In this paper, the (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">$3+1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>3</mn> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>)-dimensional Sawada–Kotera (SK) equation is introduced via the Fokas method. Through direct computation, the Nth-order Wronskian determinant and the Nth-order Pfaffian form are rigorously verified to satisfy the Hirota bilinear equation associated with the Hirota bilinear method. Consequently, we are able to construct both Wronskian and Pfaffian solutions for the (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">$3+1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>3</mn> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>)-dimensional SK equation. As illustrative examples, we explicitly discuss the Wronskian solutions for the cases <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$N = 1, 2, 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> and the Pfaffian solutions for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$N = 2, 4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>, along with a graphical analysis of the dynamical behaviours of these solutions. In the final section, starting from another bilinear form of the model, we derive a bilinear Bäcklund transformation (BT) involving three arbitrary constants. By applying a gauge transformation to this BT and selecting appropriate parameters, we obtain a new bilinear BT. Using the perturbation method on the new BT, we iteratively derive multi-soliton solutions, explicitly presenting the one-soliton, two-soliton and three-soliton solutions as illustrative examples.

The exact solutions for the three-spatial-dimensions Sawada–Kotera equation

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发表日期：

2026年6月2日

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Huanhuan Lu

摘要

In this paper, the ( $3+1$ $3 + 1$ )-dimensional Sawada–Kotera (SK) equation is introduced via the Fokas method. Through direct computation, the Nth-order Wronskian determinant and the Nth-order Pfaffian form are rigorously verified to satisfy the Hirota bilinear equation associated with the Hirota bilinear method. Consequently, we are able to construct both Wronskian and Pfaffian solutions for the ( $3+1$ $3 + 1$ )-dimensional SK equation. As illustrative examples, we explicitly discuss the Wronskian solutions for the cases $N = 1, 2, 3$ $N = 1, 2, 3$ and the Pfaffian solutions for $N = 2, 4$ $N = 2, 4$ , along with a graphical analysis of the dynamical behaviours of these solutions. In the final section, starting from another bilinear form of the model, we derive a bilinear Bäcklund transformation (BT) involving three arbitrary constants. By applying a gauge transformation to this BT and selecting appropriate parameters, we obtain a new bilinear BT. Using the perturbation method on the new BT, we iteratively derive multi-soliton solutions, explicitly presenting the one-soliton, two-soliton and three-soliton solutions as illustrative examples.