A New Generalized Heavenly Equation and the Large-Time Behavior of its Solutions

We have recently solved the inverse scattering problem for one-parameter families of vector fields, and used this result to construct the formal solution of the Cauchy problem for a class of integrable nonlinear partial differential equations connected with the commutation of multidimensional vector fields, such as the heavenly equation of Plebanski, the dispersionless Kadomtsev–Petviashvili (dKP) equation and the two-dimensional dispersionless Toda (2ddT) equation, as well as with the commutation of one-dimensional vector fields, such as the Pavlov equation. We also showed that the associated Riemann–Hilbert inverse problems are powerful tools to establish if the solutions of the Cauchy problem break at finite time, to construct their long-time behaviour and characterize classes of implicit solutions. In this paper, using the above theory, we concentrate on the heavenly and Pavlov equations, (i) establishing that their localized solutions evolve without breaking, unlike the cases of dKP and 2ddT; (ii) constructing the long-time behaviour of the solutions of their Cauchy problems; (iii) characterizing a distinguished class of implicit solutions of the heavenly equation.

In the present PhD thesis we study the initial-boundary value problem for the nonlinear evolution partial diefferential equation of Korteweg-De Vries (KDV) posed on a finite interval of the spatial variable. The method we employ is known as unified transform method. The application of the method on the IBVP under consideration consists of the so-called simultaneous spectral analysis of the Lax pair associated to the KDV equation. The first aim achieved in this contribution, is the expression of the solution of the IBVP as an integral representation in terms of the solution an appropriate Riemann-Hilbert (RH) problem in the complex plane of the spectral parameter, for a sufficiently large class of initial and boundary conditions. In particular, we provide two different integral representations for each one of two different RH problems. A second aim achieved is the invention of a procedure for the reduction of the singular RH problem to a regular one. A third aim achieved is the caracterization of the so-called generalized Dirichlet-to_Neumann map, that is, the expression of the unknown boundary functions in terms of the prescribed initial and boundary conditions. The Phd thesis is divided in 7 chapters. The first chapter is of an introductory character, while the remaining six chapters consist of the original contribution of the thesis. Analytically, the content of each chapter has as follows. The first chapter presents, among other things, the RH problem, the inverse scattering method for KDV, the dressing method for KDV and the method of simultaneous spectral analysis of the Lax pair. Chapter 2 presents the first step of the application of the method upon the IBVP, under the assumption thet KDV is solvable in the corresponding space-time region. The simultaneous spectral analysis of the Lax pair leads to the formulation of a singular homogenous RH factorization problem, which is defined in terms of six spectral functions. The last ones are expressed in terms of the initial and boundary values of the solution and of its transverse boundary derivatives up to order two. In chapter 3 we define the six spectral functions that correspond to the initial and boundary conditions and show that the inversion of these mappings can be described through appropriate RH problems. Also an appropriate “global relation” is satisfied, which characterizes the admissible initial and boundary functions. In chapter 4 we show that the asymptotic behavior of the solution of the RH problem leads actually to a solution of the IBVP. In chapter 5 we study the unique solvability of the RH problem. In chapter 6 we present an alternative RH formulation, replacing the poles by discontinuity curves. In chapter 7 we present the global relation to construct the generalized Dirichlet-to-Neumann map, that is, the expression of the unknown boundary functions (appearing in the RH formulation) in terms of the prescribed initial and boundary conditions.

The initial-boundary value problem (IBVP) for the Maxwell-Bloch equations with an arbitrary inhomogeneous broadening and periodic boundary condition is studied. This IBVP describes the propagation of an electromagnetic wave generated by periodic pumping in a resonant medium with distributed two-level atoms. We extended the inverse scattering transform method in the form of the matrix Riemann-Hilbert problem for solving the considered IBVP. Using the system of Ablowitz-Kaup-Newell-Segur equations equivalent to the system of the Maxwell-Bloch (MB) equations, we construct the associated matrix Riemann-Hilbert (RH) problem. Theorems on the existence, uniqueness and smoothness properties of a solution of the constructed RH problem are proved, and it is shown that a solution of the considered IBVP is uniquely defined by the solution of the associated RH problem. It is proved that the RH problem provides the causality principle. The representation of a solution of the MB equations in terms of a solution of the associated RH problem are given. The significance of this method also lies in the fact that, having studied the asymptotic behavior of the constructed RH problem and equivalent ones, we can obtain formulas for the asymptotics of a solution of the corresponding IBVP for the MB equations.

