Wave Structures and Nonlinear Balances in a Family of 1+1 Evolutionary PDEs

Abstract

We study the following family of evolutionary 1+1 PDEs that describe the balance between convection and stretching for small viscosity in the dynamics of 1D nonlinear waves in fluids: \[ m_t + \underbrace{um_x \} _{(-2mm)\hbox{convection}(-2mm)} + \underbrace{b u_xm \} _{(-2mm)\hbox{stretching}(-2mm)} = \underbrace{νm_{xx}\ }_{(-2mm)\hbox{viscosity}}, \quad\hbox{with}\quad u=g*m . \] Here $u=g\ast m$ denotes $u(x)={\int }_{-\infty }^{\infty }g(x-y)m(y)dy.$ We study exchanges of stability in the dynamics of solitons, peakons, ramps/cliffs, leftons, stationary solutions and other solitary wave solutions associated with this equation under changes in the nonlinear balance parameter $b$.

69 pages, 26 figures

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• An Integrable Shallow Water Equation with Linear and Nonlinear Dispersion
• Perturbative Symmetry Approach
• The Complex Geometry of Weak Piecewise Smooth Solutions of Integrable Nonlinear PDE's of Shallow Water and Dym Type
• A New Integrable Equation with Peakon Solutions

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• A note on multi-dimensional Camassa-Holm type systems on the torus
• Geometric dynamics of Vlasov kinetic theory and its moments
• Orbital stability of smooth solitary waves for the $b$-family of Camassa-Holm equations
• On the Convergence of the Convectively Filtered Burgers Equation to the Entropy Solution of the Inviscid Burgers Equation
• A General Approach for Producing Hamiltonian Numerical Schemes for Fluid Equations
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• On the modified multi-component Camassa-Holm system in higher dimensions
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• The two-dimensional periodic $b$-equation on the diffeomorphism group of the torus
• A generalised multicomponent system of Camassa-Holm-Novikov equations
• Asymptotics in shallow water waves
• Reduced Singular Solutions of EPDiff Equations on Manifolds with Symmetry
• A class of fast geodesic shooting algorithms for template matching and its applications via the $N$-particle system of the Euler-Poincaré equations
• An integrable semi-discrete Degasperis-Procesi equation
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• A stochastic variational approach to the viscous Camassa-Holm and Leray-alpha equations
• Global estimates and blow-up criteria for the generalized Hunter-Saxton system
• Cauchy problem for a generalized cross-coupled Camassa-Holm system with waltzing peakons and higher-order nonlinearities
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• On the Uniform Controllability of the Inviscid and Viscous Burgers-$\alpha$ Systems
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• On the continuity of the solution map of the Euler-Poincaré equations in Besov spaces
• Ill-posedness for the higher dimensional Camassa-Holm equations in Besov spaces
• Numerical simulation of compressible flows using two-phase observable Euler equations
• Peakon and kink solutions for a system of $0-$Holm-Staley equations
• An explicit finite difference scheme for the Camassa-Holm equation
• On the Cauchy problem for a two-component Degasperis-Procesi system
• An Examination of Homentropic Euler Equations with Averaged Characteristics
• A New Regularization of the One-Dimensional Euler and Homentropic Euler Equations
• Stability of peakons for the Degasperis-Procesi equation