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<p>Multi-geometric discrete spectral problem for Sasa–Satsuma equation is studied via Riemann–Hilbert approach. Summation representation of a new Darboux matrix is derived to remove pairs of the higher-order non-elementary zeros of the matrix Riemann–Hilbert problem simultaneously. Thus, the soliton solutions corresponding to the higher-order non-elementary zeros are constructed, which is also known as a weakly bound state. For those weakly bound states, results of asymptotic analysis show that: (1) interaction is elastic, and experiences the phase shift but no position shift; (2) asymptotic solitons are reflected instead of passing through each other during the interaction, and they diverge from each other logarithmically instead of linearly; (3) different from the solitons corresponding to the higher-order elementary zeros which describe the nonlinear superposition of weakly bound states with the same profile, profile of the solitons corresponding to the higher-order non-elementary zeros is asymmetrical about a certain straight line; (4) when the rank sequence is <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\left\{ r_n,r_{n-1},\dots ,r_1\right\} \)</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close="}" open="{"> <msub> <mi>r</mi> <mi>n</mi> </msub> <mo>,</mo> <msub> <mi>r</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>r</mi> <mn>1</mn> </msub> </mfenced> </math></EquationSource> </InlineEquation>, the velocity distribution law of asymptotic soliton is the same for both the higher-order elementary and non-elementary zeros.</p>

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Soliton solutions corresponding to the higher-order non-elementary zeros and their asymptotic analysis for Sasa–Satsuma equation

  • 期刊论文

  • 发表日期：

    2026年1月13日

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  • Su-Su Chen

摘要

Multi-geometric discrete spectral problem for Sasa–Satsuma equation is studied via Riemann–Hilbert approach. Summation representation of a new Darboux matrix is derived to remove pairs of the higher-order non-elementary zeros of the matrix Riemann–Hilbert problem simultaneously. Thus, the soliton solutions corresponding to the higher-order non-elementary zeros are constructed, which is also known as a weakly bound state. For those weakly bound states, results of asymptotic analysis show that: (1) interaction is elastic, and experiences the phase shift but no position shift; (2) asymptotic solitons are reflected instead of passing through each other during the interaction, and they diverge from each other logarithmically instead of linearly; (3) different from the solitons corresponding to the higher-order elementary zeros which describe the nonlinear superposition of weakly bound states with the same profile, profile of the solitons corresponding to the higher-order non-elementary zeros is asymmetrical about a certain straight line; (4) when the rank sequence is \(\left\{ r_n,r_{n-1},\dots ,r_1\right\} \) r n , r n - 1 , ⋯ , r 1 , the velocity distribution law of asymptotic soliton is the same for both the higher-order elementary and non-elementary zeros.