We have recently solved the inverse spectral problem for integrable partial differential equations (PDEs) in arbitrary dimensions arising as commutation of multidimensional vector fields depending on a spectral parameter λ. The associated inverse problem, in particular, can be formulated as a nonlinear Riemann–Hilbert (NRH) problem on a given contour of the complex λ plane. The most distinguished examples of integrable PDEs of this type, like the dispersionless Kadomtsev–Petviashivili (dKP), the heavenly and the two-dimensional dispersionless Toda equations, are real PDEs associated with Hamiltonian vector fields. The corresponding NRH data satisfy suitable reality and symplectic constraints. In this paper, generalizing the examples of solvable NRH problems illustrated in Manakov and Santini (2009 J. Phys. A: Math. Theor. 42 095203; 2008 J. Phys. A: Math. Theor. 41 055204; 2009 J. Phys. A: Math. Theor. 42 404013), we present a general procedure to construct solvable NRH problems for integrable real PDEs associated with Hamiltonian vector fields, allowing one to construct exact implicit solutions of such PDEs parametrized by an arbitrary number of real functions of a single variable. Then, we illustrate this theory on few distinguished examples for the dKP and heavenly equations. For the dKP case, we characterize a class of similarity solutions, of solutions constant on their parabolic wave front and breaking simultaneously on it, of localized solutions whose breaking point travels with constant speed along the wave front, and of localized solutions breaking in a point of the (x, y) plane. For the heavenly equation, we characterize two classes of symmetry reductions.

In this paper, we consider the current status of the Riemann–Hilbert problem and problems closely related to it. In the global theory of differential equations with regular singular points, the Riemann– Hilbert problem is one of the central topics. Two-dimensional models in contemporary theoretical physics use fundamental facts of the theory of Riemann surfaces. One of the central problems of this theory is the Riemann–Hilbert problem and questions related to it, for example, the linear conjugation problem. A natural language for the investigation of the Riemann–Hilbert problem is the language of holomorphic bundles with connections on Riemann surfaces. It makes clear the relationship with the linear conjugation problem, which is usually considered for Holder-class functions. We develop a different approach; namely, we consider the Riemann–Hilbert problem for the Carleman– Bers–Vekua system, replacing the Wiener–Hopf factorization of a matrix-valued function by the Φ-factorization. To study deformations of complex structures on Riemann surfaces, we also consider the Beltrami equation as a particular case of the Carleman–Bers–Vekua equation. We also discuss the algebraic formulation of the Riemann–Hilbert problem in the framework of the differential Galois theory, where this problem is known as the inverse problem and is interesting for us because of its constructive nature.

The inverse scattering transform for the vector defocusing nonlinear Schrödinger (NLS) equation with nonvanishing boundary values at infinity is constructed. The direct scattering problem is formulated on a two-sheeted covering of the complex plane. Two out of the six Jost eigenfunctions, however, do not admit an analytic extension on either sheet of the Riemann surface. Therefore, a suitable modification of both the direct and the inverse problem formulations is necessary. On the direct side, this is accomplished by constructing two additional analytic eigenfunctions which are expressed in terms of the adjoint eigenfunctions. The discrete spectrum, bound states and symmetries of the direct problem are then discussed. In the most general situation, a discrete eigenvalue corresponds to a quartet of zeros (poles) of certain scattering data. The inverse scattering problem is formulated in terms of a generalized Riemann-Hilbert (RH) problem in the upper/lower half planes of a suitable uniformization variable. Special soliton solutions are constructed from the poles in the RH problem, and include dark-dark soliton solutions, which have dark solitonic behavior in both components, as well as dark-bright soliton solutions, which have one dark and one bright component. The linear limit is obtained from the RH problem and is shown to correspond to the Fourier transform solution obtained from the linearized vector NLS system.

