Bilinear transformation method
From MaRDI portal
Publication:801464
zbMath0552.35001MaRDI QIDQ801464
Publication date: 1984
Published in: Mathematics in Science and Engineering (Search for Journal in Brave)
Cole-Hopf transformationKdV equationBäcklund transformationsBenjamin-Ono equationnonlinear evolution equationsinverse scatteringN-soliton solutionbilinear operator methodJoseph's finite depth equation
Partial differential equations of mathematical physics and other areas of application (35Q99) Geometric theory, characteristics, transformations in context of PDEs (35A30) Research exposition (monographs, survey articles) pertaining to partial differential equations (35-02)
Related Items
Algebro-geometric solutions of the generalized Burgers hierarchy associated with a \(3\times 3\) matrix spectral problem based on Riemann surface, Lump-soliton solutions to the Fokas system, Conservation laws and rational solutions of a coupled Toda equation and its related equation, Single and multiple-soliton solutions for the \((2+1)\)-dimensional KdV equation, Solitons and singular solitons for the Gardner-KP equation, Invariant measure and long time behavior of regular solutions of the Benjamin-Ono equation, Symbolic software for soliton theory, A class of doubly periodic waves for nonlinear evolution equations, Application of Hirota's bilinear formalism to a two-dimensional lattice by Leznov, Nonlinearization of the Lax pairs for discrete Ablowitz-Ladik hierarchy., Nonlinear physics: integrability, chaos and beyond, Doubly periodic patterns of modulated hydrodynamic waves: exact solutions of the Davey-Stewartson system, Bilinear form and two Bäcklund transformations for the \((3+1)\)-dimensional Jimbo-Miwa equation, Cusped solitons of the Camassa-Holm equation. II. Binary cuspon-soliton interactions, Complete integrability of the Benjamin-Ono equation on the multi-soliton manifolds, The third order Benjamin-Ono equation on the torus: well-posedness, traveling waves and stability, Soliton solution of new \((2+1)\)-dimensional nonlinear partial differential equations, Generation of nonlinear modulated waves in a modified Noguchi electrical transmission network, On the hierarchies of higher order mKdV and KdV equations, Nonlinear stability of Gardner breathers, Breathers and `black' rogue waves of coupled nonlinear Schrödinger equations with dispersion and nonlinearity of opposite signs, A numerical study of the 3-periodic wave solutions to KdV-type equations, Periodic and localized wave patterns for coupled Ablowitz-Ladik systems with negative cross phase modulation, Families of semi-rational solutions to the Kadomtsev-Petviashvili I equation, Transformations and multi-solitonic solutions for a generalized variable-coefficient Kadomtsev-Petviashvili equation, General high-order breathers and rogue waves in the \((3 + 1)\)-dimensional KP-Boussinesq equation, Nonlocal models for nonlinear, dispersive waves, The Painlevé tests, Bäcklund transformation and bilinear form for the KdV equation with a self-consistent source, Steady-states and traveling-wave solutions of the generalized Constantin-Lax-Majda equation, A bilinear Bäcklund transformation of a \((3+1)\)-dimensional generalized KP equation, Dynamics of interaction between lumps and solitons in the Mel'nikov equation, Exact periodic solutions of the nonintegrable Kawahara equation, Lax pairs, symmetries and conservation laws of a differential-difference equation -- Sato's approach, Soliton solutions to the Jimbo-Miwa equations and the Fordy-Gibbons-Jimbo-Miwa equation, Soliton solutions and symmetries of the \(2+1\) dimensional Kaup-Kupershmidt equation, \(P T\)-symmetric nonlocal Davey-Stewartson I equation: soliton solutions with nonzero background, The Hirota's bilinear method and the tanh-coth method for multiple-soliton solutions of the Sawada-Kotera-Kadomtsev-Petviashvili equation, Optimized direct Padé and HPM for solving equation of oxygen diffusion in a spherical cell, On soliton solutions of the Kaup-Kupershmidt equation. I: Direct bilinearisation and solitary wave, On soliton solutions of the Kaup-Kupershmidt equation. II: `Anomalous' \(N\)-soliton solutions, A generalized Leznov lattice: Bilinear form, Bäcklund transformation, and Lax pair, Decomposition of a \(2 + 1\)-dimensional Volterra type lattice and its quasi-periodic