A Higher-Order Water-Wave Equation and the Method for Solving It

From MaRDI portal
Publication:2752745

DOI10.1143/PTP.54.396zbMath1079.37514OpenAlexW2141172861MaRDI QIDQ2752745

D. J. Kaup

Publication date: 18 October 2001

Published in: Progress of Theoretical Physics (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1143/ptp.54.396




Related Items

Nonlocal symmetry and consistent Riccati expansion integrability of the (1+1)-dimensional integrable nonlinear dispersive-wave systemThe dynamics of flat surface internal geophysical waves with currentsVadermonde-Type Odd-Soliton Solutions for the Whitham–Broer–Kaup Model in the Shallow Water Small-Amplitude RegimeThe singular manifold method revisitedMiura transformation between two non-linear equations in 2+1 dimensionsWave scattering in one dimension with absorptionOn the complete integrability of nonlinearized Lax systems for the classical Boussinesq hierachyInfinite generation of soliton-like solutions for the (2+1) -dimensional generalized Broer-Kaup systemAsymptotics of the scattering coefficients for a generalized Schrödinger equationExact soliton solutions for the fifth-order Sawada-Kotera equationApplication of the \(\frac{G^\prime}{G}\)-expansion to travelling wave solutions of the Broer-Kaup and the approximate long water wave equationsAsymptotic soliton train solutions of Kaup --- Boussinesq equationsDarboux transformations for the supersymmetric two-boson hierarchyThe constraints of potentials and the finite-dimensional integrable systemsQuasi-periodic solutions for generalized Kaup system with quasi-periodic forcingA new class of Euler equation on the dual of the N = 1 extended Neveu-Schwarz algebraHamiltonian models for the propagation of irrotational surface gravity waves over a variable bottomNovikov Algebras and a Classification of Multicomponent Camassa-Holm EquationsMulti-Lagrangians for integrable systemsFormation of the undular bores in shallow water generalized Kaup-Boussinesq modelIntegrable systems and the boundary dynamics of higher spin gravity on \( \mathrm{AdS}_3\)Asymptotic solutions of time-space fractional coupled systems by residual power series methodUnnamed ItemAsymptotic behavior of solution of Whitham-Broer-Kaup type equations with negative dispersionThe Fokas method for the Broer-Kaup system on the half-lineLax pair, infinitely-many conservation laws and soliton solutions for a set of the time-dependent Whitham-Broer-Kaup equations for the shallow waterInvariant solutions and conservation laws of the generalized Kaup–Boussinesq equationOn the (non)removability of spectral parameters in \(\mathbb{Z}_{2}\)-graded zero-curvature representations and its applicationsAnalysis of fractional differential equations with the help of different operatorsLiouville correspondences between multicomponent integrable hierarchiesOptimal system and group invariant solutions of the Whitham-Broer-Kaup systemThe Kaup–Broer system and trilinear odes of Painlevé typeExact combined traveling wave solutions and multi-symplectic structure of the variant Boussinesq-Whitham-Broer-Kaup type equationsSoliton-shape-preserving and soliton-complex interactions for a \((1+1)\)-dimensional nonlinear dispersive-wave system in shallow waterCanonical structures for dispersive waves in shallow waterApproximate Conservation Laws for an Integrable Boussinesq SystemFully resonant soliton interactions in the Whitham-Broer-Kaup system based on the double Wronskian solutionsOrbital stability of periodic traveling waves for the ``abcd Boussinesq systemsInverse scattering on the half-line for energy-dependent Schrödinger equationsSymmetries, explicit solutions and conservation laws for some time space fractional nonlinear systemsExtended Equations of Long WavesAnalytic model for a weakly dissipative shallow-water undular boreResolution of a shock in hyperbolic systems modified by weak dispersionStanding solutions of one-dimensional Boiti-Leon-Pempinelli-Spire system localized in space and periodical in timeOn matrix fourth Painlevé hierarchiesAsymptotic soliton-like solutions to the singularly perturbed Benjamin-Bona-Mahony equation with variable coefficientsModified constrained KP hierarchy and bi-Hamiltonian structuresIntegration of equations of Kaup system kind with self-consistent source in class of periodic functionsSuper Camassa–Holm-type systems associated to the Kuper–Ramond–Schwarz superalgebraAUXILIARY EQUATION METHOD AND ITS APPLICATIONS TO NONLINEAR EVOLUTION