This paper aims at developing the Riemann–Hilbert problem approach to the modified Camassa–Holm (mCH) equation in the case when the solution is assumed to approach a non-zero constant at both infinities of the space variable. In this case, the spectral problem for the associated Lax pair equation has a continuous spectrum, which allows formulating the inverse spectral problem as a Riemann–Hilbert factorization problem with jump conditions across the real axis. We obtain a representation for the solution of the Cauchy problem for the mCH equation and also a description of certain soliton-type solutions, both regular and non-regular.

We construct a new method for approximating Hilbert transforms and their inverse throughout the complex plane. Both problems can be formulated as Riemann–Hilbert problems via Plemelj’s lemma. Using this framework, we rederive existing approaches for computing Hilbert transforms over the real line and unit interval, with the added benefit that we can compute the Hilbert transform in the complex plane. We then demonstrate the power of this approach by generalizing to the half line. Combining two half lines, we can compute the Hilbert transform of a more general class of functions on the real line than is possible with existing methods.

We start with a Riemann-Hilbert problem (RHP) related to a BD.I-type symmetric spaces $SO(2r+1)/S(O(2r-2s +1)\otimes O(2s))$, $s\geq 1$. We consider two Riemann-Hilbert problems: the first formulated on the real axis $\mathbb{R}$ in the complex $\lambda$-plane; the second one is formulated on $\mathbb{R} \oplus i\mathbb{R}$. The first RHP for $s=1$ allows one to solve the Kulish-Sklyanin (KS) model; the second RHP is relevant for a new type of KS model. An important example for nontrivial deep reductions of KS model is given. Its effect on the scattering matrix is formulated. In particular we obtain new 2-component NLS equations. Finally, using the Wronskian relations we demonstrate that the inverse scattering method for KS models may be understood as a generalized Fourier transforms. Thus we have a tool to derive all their fundamental properties, including the hierarchy of equations and the hierarchy of their Hamiltonian structures.

We have recently solved the inverse scattering problem for one-parameter families of vector fields, and used this result to construct the formal solution of the Cauchy problem for a class of integrable nonlinear partial differential equations in multidimensions, including the second heavenly equation of Plebanski and the dispersionless Kadomtsev–Petviashvili (dKP) equation. We showed, in particular, that the associated inverse problems can be expressed in terms of nonlinear Riemann–Hilbert problems on the real axis. In this paper, we make use of the nonlinear Riemann–Hilbert problem of dKP (i) to construct the longtime behaviour of the solutions of its Cauchy problem; (ii) to characterize a class of implicit solutions; (iii) to elucidate the spectral mechanism causing the gradient catastrophe of localized solutions of dKP, at finite time as well as in the longtime regime, and the corresponding universal behaviours near breaking.

In this paper, we develop the numerical inverse scattering transform (NIST) for solving the derivative nonlinear Schrödinger (DNLS) equation. The key technique involves formulating a Riemann–Hilbert problem that is associated with the initial value problem and solving it numerically. Before solving the Riemann–Hilbert problem (RHP), two essential operations need to be carried out. Firstly, high-precision numerical calculations are performed on the scattering data. Secondly, the RHP is deformed using the Deift–Zhou nonlinear steepest descent method. The DNLS equation has a continuous spectrum consisting of the real and imaginary axes and features three saddle points, which introduces complexity not encountered in previous NIST approaches. In our numerical inverse scattering method, we divide the (x, t)-plane into three regions and propose specific deformations for each region. These strategies not only help reduce computational costs but also minimise errors in the calculations. Unlike traditional numerical methods, the NIST does not rely on time-stepping to compute the solution. Instead, it directly solves the associated Riemann–Hilbert problem. This unique characteristic of the NIST eliminates convergence issues typically encountered in other numerical approaches and proves to be more effective, especially for long-time simulations.

