Some Non-Linear Evolution Equations and Their Explicit Smooth Solutions with Exponential Growth Written into Integral Form
In this paper, exact solutions of semilinear equations having exponential growth in the space variable <i>x</i> are found. Semilinear Schrödinger equation with logarithmic nonlinearity and third-order evolution equations arising in optics with logarithmic and power-logarithmic nonlineari...
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MDPI AG
2024-03-01
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| Online Access: | https://www.mdpi.com/2227-7390/12/7/1003 |
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| author | Petar Popivanov Angela Slavova |
| author_facet | Petar Popivanov Angela Slavova |
| author_sort | Petar Popivanov |
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| description | In this paper, exact solutions of semilinear equations having exponential growth in the space variable <i>x</i> are found. Semilinear Schrödinger equation with logarithmic nonlinearity and third-order evolution equations arising in optics with logarithmic and power-logarithmic nonlinearities are investigated. In the parabolic case, the solution <i>u</i> is written as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mo>=</mo><mi>b</mi><msup><mi>e</mi><mrow><mo>−</mo><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup></mrow></msup></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo><</mo><mn>0</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>,</mo><mi>b</mi></mrow></semantics></math></inline-formula> being real-valued functions. We are looking for the solutions <i>u</i> of Schrödinger-type equation of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mo>=</mo><mi>b</mi><msup><mi>e</mi><mrow><mo>−</mo><mi>a</mi><mfrac><msup><mi>x</mi><mn>2</mn></msup><mn>2</mn></mfrac></mrow></msup></mrow></semantics></math></inline-formula>, respectively, for the third-order PDE, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mo>=</mo><mi>A</mi><msup><mi>e</mi><mrow><mi>i</mi><mo>Φ</mo></mrow></msup></mrow></semantics></math></inline-formula>, where the amplitude <i>b</i> and the phase function <i>a</i> are complex-valued functions, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Φ</mo></semantics></math></inline-formula> is real-valued. In our proofs, the method of the first integral is used, not Hirota’s approach or the method of simplest equation. |
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| spelling | doaj.art-cafef7683d5c4b6da12b1faa9ac21d2c2024-04-12T13:22:36ZengMDPI AGMathematics2227-73902024-03-01127100310.3390/math12071003Some Non-Linear Evolution Equations and Their Explicit Smooth Solutions with Exponential Growth Written into Integral FormPetar Popivanov0Angela Slavova1Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, BulgariaInstitute of Mechanics, Bulgarian Academy of Sciences, 1113 Sofia, BulgariaIn this paper, exact solutions of semilinear equations having exponential growth in the space variable <i>x</i> are found. Semilinear Schrödinger equation with logarithmic nonlinearity and third-order evolution equations arising in optics with logarithmic and power-logarithmic nonlinearities are investigated. In the parabolic case, the solution <i>u</i> is written as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mo>=</mo><mi>b</mi><msup><mi>e</mi><mrow><mo>−</mo><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup></mrow></msup></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo><</mo><mn>0</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>,</mo><mi>b</mi></mrow></semantics></math></inline-formula> being real-valued functions. We are looking for the solutions <i>u</i> of Schrödinger-type equation of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mo>=</mo><mi>b</mi><msup><mi>e</mi><mrow><mo>−</mo><mi>a</mi><mfrac><msup><mi>x</mi><mn>2</mn></msup><mn>2</mn></mfrac></mrow></msup></mrow></semantics></math></inline-formula>, respectively, for the third-order PDE, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mo>=</mo><mi>A</mi><msup><mi>e</mi><mrow><mi>i</mi><mo>Φ</mo></mrow></msup></mrow></semantics></math></inline-formula>, where the amplitude <i>b</i> and the phase function <i>a</i> are complex-valued functions, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Φ</mo></semantics></math></inline-formula> is real-valued. In our proofs, the method of the first integral is used, not Hirota’s approach or the method of simplest equation.https://www.mdpi.com/2227-7390/12/7/1003semilinear parabolic equationsemilinear Schrödinger equationlogarithmic nonlinearityparabolic equations with solutions of exponential growthsolutions into explicit formspecial functions of Jacobi type |
| spellingShingle | Petar Popivanov Angela Slavova Some Non-Linear Evolution Equations and Their Explicit Smooth Solutions with Exponential Growth Written into Integral Form Mathematics semilinear parabolic equation semilinear Schrödinger equation logarithmic nonlinearity parabolic equations with solutions of exponential growth solutions into explicit form special functions of Jacobi type |
| title | Some Non-Linear Evolution Equations and Their Explicit Smooth Solutions with Exponential Growth Written into Integral Form |
| title_full | Some Non-Linear Evolution Equations and Their Explicit Smooth Solutions with Exponential Growth Written into Integral Form |
| title_fullStr | Some Non-Linear Evolution Equations and Their Explicit Smooth Solutions with Exponential Growth Written into Integral Form |
| title_full_unstemmed | Some Non-Linear Evolution Equations and Their Explicit Smooth Solutions with Exponential Growth Written into Integral Form |
| title_short | Some Non-Linear Evolution Equations and Their Explicit Smooth Solutions with Exponential Growth Written into Integral Form |
| title_sort | some non linear evolution equations and their explicit smooth solutions with exponential growth written into integral form |
| topic | semilinear parabolic equation semilinear Schrödinger equation logarithmic nonlinearity parabolic equations with solutions of exponential growth solutions into explicit form special functions of Jacobi type |
| url | https://www.mdpi.com/2227-7390/12/7/1003 |
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