EQUATIONSIntroductionUnnamed ItemAsymptotic Behavior of Solutions of a System of KdV Type Associated with the Schrödinger Operator with an Energy-Dependent PotentialThe water wave problem and Hamiltonian transformation theorySolitary-wave and multi-pulsed traveling-wave solutions of boussinesq systemsOn soliton-type solutions of equations associated with N-component systemsEvolution of initial discontinuities in the Riemann problem for the Kaup-Boussinesq equation with positive dispersionExplicit \(N\)-fold Darboux transformation and multi-soliton solutions for the \((1 + 1)\)-dimensional higher-order Broer-Kaup systemDerivative non-linear Schrödinger equation: singular manifold method and Lie symmetriesMiura and auto-Bäcklund transformations for the cKP and cmKP hierarchiesBroer-Kaup system with corrections via inverse scattering transformAn inversion formula in energy dependent scatteringExact solution and semifolded structures of generalized Broer-Kaup system in \((2+1)\)-dimensionsExplicit and exact traveling wave solutions of Whitham-Broer-Kaup shallow water equationsDarboux transformations of classical Boussinesq system and its multi-soliton solutionsBifurcation method and traveling wave solution to Whitham-Broer-Kaup equationInverse scattering for a Schrödinger equation with energy dependent potentialReduction of dispersionless coupled Korteweg–de Vries equations to the Euler–Darboux equationHamiltonian and super-Hamiltonian extensions related to Broer-Kaup-Kupershmidt systemFission, fusion and annihilation in the interaction of localized structures for the \((2 + 1)\)-dimensional generalized Broer-Kaup systemMathematics of dispersive water wavesAsymptotic \(\Sigma\)-solutions of a singularly perturbed Benjamin-Bona-Mahony equation with variable coefficientsGroup theoretical analysis of dispersive long wave equations in two space dimensionsThe generalized KaupߝBoussinesq equation: multiple soliton solutionsSOLITONS AND LOCALIZED EXCITATIONS FOR THE (2+1)-DIMENSIONAL DISPERSIVE LONG WAVE SYSTEM VIA SYMBOLIC COMPUTATIONMultilinear forms associated with the Yajima-Oikawa systemSymmetry analysis for Whitham-Broer-Kaup equationsSupersymmetric Euler equations associated to the N ≤ 3 Neveu-Schwarz algebraGlobal solutions and ill-posedness for the Kaup system and related Boussinesq systemsA reliable technique for fractional modified Boussinesq and approximate long wave equationsNew Double Wronskian Solutions of the Whitham-Broer-Kaup System: Asymptotic Analysis and Resonant Soliton InteractionsFactorisation of recursion operators of some Lagrangian systemsOn a 2+1-Dimensional Whitham-Broer-Kaup System: A Resonant NLS ConnectionTime Evolution of the Scattering Data for the Forced Toda LatticeLong wave equations and metrizable Lie algebrasBäcklund transformations for fourth Painlevé hierarchiesExact and numerical traveling wave solutions of Whitham-Broer-Kaup equationsUnnamed ItemNonlinear evolutions of surface gravity waves on fluid of finite depthAsymptotic Integrability of Water WavesA variety of soliton solutions for the Boussinesq-Burgers equation and the higher-order Boussinesq-Burgers equationBifurcations and Exact Traveling Wave Solutions of Two Shallow Water Two-Component SystemsOn the nonlocal Boussinesq equation: multiple-soliton solutionsUnified approach to Miura, Bäcklund and Darboux Transformations for Nonlinear Partial Differential EquationsSoliton gas in integrable dispersive hydrodynamicsA novel method for travelling wave solutions of fractional Whitham–Broer–Kaup, fractional modified Boussinesq and fractional approximate long wave equations in shallow waterExact solutions of nonlinear wave equations using \((G^\prime/G, 1/G)\)-expansion methodQuadratic operator pencils associated with the conservative Camassa-Holm flowIntegrable discretizations for classical Boussinesq systemThe Cauchy Problem on Large Time for Surface-Waves-Type Boussinesq Systems IIThe Boussinesq systems on the background of a line solitary waveTravelling wave solutions of Drinfel'd-Sokolov-Wilson, Whitham-Broer-Kaup and \((2+1)\)-dimensional Broer-Kaup-Kupershmit equations and their applicationsWell-posedness of Whitham-Broer-Kaup equation with negative dispersionIntegration of the Kaup-Boussinesq system with time-dependent coefficientsSimple two-layer dispersive models in the Hamiltonian reduction formalismNumerical simulation of coupled fractional‐order Whitham‐Broer‐Kaup equations arising in shallow water with Atangana‐Baleanu derivativeNew explicit