Previous article Next article Determination of an Unknown Heat Source from Overspecified Boundary DataJ. R. CannonJ. R. Cannonhttps://doi.org/10.1137/0705024PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] J. R. Cannon, Determination of an unknown coefficient in a parabolic differential equation, Duke Math. J., 30 (1963), 313–323 10.1215/S0012-7094-63-03033-3 MR0157121 (28:358) 0117.06901 CrossrefISIGoogle Scholar[2] J. R. Cannon, Determination of certain parameters in heat conduction problems, J. Math. Anal. Appl., 8 (1964), 188–201 10.1016/0022-247X(64)90061-7 MR0160047 (28:3261) 0131.32104 CrossrefGoogle Scholar[3] J. R. Cannon, Determination of the unknown coefficient $k(u)$ in the equation $\nabla \cdot k(u)\nabla u=0$ from overspecified boundary data, J. Math. Anal. Appl., 18 (1967), 112–114 10.1016/0022-247X(67)90185-0 MR0209634 (35:531) 0151.15901 CrossrefISIGoogle Scholar[4] J. R. Cannon and , D. L. Filmer, The determination of unknown parameters in analytic systems of ordinary differential equations, SIAM J. Appl. Math., 15 (1967), 799–809 10.1137/0115069 MR0218632 (36:1716) 0251.34002 LinkISIGoogle Scholar[5] J. R. Cannon, , Jim Douglas, Jr. and , B. Frank Jones, Jr., Determination of the diffusivity of an isotropic medium, Internat. J. Engrg. Sci., 1 (1963), 453–455 10.1016/0020-7225(63)90002-8 MR0160045 (28:3259) CrossrefGoogle Scholar[6] J. R. Cannon and , B. Frank Jones, Jr., Determination of the diffusivity of an anisotropic medium, Internat. J. Engrg. Sci., 1 (1963), 457–460 10.1016/0020-7225(63)90003-X MR0160046 (28:3260) CrossrefGoogle Scholar[7] J. R. Cannon and , J. H. Halton, The irrotational solution of an elliptic differential equation with an unknown coefficient, Proc. Cambridge Philos. Soc., 59 (1963), 680–682 MR0149064 (26:6560) 0117.07101 CrossrefISIGoogle Scholar[8] Jim Douglas, Jr. and , B. Frank Jones, Jr., The determination of a coefficient in a parabolic differential equation. II. Numerical approximation, J. Math. Mech., 11 (1962), 919–926 MR0153988 (27:3949) 0112.32603 ISIGoogle Scholar[9] B. Frank Jones, Jr., The determination of a coefficient in a parabolic differential equation. I. Existence and uniqueness, J. Math. Mech., 11 (1962), 907–918 MR0153987 (27:3948) 0112.32602 ISIGoogle Scholar[10] B. Frank Jones, Jr., Various methods for finding unknown coefficients in parabolic differential equations, Comm. Pure Appl. Math., 16 (1963), 33–44 MR0152760 (27:2735) 0119.08302 CrossrefISIGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Identifying a space-dependent source term in distributed order time-fractional diffusion equationsMathematical Control and Related Fields, Vol. 0, No. 0 | 1 Jan 2022 Cross Ref Identification of stationary source in the anomalous diffusion equationInverse Problems in Science and Engineering, Vol. 29, No. 13 | 21 November 2021 Cross Ref A modified quasi-reversibility method for inverse source problem of Poisson equationInverse Problems in Science and Engineering, Vol. 29, No. 12 | 22 March 2021 Cross Ref Inverse modeling of contaminant transport for pollution source identification in surface and groundwaters: a reviewGroundwater for Sustainable Development, Vol. 15 | 1 Nov 2021 Cross Ref Convergence Analysis of a Crank–Nicolson Galerkin Method for an Inverse Source Problem for Parabolic Equations with Boundary ObservationsApplied 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Ref A self-adaptive LGSM to recover initial condition or heat source of one-dimensional heat conduction equation by using only minimal boundary thermal dataInternational Journal of Heat and Mass Transfer, Vol. 54, No. 7-8 | 1 Mar 2011 Cross Ref Appendix B: References to the Literature on the IHCPThe Mollification Method and the Numerical Solution of Ill-Posed Problems | 22 February 2011 Cross Ref Identification of moving pointwise sources in an advection–dispersion–reaction equationInverse Problems, Vol. 27, No. 2 | 18 January 