solutions, Interactions of breathers and solitons in the extended Korteweg-de Vries equation, Solitary wave solutions of the generalized shallow water wave (GSWW) equation by Hirota's method, tanh-coth method and exp-function method, A system of coupled partial differential equations exhibiting both elevation and depression rogue wave modes, Multiple-soliton solutions for the generalized \((1+1)\)-dimensional and the generalized \((2+1)\)-dimensional Itô equations, Invariant measures and long time behaviour for the Benjamin-Ono equation. III, Multiple-soliton solutions for the Lax-Kadomtsev-Petviashvili (Lax-KP) equation, The Hirota's direct method for multiple-soliton solutions for three model equations of shallow water waves, Multiple-soliton solutions of two extended model equations for shallow water waves, Doubly localized two-dimensional rogue waves in the Davey-Stewartson I equation, Linear stability analysis for the multikink solution of the sine-Hilbert equation, Hetero-Bäcklund transformation, bilinear forms and \(N\) solitons for a generalized three-coupled Korteweg-de Vries system, Multiple solution profiles to the higher-dimensional Kadomtsev-Petviashvilli equations via Wronskian determinant, Homoclinic and heteroclinic flows in global attractor for Davey-Stewartson II equation, Rational function representations of wave patterns in higher-dimensional and discrete evolution equations, Local well-posedness for fourth order Benjamin-Ono type equations, Magneto-optical/ferromagnetic-material computation: Bäcklund transformations, bilinear forms and \(N\) solitons for a generalized \((3+1)\)-dimensional variable-coefficient modified Kadomtsev-Petviashvili system, Periodic solutions of a derivative nonlinear Schrödinger equation: Elliptic integrals of the third kind, \(N\)-soliton solutions for shallow water waves equations in \((1+1)\) and \((2+1)\) dimensions, On properties of new parameterized 5th-order nonlinear evolution equation, A new representation of \(N\)-soliton solution and limiting solutions for the fifth order KdV equation, Quasi-periodic wave solution and asymptotic behavior for the \((2+1)\)-dimensional Toda lattice equation, Asymptotic behavior of the periodic wave solution for the \((3+1)\)-dimensional Kadomtsev-Petviashvili equation, Families of nonsingular soliton solutions of a nonlocal Schrödinger-Boussinesq equation, Bright and dark soliton solutions to the partial reverse space-time nonlocal Mel'nikov equation, Homoclinic breather-wave solutions for sine-Gordon equation, New \((2 + 1)\)-dimensional sine-Gordon equation with self-consistent sources derived by the nonlinear variable separation method, IST Versus PDE: A Comparative Study, On integrability of a (2+1)-dimensional MKdV-type equation, Multiple-soliton solutions of the perturbed KdV equation, SOLITONS AND LOCALIZED EXCITATIONS FOR THE (2+1)-DIMENSIONAL DISPERSIVE LONG WAVE SYSTEM VIA SYMBOLIC COMPUTATION, Two integrable differential-difference equations exhibiting soliton solutions of the Kaup-Kupershmidt equation type, On the zero-dispersion limit of the benjamin-ono cauchy problem for positive initial data, The Davey-Stewartson equation with sources derived from nonlinear variable separation method, General soliton solutions to a nonlocal long-wave-short-wave resonance interaction equation with nonzero boundary condition, Interaction of lumps and dark solitons in the Mel'nikov equation, General soliton solutions to a \((2+1)\)-dimensional nonlocal nonlinear Schrödinger equation with zero and nonzero boundary conditions, Multiple-soliton solutions for the ninth-order KdV equation and sixth-order Boussinesq equation, \(N\)-soliton solutions for the combined KdV-CDG equation and the KdV-Lax equation, Algebro-geometric solutions for some \((2+1)\)-dimensional discrete systems, Fission and fusion collision of high-order lumps and solitons in a \((3+1)\)-dimensional nonlinear evolution equation, Multi soliton solution for the system of coupled Korteweg-de Vries equations, Degenerate Solutions of the Nonlinear Self-Dual Network Equation, Certain electromagnetic waves in a ferromagnetic film, Nonlinear variable separation solutions of some mKdV-type equations and the unified construction method, Lifshitz scaling, microstate counting from number theory and black hole entropy, Integrable multi-component