solitary wave solutions and periodic wave solutions for Whitham-Broer-Kaup equation in shallow waterProliferation scheme for Kaup-Boussinesq systemA linear algebraic nonlinear superposition formulaVariable separation solutions to the coupled integrable dispersionless equationsOn the integrability of a new generalized Gurevich-Zybin dynamical system, its Hunter-Saxton type reduction and related mysterious symmetriesSolutions to the Kaup-Broer system and its \((2+1)\) dimensional integrable generalization via the dressing methodViewing the solar system via a variable-coefficient nonlinear dispersive-wave systemOceanic studies via a variable-coefficient nonlinear dispersive-wave system in the solar systemOn solutions to the nonlocal \(\overline{\partial}\)-problem and (2+1) dimensional completely integrable systemsThe inverse spectral problem for indefinite stringsEnergy-dependent reflectionless inverse scatteringReal finite-gap regular solutions of the Kaup-Boussinesq equationA two-stage physics-informed neural network method based on conserved quantities and applications in localized wave solutionsSome symmetries, similarity solutions and various conservation laws of a type of dispersive water wavesSome invariant solutions and conservation laws of a type of long-water wave systemComments on the inverse scattering transform for the forced Toda latticeLinear stability of exact solutions for the generalized Kaup-Boussinesq equation and their dynamical evolutionsSingularity formation for the Serre-Green-Naghdi equations and applications to \textit{abcd}-Boussinesq systemsInverse scattering method with variable spectral parameterResidual symmetries, \(n\)th Bäcklund transformation and interaction solutions for (2+1)-dimensional generalized Broer-Kaup equationsHamiltonian model for coupled surface and internal waves in the presence of currentsAsymptotic representation of surface waves in the form of two traveling Burgers wavesSolitary wave solutions for a general Boussinesq type fluid modelBoussinesq-type system of equations in the Bénard-Marangoni systemResidual symmetries and interaction solutions for the Whitham-Broer-Kaup equationInverse scattering in one-dimensional nonconservative mediaTrilinear equations, Bell polynomials, and resonant solutionsOn the Boussinesq-Burgers equations driven by dynamic boundary conditionsNew exact solutions to breaking soliton equations and Whitham-Broer-Kaup equationsElastic and inelastic interactions of solitons for a variable-coefficient generalized dispersive water-wave systemEnvelope soliton resonances and Broer-Kaup-type non-Madelung fluidsExtended double Wronskian solutions to the Whitham-Broer-Kaup equations in shallow waterHamiltonian long-wave approximations to the water-wave problemQuasiperiodic solutions of the Kadomtsev-Petviashvili equation via the multidimensional Baker-Akhiezer function generated by the Broer-Kaup hierarchySolitonic interactions and double-Wronskian-type solutions for a variable-coefficient variant Boussinesq model in the long gravity water wavesClassification and bifurcation of a class of second-order ODEs and its application to nonlinear PDEsA new kind of Bäcklund transformation for partial differential equations\(N\)-fold Darboux transformation and solitonic interactions of a variable-coefficient generalized Boussinesq system in shallow waterGeneralization of the relativistic string model in frames of geometric approachGeometry of jet spaces and integrable systemsNew examples of the auto-Bäcklund transformations and nonlinear superposition formulas for vector evolution systemsGauge transformation, elastic and inelastic interactions for the Whitham-Broer-Kaup shallow-water modelOn the integrable variant of the Boussinesq system: Painlevé property, rational solutions, a related many-body system, and equivalence with the AKNS hierarchyApproximate traveling wave solution for shallow water wave equationChiral resonant solitons in Chern-Simons theory and Broer-Kaup type new hydrodynamic systemsOn the inverse scattering problem for the one-dimensional Schrödinger equation with an energy dependent potentialSoliton excitations and periodic waves without dispersion relation in shallow water system.Conservation laws, soliton-like and stability analysis for the time fractional dispersive long-wave equationIs free-surface hydrodynamics an integrable system?Painlevé property of the inhomogeneous deformed Kaup systemNew exact solutions for some nonlinear differential equations