2011 Cross Ref Application of the method of fundamental solutions and radial basis functions for inverse transient heat source problemComputer Physics Communications, Vol. 181, No. 12 | 1 Dec 2010 Cross Ref Hölder-Type Approximation for the Spatial Source Term of a Backward Heat EquationNumerical Functional Analysis and Optimization, Vol. 31, No. 12 | 19 Nov 2010 Cross Ref Source term identification for an axisymmetric inverse heat conduction problemComputers & Mathematics with Applications, Vol. 59, No. 1 | 1 Jan 2010 Cross Ref A method for identifying a spacewise-dependent heat source under stochastic noise interferenceInverse Problems in Science and Engineering, Vol. 18, No. 1 | 4 November 2009 Cross Ref Identification of a time-varying point source in a system of two coupled linear diffusion-advection- reaction equations: application to surface water pollutionInverse Problems, Vol. 25, No. 11 | 29 October 2009 Cross Ref The method of fundamental solutions for the inverse space-dependent heat source problemEngineering Analysis with Boundary Elements, Vol. 33, No. 10 | 1 Oct 2009 Cross Ref Optimization method for the inverse problem of reconstructing the source term in a parabolic equationMathematics and Computers in Simulation, Vol. 80, No. 2 | 1 Oct 2009 Cross Ref Two regularization strategies for an evolutional type inverse heat source problemJournal of Physics A: Mathematical and Theoretical, Vol. 42, No. 36 | 19 August 2009 Cross Ref 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Stationary caseInverse Problems in Science and Engineering, Vol. 15, No. 8 | 18 December 2007 Cross Ref The boundary-element method for the determination of a heat source dependent on one variableJournal of Engineering Mathematics, Vol. 54, No. 4 | 3 January 2006 Cross Ref Inverse source problem for a transmission problem for a parabolic equationJournal of Inverse and Ill-posed Problems, Vol. 14, No. 1 | 1 Jan 2006 Cross Ref Identification of a point source in a linear advection–dispersion–reaction equation: application to a pollution source problemInverse Problems, Vol. 21, No. 3 | 9 May 2005 Cross Ref Conditional stability in determining a heat sourceJournal of Inverse and Ill-posed Problems, Vol. 12, No. 3 | 1 Jun 2004 Cross Ref Lipschitz stability in inverse parabolic problems by the Carleman estimateInverse Problems, Vol. 14, No. 5 | 1 January 1999 Cross Ref Determination of the unknown sources in the heat-conduction equationComputational Mathematics and Modeling, Vol. 8, No. 4 | 1 Oct 1997 Cross Ref An estimation method for point sources of multidimensional diffusion equationApplied Mathematical Modelling, Vol. 21, No. 2 | 1 Feb 1997 Cross Ref A reliable estimation method for locations of point sources for an n-dimensional Poisson equationApplied Mathematical Modelling, Vol. 20, No. 11 | 1 Nov 1996 Cross Ref Uniqueness and stable determination of forcing terms in linear partial differential equations with overspecified boundary dataInverse Problems, Vol. 10, No. 6 | 1 January 1999 Cross Ref Conditional Stability in Determination of Densities of Heat Sources in a Bounded DomainControl and Estimation of Distributed Parameter Systems: Nonlinear Phenomena | 1 Jan 1994 Cross Ref Conditional stability in determination of force terms of heat equations in a rectangleMathematical and Computer Modelling, Vol. 18, No. 1 | 1 Jul 1993 Cross Ref Inverse problems and Carleman estimatesInverse Problems, Vol. 8, No. 4 | 1 January 1999 Cross Ref Determination of an unknown function in a parabolic equation with an overspecified conditionMathematical Methods in the Applied Sciences, Vol. 13, No. 5 | 1 Nov 1990 Cross Ref Some stability estimates for a heat source in terms of overspecified data in the 3-D heat equationJournal of Mathematical Analysis and Applications, Vol. 147, No. 2 | 1 Apr 1990 Cross Ref Parameter identification in hyperbolic and parabolic partial differential equations of cylindrical