generalization of a modified short pulse equation, Integrability, conservation laws and solitons of a many-body dynamical system associated with the half-wave maps equation, Mechanisms of stationary converted waves and their complexes in the multi-component AB system, New exact solutions to the \((2 + 1)\)-dimensional Ito equation: Extended homoclinic test technique, Lax pairs and Bäcklund transformations for a coupled Ramani equation and its related system, Two integrable differential-difference equations derived from the discrete BKP equation and their related equations, Magnetooptic studies on a ferromagnetic material via an extended \((3+1)\)-dimensional variable-coefficient modified Kadomtsev-Petviashvili system, Quasi-periodic solutions for some \(2+1\)-dimensional discrete models, Periodic solutions of the non-integrable convective fluid equation, A combinatorial formula relating to Hirota's bilinear equations, Soliton solutions and Bäcklund transformation for the Kupershmidt five-field lattice: A bilinear approach, Breathers and rogue waves for a third order nonlocal partial differential equation by a bilinear transformation, The higher-order KdV equation with a source and nonlinear superposition formula., Dynamics of lumps and dark–dark solitons in the multi-component long-wave–short-wave resonance interaction system, Some connections between inverse scattering and solution hierarchies, A bidirectional Kaup–Kupershmidt equation and directionally dependent solitons, Rapidly Decreasing Solutions of the KdV Hierarchy, Quasi-periodic waves and asymptotic behavior for a coupled nonlinear Klein-Gordon equation, A coupled “AB” system: Rogue waves and modulation instabilities, Multiphase solutions and their reductions for a nonlocal nonlinear Schrödinger equation with focusing nonlinearity, SOLITON SOLUTIONS OF 10th ORDER 2-D BOUSSINESQ EQUATION, Dynamic behaviors of the lump solutions and mixed solutions to a (2+1)-dimensional nonlinear model, Dynamics of breathers in the Gardner hierarchy: universality of the variational characterization, Benjamin-Ono and Intermediate Long Wave Equations: Modeling, IST and PDE, Coupled dispersionless and generalized Heisenberg ferromagnet equations with self-consistent sources: Geometry and equivalence, Multisoliton solutions of the two-component Camassa–Holm system and their reductions, Unnamed Item, Finite gap conditions and small dispersion asymptotics for the classical periodic Benjamin–Ono equation, INTEGRABLE PROPERTIES FOR A GENERALIZED NON-ISOSPECTRAL AND VARIABLE-COEFFICIENT KORTEWEG–DE VRIES MODEL, BILINEAR FORM AND SOLITON SOLUTIONS FOR THE COUPLED NONLINEAR KLEIN–GORDON EQUATIONS, Nonlinear evolution equations admitting multilinear variable separable solutions, On the Camassa–Holm equation and a direct method of solution. II. Soliton solutions, On the Camassa–Holm equation and a direct method of solution. III. N -soliton solutions, Explicit solutions to three nonlinear physical models, Some new results on the Błaszak-Marciniak, 3-field and 4-field lattices, On `regular' and `anomalous' solitons of fifth-order equations and an iteration method, Higher-Order Nonlinear Dispersive Equations, Multisoliton solution of a class of coupled soliton equations, Application of Hirota's bilinear formalism to the Toeplitz lattice - some special soliton-like solutions, On the Long Time Behavior of Solutions to the Intermediate Long Wave Equation, EXACT ONE-PERIODIC AND TWO-PERIODIC WAVE SOLUTIONS TO HIROTA BILINEAR EQUATIONS IN (2+1) DIMENSIONS, N-SOLITON SOLUTIONS, AUTO-BÄCKLUND TRANSFORMATIONS AND LAX PAIR FOR A NONISOSPECTRAL AND VARIABLE-COEFFICIENT KORTEWEG-DE VRIES EQUATION VIA SYMBOLIC COMPUTATION, A modified complex short pulse equation of defocusing type, Finite genus solutions for Geng hierarchy, Convergence of Petviashvili's Method near Periodic Waves in the Fractional Korteweg--de Vries Equation, THE PAINLEVÉ INTEGRABILITY AND N-SOLITONIC SOLUTION IN TERMS OF THE WRONSKIAN DETERMINANT FOR A VARIABLE-COEFFICIENT VARIANT BOUSSINESQ MODEL OF NONLINEAR WAVES, From nonlinear Schrödinger hierarchy to some (2+1)-dimensional nonlinear pseudodifferential equations, Bäcklund transformation, Lax pair, and solutions for the Caudrey–Dodd–Gibbon equation, An application of the inverse scattering transform to the modified intermediate long wave equation