using symbolic computation.Lax pair, Bäcklund transformation and multi-soliton solutions for the Boussinesq-Burgers equations from shallow water wavesDarboux transformation and multi-soliton solutions of Boussinesq-Burgers equationReciprocal transformations, Painlevé property and solutions of energy-dependent Schrödinger hierarchies.Practical use of variational principles for modeling water wavesOn Bäcklund transformation for supersymmetric two-boson equationNew exact solutions for the \((2 + 1)\)-dimensional generalized Broer-Kaup systemApplication of the variational iteration method to the Whitham-Broer-Kaup equationsHomotopy series solutions to time-space fractional coupled systemsOn the quasi-periodic solutions of generalized Kaup systemsTwo-component integrable systems modelling shallow water waves: the constant vorticity caseThe explicit solution of the \((2+1)\)-dimensional modified Broer-Kaup-Kupershmidt soliton equationSome exact Wick type stochastic generalized Boussinesq equation solutionsLie analysis and novel analytical solutions for the time-fractional coupled Whitham-Broer-Kaup equationsOn the Cauchy problem for nonlinear dissipative systemsOn Kupershmidt's extended equation of dispersive water wavesFinite symmetry transformation groups and exact solutions of Lax integrable systemsNonlocal residual symmetries and exact interaction solutions for the generalized dispersive water waves systemApproximate traveling wave solutions of coupled Whitham-Broer-Kaup shallow water equations by homotopy analysis methodBroer-Kaup system revisit: inelastic interaction and blowup solutionsOn the Cauchy problem for a K. P. -Boussinesq-type systemSupersymmetric two-boson equation: bilinearization and solutionsElastic-inelastic-interaction coexistence and double Wronskian solutions for the Whitham-Broer-Kaup shallow-water-wave modelNew explicit and exact travelling wave solutions for a system of dispersive long wave equationsApproximate analytical solutions of fractional gas dynamic equationsVariable-coefficient projective Riccati equation method and its application to a new (2 + 1)-dimensional simplified generalized Broer-Kaup systemThe improved \((\frac{G^{\prime}}{G})\)-expansion method and its applications to the Broer-Kaup equations and approximate long water wave equationsStanding waves for a two-way model system for water wavesApproximate soliton solutions for a \((2+1)\)-dimensional Broer-Kaup system by He's methodsMechanical balance laws for Boussinesq models of surface water wavesNew solitary wave and periodic wave solutions for general types of KdV and KdV-Burgers equationsA new lattice Boltzmann model for solving the coupled viscous Burgers' equationApproximate traveling wave solutions for coupled Whitham-Broer-Kaup shallow waterDispersionless sTBDarboux transformations of classical Boussinesq system and its new solutionsIntegrable systems, multicomponent twisted Heisenberg-Virasoro algebra and its central extensionsExotic localized structures based on a variable separation solution of the \((2+1)\)-dimensional higher-order Broer-Kaup systemA smooth center manifold theorem which applies to some ill-posed partial differential equations with unbounded nonlinearitiesEffects of viscosity and surface tension on soliton dynamics in the generalized KdV equation for shallow water wavesComputation of Lie supersymmetries for the supersymmetric two bosons equationsBäcklund transformations and soliton solutions for a \((3+1)\)-dimensional B-type Kadomtsev-Petviashvili equation in fluid dynamicsA new auxiliary equation and exact travelling wave solutions of nonlinear equationsBifurcation solitons and breathers for the nonlocal Boussinesq equationsRational solutions of the classical Boussinesq systemBäcklund transformation and exact solutions for Whitham-Broer-Kaup equations in shallow waterGlobal well-posedness of the BCL system with viscositySolitonic solutions for a variable-coefficient variant Boussinesq system in the long gravity wavesConstraints of the KP hierarchy and multilinear formsLax pairs and Bäcklund transformations for a coupled Ramani equation and its related systemQuasi-periodic solution of the \((2+1)\)-dimensional Boussinesq-Burgers soliton equationDarboux transformation and soliton solutions for Boussinesq-Burgers equation.New exact solutions and fractal patterns of generalized Broer-Kaup system via a mapping approachPainlevé analysis and the integrability properties of coupled integrable dispersionless equations.