one-dimensional, plane wave solution, that the KP equation can be rotated to a solution of the KdV equation, where the coefficients of KdV are now functions of the rotation angle. I then show how the rotated KdV equation can be used to compute the spectral solutions of the KP equation itself. Finally, I write the spectral solutions of the KP equation as a finite gap solution in terms of Riemann theta functions. By virtue of the fact that I am able to write a theta function formulation of the KP equation, it is clear that the wave dynamics lie on tori and constitute parallel dynamics on the tori in the integrable cases and non-parallel dynamics on the tori for certain perturbed quasi-integrable cases. Therefore, we are dealing with a Kolmogorov-Arnold-Moser KAM theory for nonlinear partial differential wave equations. The nonlinear Fourier series have particular nonlinear Fourier modes, including: sine waves, Stokes waves and solitons. Indeed the theoretical formulation I have developed is a kind of exact two-dimensional “coherent wave turbulence” or “integrable wave turbulence” for the KP equation, for which the Stokes waves and solitons are the coherent structures. I discuss how NLFA provides a number of new tools that apply to a wide range of problems in offshore engineering and coastal dynamics: This includes nonlinear Fourier space and time series analysis, nonlinear Fourier wave field analysis, a nonlinear random phase approximation, the study of nonlinear coherent functions and nonlinear bi and tri spectral analysis.
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