<p>We propose and analyze a Crank-Nicolson finite difference scheme for the (2+1)-dimensional regularized logarithmic nonlinear Schrödinger equation with general nonlinear damping. The singular logarithmic term <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\ln (|u|^2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>ln</mo> <mo stretchy="false">(</mo> <mo stretchy="false">|</mo> <mi>u</mi> <msup> <mo stretchy="false">|</mo> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is handled through a suitable regularization. We prove existence, uniqueness, and uniform boundedness of the discrete solution in the discrete <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$L^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$H_0^1$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>H</mi> <mn>0</mn> <mn>1</mn> </msubsup> </math></EquationSource> </InlineEquation> norms. A sharp Lipschitz-type estimate enables rigorous error analysis, yielding optimal convergence of order <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$O(\tau ^2 + h^2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>τ</mi> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>h</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$L^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> norm. Numerical experiments confirm the theoretical results and demonstrate accurate resolution of dissipative soliton dynamics under varying damping and regularization parameters.</p>

A stable Crank-Nicolson scheme for the $(2+1)$D regularized logarithmic nonlinear Schrödinger equation with nonlinear damping

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发表日期：

2026年7月1日

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Thi Hoai Thuong Nguyen,
Quan M. Nguyen,
Anh Ha Le

摘要

We propose and analyze a Crank-Nicolson finite difference scheme for the (2+1)-dimensional regularized logarithmic nonlinear Schrödinger equation with general nonlinear damping. The singular logarithmic term $\ln (|u|^2)$ $ln (| u |^{2})$ is handled through a suitable regularization. We prove existence, uniqueness, and uniform boundedness of the discrete solution in the discrete $L^2$ $L^{2}$ and $H_0^1$ $H_{0}^{1}$ norms. A sharp Lipschitz-type estimate enables rigorous error analysis, yielding optimal convergence of order $O(\tau ^2 + h^2)$ $O (τ^{2} + h^{2})$ in $L^2$ $L^{2}$ norm. Numerical experiments confirm the theoretical results and demonstrate accurate resolution of dissipative soliton dynamics under varying damping and regularization parameters.

A stable Crank-Nicolson scheme for the \((2+1)\)D regularized logarithmic nonlinear Schrödinger equation with nonlinear damping

摘要