We study a generalized $ (2{+}1) $-dimensional Davey–Stewartson system with arbitrary-power and double-power nonlinearities. For a general propagation direction, the amplitude coordinate $ \rho = |u|^p $ reduces the traveling-wave problem to a planar system that is singular at $ \rho = 0 $ for every real $ p > 1 $. A time rescaling yields a polynomial system with an invariant boundary. We clarify the physical domain and uniqueness properties of this regularization, rederive the known arbitrary-power first integral on $ \rho > 0 $, and classify seven complete configurations of positive interior roots, including the no-positive-root case and two double-root boundaries. A representative $ p = 2 $ solitary wave is verified analytically and by direct pseudospectral simulation of the original partial differential equations, with relative $ L^2 $ error $ 3.31\times10^{-7} $ at $ t = 1 $. Linearization about a continuous wave gives a directional modulational-instability criterion that separates the effects of local nonlinearities and mean-field coupling. A two-dimensional simulation with oblique perturbation vector $ (2, 1) $ gives growth rate $ 3.60555 $, agreeing with the theoretical value $ \sqrt{13} $ at $ p = 2 $; a separate $ p = 3/2 $ test also agrees with its prediction. Equal-magnitude oblique perturbations can grow or oscillate solely because of their orientation. Boundary-reaching orbit segments are identified, but extending them by zero generally does not produce classical compactly supported solutions. A weak-solution analysis and nonlinear stability theory remain open.
Citation: Chun Tong. Boundary geometry and directional modulational instability in a generalized $ (2{+}1) $-dimensional Davey-Stewartson system[J]. AIMS Mathematics, 2026, 11(10): 35818-35831. doi: 10.3934/math.20261388
We study a generalized $ (2{+}1) $-dimensional Davey–Stewartson system with arbitrary-power and double-power nonlinearities. For a general propagation direction, the amplitude coordinate $ \rho = |u|^p $ reduces the traveling-wave problem to a planar system that is singular at $ \rho = 0 $ for every real $ p > 1 $. A time rescaling yields a polynomial system with an invariant boundary. We clarify the physical domain and uniqueness properties of this regularization, rederive the known arbitrary-power first integral on $ \rho > 0 $, and classify seven complete configurations of positive interior roots, including the no-positive-root case and two double-root boundaries. A representative $ p = 2 $ solitary wave is verified analytically and by direct pseudospectral simulation of the original partial differential equations, with relative $ L^2 $ error $ 3.31\times10^{-7} $ at $ t = 1 $. Linearization about a continuous wave gives a directional modulational-instability criterion that separates the effects of local nonlinearities and mean-field coupling. A two-dimensional simulation with oblique perturbation vector $ (2, 1) $ gives growth rate $ 3.60555 $, agreeing with the theoretical value $ \sqrt{13} $ at $ p = 2 $; a separate $ p = 3/2 $ test also agrees with its prediction. Equal-magnitude oblique perturbations can grow or oscillate solely because of their orientation. Boundary-reaching orbit segments are identified, but extending them by zero generally does not produce classical compactly supported solutions. A weak-solution analysis and nonlinear stability theory remain open.
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Figures(2) / Tables(3)
Chun Tong. Boundary geometry and directional modulational instability in a generalized $ (2{+}1) $-dimensional Davey-Stewartson system[J]. AIMS Mathematics, 2026, 11(10): 35818-35831. doi: 10.3934/math.20261388
Number of positive roots of $ f(\rho) = A\rho^2+B\rho+C $ for (a) $ A = -1 $ and (b) $ A = 1 $. The solid parabola is $ \Delta = 0 $, the dashed line is $ C = 0 $, and the dotted line is $ B = 0 $. For $ A = -1 $, a single positive root is a center and two positive roots are ordered saddle-center. For $ A = 1 $, a single positive root is a saddle and two positive roots are ordered center-saddle. The grayscale shading is keyed below the panels
Direct simulations of the original PDE. (a) The exact $ p = 2 $ solitary amplitude (6.6) (black solid line) and the numerical amplitude at $ t = 1 $ (open circles) for the parameters (6.5). (b) Growth of the oblique $ (q_x, q_y) = (2, 1) $ Fourier sideband for $ p = 2 $, $ \mathcal A = 1 $, and $ \gamma = \delta = \alpha = \beta = 1 $: the full-PDE result is shown by a black solid line with open squares, and the linear prediction $ a_1(0)\cosh(\sqrt{13}\, t) $ by a thick gray dashed line. Only selected numerical samples are marked. The agreement causes the curves to overlap at plotting resolution; the distinct markers and explicit legends identify both data sets in grayscale
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