| Summary: | A smooth map <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>γ</mi></semantics></math></inline-formula> in the symplectic space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mrow><mn>2</mn><mi>n</mi></mrow></msup></semantics></math></inline-formula> is <i>Lagrangian</i> if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>γ</mi><mo>,</mo><msub><mi>γ</mi><mi>x</mi></msub><mo>,</mo><mo>…</mo></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>γ</mi><mi>x</mi><mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></msubsup></semantics></math></inline-formula> are linearly independent and the span of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>γ</mi><mo>,</mo><msub><mi>γ</mi><mi>x</mi></msub><mo>,</mo><mo>…</mo><mo>,</mo><msubsup><mi>γ</mi><mi>x</mi><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></msubsup></mrow></semantics></math></inline-formula> is a Lagrangian subspace of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mrow><mn>2</mn><mi>n</mi></mrow></msup></semantics></math></inline-formula>. In this paper, we (i) construct a complete set of differential invariants for Lagrangian curves in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mrow><mn>2</mn><mi>n</mi></mrow></msup></semantics></math></inline-formula> with respect to the symplectic group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>p</mi><mo>(</mo><mn>2</mn><mi>n</mi><mo>)</mo></mrow></semantics></math></inline-formula>, (ii) construct two hierarchies of commuting Hamiltonian Lagrangian curve flows of C-type and A-type, (iii) show that the differential invariants of solutions of Lagrangian curve flows of C-type and A-type are solutions of the Drinfeld-Sokolov’s <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mover accent="true"><mi>C</mi><mo stretchy="false">^</mo></mover><mi>n</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msubsup></semantics></math></inline-formula>-KdV flows and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mover accent="true"><mi>A</mi><mo stretchy="false">^</mo></mover><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></msubsup></semantics></math></inline-formula>-KdV flows respectively, (iv) construct Darboux transforms, Permutability formulas, and scaling transforms, and give an algorithm to construct explicit soliton solutions, (v) give bi-Hamiltonian structures and commuting conservation laws for these curve flows.
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