
Numerical study of ZakharovKuznetsov equations in two dimensions
We present a detailed numerical study of solutions to the (generalized) ...
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Numerical verification for asymmetric solutions of the Hénon equation on the unit square
The Hénon equation, a generalized form of the Emden equation, admits sym...
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Discontinuous Galerkin methods for short pulse type equations via hodograph transformations
In the present paper, we consider the discontinuous Galerkin (DG) method...
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Existence and nonexistence results of radial solutions to singular BVPs arising in epitaxial growth theory
The existence and nonexistence of stationary radial solutions to the ell...
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New insights on MultiSolution Distribution of the P3P Problem
Traditionally, the P3P problem is solved by firstly transforming its 3 q...
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Discontinuous Galerkin methods for the OstrovskyVakhnenko equation
In this paper, we develop discontinuous Galerkin (DG) methods for the Os...
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Semiclassical limit for the varyingmass Schrödinger equation with random inhomogeneities
The varyingmass Schrödinger equation (VMSE) has been successfully appli...
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Stable blowup dynamics in the L^2critical and L^2supercritical generalized Hartree equation
We study stable blowup dynamics in the generalized Hartree equation with radial symmetry, a Schrödingertype equation with a nonlocal, convolutiontype nonlinearity: iu_t+Δ u +(x^(d2)∗ u^p) u^p2u = 0, x ∈R^d. First, we consider the L^2critical case in dimensions d=3, 4, 5, 6, 7 and obtain that a generic blowup has a selfsimilar structure and exhibits not only the square root blowup rate but also the loglog correction (via asymptotic analysis and functional fitting). In this setting we also study blowup profiles and show that generic blowup solutions converge to the rescaled Q, a ground state solution of the elliptic equation Δ Q+Q (x^(d2)∗ Q^p ) Q^p2 Q =0. We also consider the L^2supercritical case in dimensions d=3,4. We derive the profile equation for the selfsimilar blowup and establish the existence and local uniqueness of its solutions. As in the NLS L^2supercritical regime, the profile equation exhibits branches of nonoscillating, polynomially decaying (multibump) solutions. A numerical scheme of putting constraints into solving the corresponding ODE is applied during the process of finding the multibump solutions. Direct numerical simulation of solutions to the generalized Hartree equation by the dynamic rescaling method indicates that the Q_1,0 is the profile for the stable blowup. In this supercritical case, we obtain the blowup rate without any correction. This blowup happens at the focusing level 10^5, and thus, numerically observable (unlike the L^2critical case). In summary, we find that the results are similar to the behavior of stable blowup dynamics in the corresponding NLS settings. Consequently, one may expect that the form of the nonlinearity in the Schrödingertype equations is not essential in stable blowup.
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