geometry from overspecified boundary dataInternational Journal of Engineering Science, Vol. 28, No. 10 | 1 Jan 1990 Cross Ref Identification of certain physical parameters in hyperbolic boundary value problems from overspecified boundary dataInternational Journal of Engineering Science, Vol. 26, No. 8 | 1 Jan 1988 Cross Ref An Inverse Problem for a Nonlinear Elliptic Differential EquationMichael Pilant and William RundellSIAM Journal on Mathematical Analysis, Vol. 18, No. 6 | 1 August 2006AbstractPDF (793 KB)An inverse problem for the heat equationInverse Problems, Vol. 2, No. 4 | 1 January 1999 Cross Ref A Note on an Inverse Problem Related to the 3-D Heat EquationInverse Problems | 1 Jan 1986 Cross Ref Determination of a Source Term in a Linear Parabolic Differential Equation with Mixed Boundary ConditionsInverse Problems | 1 Jan 1986 Cross Ref Nonparametric algorithm for input signals identification in static distributed-parameter systemsIEEE Transactions on Automatic Control, Vol. 29, No. 7 | 1 Jul 1984 Cross Ref Parameter determination in parabolic partial differential equations from overspecified boundary dataInternational Journal of Engineering Science, Vol. 20, No. 6 | 1 Jan 1982 Cross Ref Determination of an unknown non-homogeneous term in a linear partial differential equation .from overspecified boundary dataApplicable Analysis, Vol. 10, No. 3 | 2 May 2007 Cross Ref Distributed parameter system indentification A survey†International Journal of Control, Vol. 26, No. 4 | 16 May 2007 Cross Ref Determination of a source term in a linear parabolic partial differential equationZeitschrift für angewandte Mathematik und Physik ZAMP, Vol. 27, No. 3 | 1 May 1976 Cross Ref Some general remarks on improperly posed problems for partial differential equationsSymposium on Non-Well-Posed Problems and Logarithmic Convexity | 22 August 2006 Cross Ref Estimation of parameters in partial differential equations from noisy experimental dataChemical Engineering Science, Vol. 26, No. 6 | 1 Jun 1971 Cross Ref Determination of an unknown forcing function in a hyperbolic equation from overspecified dataAnnali di Matematica Pura ed Applicata, Vol. 85, No. 1 | 1 Dec 1970 Cross Ref The character of non-uniqueness in the conductivity modelling problem for the earthPure and Applied Geophysics PAGEOPH, Vol. 80, No. 1 | 1 Jan 1970 Cross Ref Volume 5, Issue 2| 1968SIAM Journal on Numerical Analysis199-459 History Submitted:29 September 1967Published online:03 August 2006 InformationCopyright © 1968 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0705024Article page range:pp. 275-286ISSN (print):0036-1429ISSN (online):1095-7170Publisher:Society for Industrial and Applied Mathematics

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The inverse dynamic problem for the wave equation with a potential on a real line is considered. The forward initial-boundary value problem is set up with the help of boundary triplets. As an inverse data, an analog of the response operator (dynamic Dirichlet-to-Neumann map) is used. Equations of the inverse problem are derived; also, a relationship between the dynamic inverse problem and the spectral inverse problem from a matrix-valued measure is pointed out.

ABSTRACTThis paper presents a Riemann–Hilbert (RH) problem formalism for the initial value problem of the Sawada–Kotera equation defined on the real line. Assuming the existence of a solution, we establish that this solution can be effectively represented by solving a matrix RH problem. Notably, the formulation of this RH problem involves four spectral functions: , , , and , which are obtained via a nonlinear Fourier transform applied to the initial data. Furthermore, this study conducts a detailed spectral analysis, providing a foundation for the application of the nonlinear steepest descent method to determine the long‐time asymptotic behavior of solutions to the Sawada–Kotera equation on the real line.