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Spatially-bounded rogue waves in the Davey-Stewartson I equation

Bo Yang, Jianke Yang

TL;DR

This work identifies and characterizes spatially-bounded rogue waves in the Davey-Stewartson I equation by focusing on higher-order rational solutions with multiple large real internal parameters. The authors show that boundedness and crest geometry arise when the order-index vector $\Lambda$ is a concatenation of pairs $(2n,2n+1)$ and the associated double-real-variable root curve, given by $\mathcal{P}_{\Lambda}(z_1,z_2)=0$, is nondegenerate. They derive uniformly-valid asymptotic approximations in the large-parameter regime, revealing that crests align with the root curve and that near critical curves the solutions reduce to Peregrine-type profiles or to second-order rational solutions near horizontal edges. Numerical experiments confirm the analytic predictions, demonstrating spatially-bounded rings or other closed crest curves on a quasi-uniform background, with potential implications for experimental generation and for understanding rogue-wave mechanisms in multidimensional integrable systems.

Abstract

Spatially-bounded rogue waves, i.e., rogue waves that arise in a limited region of a multi-dimensional space, are interesting and important from both theoretical and applied points of view. In this paper, we determine spatially-bounded rogue waves in the Davey-Stewartson I equation. We show that these rogue waves can be obtained when a single or multiple internal parameters in the higher-order rational solution of the Davey-Stewartson I equation are real and large, and the order-index vector of this higher-order rational solution has even length and comprises pairs of the form (2n, 2n+1), where n is a positive integer. Under these conditions and another nondegeneracy condition on the root curve of a certain double-real-variable polynomial, the higher-order rational solution will exhibit spatially-bounded rogue waves that arise from a uniform background with some time-varying lumps on it, reach high amplitude in limited space, and then disappear into the same background again. The crests of these rogue waves form a single or multiple closed curves that are generically disconnected from each other on the spatial plane, and are analytically predicted by the root curve mentioned above. We also derive uniformly-valid asymptotic approximations for these spatially-bounded rogue waves in the large-parameter regime. Near the crests of these rogue waves, these asymptotic approximations reduce to simple expressions. Our asymptotic approximations of these rogue waves are compared to true solutions and good agreement is demonstrated.

Spatially-bounded rogue waves in the Davey-Stewartson I equation

TL;DR

This work identifies and characterizes spatially-bounded rogue waves in the Davey-Stewartson I equation by focusing on higher-order rational solutions with multiple large real internal parameters. The authors show that boundedness and crest geometry arise when the order-index vector is a concatenation of pairs and the associated double-real-variable root curve, given by , is nondegenerate. They derive uniformly-valid asymptotic approximations in the large-parameter regime, revealing that crests align with the root curve and that near critical curves the solutions reduce to Peregrine-type profiles or to second-order rational solutions near horizontal edges. Numerical experiments confirm the analytic predictions, demonstrating spatially-bounded rings or other closed crest curves on a quasi-uniform background, with potential implications for experimental generation and for understanding rogue-wave mechanisms in multidimensional integrable systems.

Abstract

Spatially-bounded rogue waves, i.e., rogue waves that arise in a limited region of a multi-dimensional space, are interesting and important from both theoretical and applied points of view. In this paper, we determine spatially-bounded rogue waves in the Davey-Stewartson I equation. We show that these rogue waves can be obtained when a single or multiple internal parameters in the higher-order rational solution of the Davey-Stewartson I equation are real and large, and the order-index vector of this higher-order rational solution has even length and comprises pairs of the form (2n, 2n+1), where n is a positive integer. Under these conditions and another nondegeneracy condition on the root curve of a certain double-real-variable polynomial, the higher-order rational solution will exhibit spatially-bounded rogue waves that arise from a uniform background with some time-varying lumps on it, reach high amplitude in limited space, and then disappear into the same background again. The crests of these rogue waves form a single or multiple closed curves that are generically disconnected from each other on the spatial plane, and are analytically predicted by the root curve mentioned above. We also derive uniformly-valid asymptotic approximations for these spatially-bounded rogue waves in the large-parameter regime. Near the crests of these rogue waves, these asymptotic approximations reduce to simple expressions. Our asymptotic approximations of these rogue waves are compared to true solutions and good agreement is demonstrated.

Paper Structure

This paper contains 13 sections, 1 theorem, 90 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Spatially-bounded rogue waves
  4. Numerical confirmation of Theorem \ref{['Theorem1']}
  5. Proof of Theorem \ref{['Theorem1']}
  6. Conditions for bounded root curves
  7. Conditions for nonexistence of real or purely-imaginary roots in Wronskian-Hermite polynomials
  8. Asymptotic reductions of rogue waves
  9. A spatially uniformly valid asymptotic approximation
  10. Profiles of the rational solution near the critical curve
  11. Rogue profiles near the critical curve but away from its exceptional points
  12. Solution profiles near horizontal local edges of the critical curve
  13. Summary and discussion

Key Result

Theorem 1

The higher-order rational solution (DSAQ) under parameter conditions (acond) would exhibit spatially-bounded rogue waves if it satisfies the following two conditions:

Figures (8)

  • Figure 1: The second-order rational solution $|A(x, y, t)|$ from Eqs. (\ref{['2ndrogue']})-(\ref{['2ndroguetauk']}) with $a_2=0$ at four time values of $t=-4, -2, 0$ and 4. In all panels, $-200\le x\le 200$ and $-20\le y\le 20$.
  • Figure 2: The simplest spatially-bounded rogue wave $|A(x, y, t)|$ from Eqs. (\ref{['simpleLRogue']})-(\ref{['simpleLRogue2']}) with $a_3=1000$ at four time values of $t=-4, -2, 0$ and 4. In all panels, $-400\le x\le 400$, and $-15\le y\le 38$.
  • Figure 3: Root curves of Eq. (\ref{['Pmz1z2']}) for parameters (\ref{['paraFig3a']})-(\ref{['paraFig4b']}) of Examples 1-4 respectively.
  • Figure 4: Two spatially-bounded rogue waves $|A(x, y, t)|$ (Examples 1 and 2) to confirm Theorem \ref{['Theorem1']}. These graphs are obtained by plotting rational solutions (\ref{['DSAQ']}) with parameter values (\ref{['paraFig3a']}) (upper row) and (\ref{['paraFig3b']}) (lower row) at four time values of $t=-4, -2, 0$ and 4. In all panels, $-700\le x\le 700$ and $-42\le y\le 50$.
  • Figure 5: Two more spatially-bounded rogue waves $|A(x, y, t)|$ (Examples 3 and 4) to confirm Theorem \ref{['Theorem1']}. These graphs are obtained by plotting rational solutions (\ref{['DSAQ']}) with parameter values (\ref{['paraFig4a']}) (upper row) and (\ref{['paraFig4b']}) (lower row) at four time values of $t=-4, -2, 0$ and 4. In upper panels, $-900\le x\le 900$ and $-80\le y\le 100$; in lower panels, $-800\le x\le 800$ and $-100\le y\le 100$.
  • ...and 3 more figures

Theorems & Definitions (1)

  • Theorem 1